Cartesian tensor

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Two different 3d orthonormal bases: each basis consists of unit vectors that are mutually perpendicular.

In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis to another is through an orthogonal transformation.

The most familiar coordinate systems are the two-dimensional and three-dimensional Cartesian coordinate systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional vector space over the field of real numbers that has an inner product.

Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor in rigid body dynamics. Sometimes general curvilinear coordinates are convenient, as in high-deformation continuum mechanics, or even necessary, as in general relativity. While orthonormal bases may be found for some such coordinate systems (e.g. tangent to spherical coordinates), Cartesian tensors may provide considerable simplification for applications in which rotations of rectilinear coordinate axes suffice. The transformation is a passive transformation, since the coordinates are changed and not the physical system.

Cartesian basis and related terminology

Vectors in three dimensions

In 3d Euclidean space, ℝ³, the standard basis is e_x, e_y, e_z. Each basis vector points along the x-, y-, and z-axes, and the vectors are all unit vectors (or normalized), so the basis is orthonormal.
Throughout, when referring to Cartesian coordinates in three dimensions, a right-handed system is assumed and this is much more common than a left-handed system in practice.

For Cartesian tensors of order 1, a Cartesian vector a can be written algebraically as a linear combination of the basis vectors e_x, e_y, e_z:

${\displaystyle \mathbf {a} =a_{\text{x}}\mathbf {e} _{\text{x}}+a_{\text{y}}\mathbf {e} _{\text{y}}+a_{\text{z}}\mathbf {e} _{\text{z}}}$

where the coordinates of the vector with respect to the Cartesian basis are denoted a_x, a_y, a_z. It is common and helpful to display the basis vectors as column vectors

${\displaystyle \mathbf {e} _{\text{x}}={\begin{pmatrix}1\\0\\0\end{pmatrix}}\,,\quad \mathbf {e} _{\text{y}}={\begin{pmatrix}0\\1\\0\end{pmatrix}}\,,\quad \mathbf {e} _{\text{z}}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}$

when we have a coordinate vector in a column vector representation:

${\displaystyle \mathbf {a} ={\begin{pmatrix}a_{\text{x}}\\a_{\text{y}}\\a_{\text{z}}\end{pmatrix}}}$

A row vector representation is also legitimate, although in the context of general curvilinear coordinate systems the row and column vector representations are used separately for specific reasons – see Einstein notation and covariance and contravariance of vectors for why.

The term "component" of a vector is ambiguous: it could refer to:

• a specific coordinate of the vector such as a_z (a scalar), and similarly for x and y, or
• the coordinate scalar-multiplying the corresponding basis vector, in which case the "y-component" of a is a_ye_y (a vector), and similarly for y and z.

A more general notation is tensor index notation, which has the flexibility of numerical values rather than fixed coordinate labels. The Cartesian labels are replaced by tensor indices in the basis vectors e_x ↦ e₁, e_y ↦ e₂, e_z ↦ e₃ and coordinates A_x ↦ A₁, A_y ↦ A₂, A_z ↦ A₃. In general, the notation e₁, e₂, e₃ refers to any basis, and A₁, A₂, A₃ refers to the corresponding coordinate system; although here they are restricted to the Cartesian system. Then:

${\displaystyle \mathbf {a} =a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}=\sum _{i=1}^{3}a_{i}\mathbf {e} _{i}}$

It is standard to use the Einstein notation – the summation sign for summation over an index repeated only twice within a term may be suppressed for notational conciseness:

${\displaystyle \mathbf {a} =\sum _{i=1}^{3}a_{i}\mathbf {e} _{i}\equiv a_{i}\mathbf {e} _{i}}$

An advantage of the index notation over coordinate-specific notations is the independence of the dimension of the underlying vector space, i.e. the same expression on the right hand side takes the same form in higher dimensions (see below). Previously, the Cartesian labels x, y, z were just labels and not indices. (It is informal to say "i = x, y, z").

Second order tensors in three dimensions

A dyadic tensor T is an order 2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = a ⊗ b. Analogous to vectors, it can be written as a linear combination of the tensor basis e_x ⊗ e_x ≡ e_xx, e_x ⊗ e_y ≡ e_xy, ..., e_z ⊗ e_z ≡ e_zz (the right hand side of each identity is only an abbreviation, nothing more):

${\displaystyle {\begin{array}{ccl}\mathbf {T} &=&\left(a_{\text{x}}\mathbf {e} _{\text{x}}+a_{\text{y}}\mathbf {e} _{\text{y}}+a_{\text{z}}\mathbf {e} _{\text{z}}\right)\otimes \left(b_{\text{x}}\mathbf {e} _{\text{x}}+b_{\text{y}}\mathbf {e} _{\text{y}}+b_{\text{z}}\mathbf {e} _{\text{z}}\right)\\&&\\&=&a_{\text{x}}b_{\text{x}}\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{x}}+a_{\text{x}}b_{\text{y}}\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{y}}+a_{\text{x}}b_{\text{z}}\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{z}}\\&&{}+a_{\text{y}}b_{\text{x}}\mathbf {e} _{\text{y}}\otimes \mathbf {e} _{\text{x}}+a_{\text{y}}b_{\text{y}}\mathbf {e} _{\text{y}}\otimes \mathbf {e} _{\text{y}}+a_{\text{y}}b_{\text{z}}\mathbf {e} _{\text{y}}\otimes \mathbf {e} _{\text{z}}\\&&{}+a_{\text{z}}b_{\text{x}}\mathbf {e} _{\text{z}}\otimes \mathbf {e} _{\text{x}}+a_{\text{z}}b_{\text{y}}\mathbf {e} _{\text{z}}\otimes \mathbf {e} _{\text{y}}+a_{\text{z}}b_{\text{z}}\mathbf {e} _{\text{z}}\otimes \mathbf {e} _{\text{z}}\\\end{array}}}$

Representing each basis tensor as a matrix:

${\displaystyle {\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{x}}}\equiv \mathbf {e} _{\text{xx}}={\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}}\,,\quad {\mathbf {e} _{\text{x}}\otimes \mathbf {e} _{\text{y}}}\equiv \mathbf {e} _{\text{xy}}={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}}\,,\cdots \quad {\mathbf {e} _{\text{z}}\otimes \mathbf {e} _{\text{z}}}\equiv \mathbf {e} _{\text{zz}}={\begin{pmatrix}0&0&0\\0&0&0\\0&0&1\end{pmatrix}}}$

then T can be represented more systematically as a matrix:

${\displaystyle \mathbf {T} ={\begin{pmatrix}a_{\text{x}}b_{\text{x}}&a_{\text{x}}b_{\text{y}}&a_{\text{x}}b_{\text{z}}\\a_{\text{y}}b_{\text{x}}&a_{\text{y}}b_{\text{y}}&a_{\text{y}}b_{\text{z}}\\a_{\text{z}}b_{\text{x}}&a_{\text{z}}b_{\text{y}}&a_{\text{z}}b_{\text{z}}\end{pmatrix}}}$

See matrix multiplication for the notational correspondence between matrices and the dot and tensor products.

More generally, whether or not T is a tensor product of two vectors, it is always a linear combination of the basis tensors with coordinates T_xx, T_xy, ... T_zz:

${\displaystyle {\begin{array}{ccl}\mathbf {T} &=&T_{\text{xx}}\mathbf {e} _{\text{xx}}+T_{\text{xy}}\mathbf {e} _{\text{xy}}+T_{\text{xz}}\mathbf {e} _{\text{xz}}\\&&{}+T_{\text{yx}}\mathbf {e} _{\text{yx}}+T_{\text{yy}}\mathbf {e} _{\text{yy}}+T_{\text{yz}}\mathbf {e} _{\text{yz}}\\&&{}+T_{\text{zx}}\mathbf {e} _{\text{zx}}+T_{\text{zy}}\mathbf {e} _{\text{zy}}+T_{\text{zz}}\mathbf {e} _{\text{zz}}\end{array}}}$

while in terms of tensor indices:

${\displaystyle \mathbf {T} =T_{ij}\mathbf {e} _{ij}\equiv \sum _{ij}T_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,,}$

and in matrix form:

${\displaystyle \mathbf {T} ={\begin{pmatrix}T_{\text{xx}}&T_{\text{xy}}&T_{\text{xz}}\\T_{\text{yx}}&T_{\text{yy}}&T_{\text{yz}}\\T_{\text{zx}}&T_{\text{zy}}&T_{\text{zz}}\end{pmatrix}}}$

Second order tensors occur naturally in physics and engineering when physical quantities have directional dependence in the system, often in a "stimulus-response" way. This can be mathematically seen through one aspect of tensors - they are multilinear functions. A second order tensor T which takes in a vector u of some magnitude and direction will return a vector v; of a different magnitude and in a different direction to u, in general. The notation used for functions in mathematical analysis leads us to write v = T(u), while the same idea can be expressed in matrix and index notations (including the summation convention), respectively:

${\displaystyle {\begin{pmatrix}v_{\text{x}}\\v_{\text{y}}\\v_{\text{z}}\end{pmatrix}}={\begin{pmatrix}T_{\text{xx}}&T_{\text{xy}}&T_{\text{xz}}\\T_{\text{yx}}&T_{\text{yy}}&T_{\text{yz}}\\T_{\text{zx}}&T_{\text{zy}}&T_{\text{zz}}\end{pmatrix}}{\begin{pmatrix}u_{\text{x}}\\u_{\text{y}}\\u_{\text{z}}\end{pmatrix}}\,,\quad v_{i}=T_{ij}u_{j}}$

By "linear", if u = ρr + σs for two scalars ρ and σ and vectors r and s, then in function and index notations:

${\displaystyle \mathbf {v} =\mathbf {T} (\rho \mathbf {r} +\sigma \mathbf {s} )=\rho \mathbf {T} (\mathbf {r} )+\sigma \mathbf {T} (\mathbf {s} )}$

${\displaystyle v_{i}=T_{ij}(\rho r_{j}+\sigma s_{j})=\rho T_{ij}r_{j}+\sigma T_{ij}s_{j}}$

and similarly for the matrix notation. The function, matrix, and index notations all mean the same thing. The matrix forms provide a clear display of the components, while the index form allows easier tensor-algebraic manipulation of the formulae in a compact manner. Both provide the physical interpretation of directions; vectors have one direction, while second order tensors connect two directions together. One can associate a tensor index or coordinate label with a basis vector direction.

The use of second order tensors are the minimum to describe changes in magnitudes and directions of vectors, as the dot product of two vectors is always a scalar, while the cross product of two vectors is always a pseudovector perpendicular to the plane defined by the vectors, so these products of vectors alone cannot obtain a new vector of any magnitude in any direction. (See also below for more on the dot and cross products). The tensor product of two vectors is a second order tensor, although this has no obvious directional interpretation by itself.

The previous idea can be continued: if T takes in two vectors p and q, it will return a scalar r. In function notation we write r = T(p, q), while in matrix and index notations (including the summation convention) respectively:

${\displaystyle r={\begin{pmatrix}p_{\text{x}}&p_{\text{y}}&p_{\text{z}}\end{pmatrix}}{\begin{pmatrix}T_{\text{xx}}&T_{\text{xy}}&T_{\text{xz}}\\T_{\text{yx}}&T_{\text{yy}}&T_{\text{yz}}\\T_{\text{zx}}&T_{\text{zy}}&T_{\text{zz}}\end{pmatrix}}{\begin{pmatrix}q_{\text{x}}\\q_{\text{y}}\\q_{\text{z}}\end{pmatrix}}\,,\quad r=p_{i}T_{ij}q_{j}}$

The tensor T is linear in both input vectors. When vectors and tensors are written without reference to components, and indices are not used, sometimes a dot · is placed where summations over indices (known as tensor contractions) are taken. For the above cases:

${\displaystyle \mathbf {v} =\mathbf {T} \cdot \mathbf {u} }$

${\displaystyle r=\mathbf {p} \cdot \mathbf {T} \cdot \mathbf {q} }$

motivated by the dot product notation:

${\displaystyle \mathbf {a} \cdot \mathbf {b} \equiv a_{i}b_{i}}$

More generally, a tensor of order m which takes in n vectors (where n is between 0 and m inclusive) will return a tensor of order m − n, see Tensor: As multilinear maps for further generalizations and details. The concepts above also apply to pseudovectors in the same way as for vectors. The vectors and tensors themselves can vary within throughout space, in which case we have vector fields and tensor fields, and can also depend on time.

Following are some examples:

For the electrical conduction example, the index and matrix notations would be:

${\displaystyle J_{i}=\sigma _{ij}E_{j}\equiv \sum _{j}\sigma _{ij}E_{j}}$

${\displaystyle {\begin{pmatrix}J_{\text{x}}\\J_{\text{y}}\\J_{\text{z}}\end{pmatrix}}={\begin{pmatrix}\sigma _{\text{xx}}&\sigma _{\text{xy}}&\sigma _{\text{xz}}\\\sigma _{\text{yx}}&\sigma _{\text{yy}}&\sigma _{\text{yz}}\\\sigma _{\text{zx}}&\sigma _{\text{zy}}&\sigma _{\text{zz}}\end{pmatrix}}{\begin{pmatrix}E_{\text{x}}\\E_{\text{y}}\\E_{\text{z}}\end{pmatrix}}}$

while for the rotational kinetic energy T:

${\displaystyle T={\frac {1}{2}}\omega _{i}I_{ij}\omega _{j}\equiv {\frac {1}{2}}\sum _{ij}\omega _{i}I_{ij}\omega _{j}\,,}$

${\displaystyle T={\frac {1}{2}}{\begin{pmatrix}\omega _{\text{x}}&\omega _{\text{y}}&\omega _{\text{z}}\end{pmatrix}}{\begin{pmatrix}I_{\text{xx}}&I_{\text{xy}}&I_{\text{xz}}\\I_{\text{yx}}&I_{\text{yy}}&I_{\text{yz}}\\I_{\text{zx}}&I_{\text{zy}}&I_{\text{zz}}\end{pmatrix}}{\begin{pmatrix}\omega _{\text{x}}\\\omega _{\text{y}}\\\omega _{\text{z}}\end{pmatrix}}\,.}$

See also constitutive equation for more specialized examples.

Vectors and tensors in n dimensions

In n-dimensional Euclidean space over the real numbers, ℝⁿ, the standard basis is denoted e₁, e₂, e₃, ... e_n. Each basis vector e_i points along the positive x_i axis, with the basis being orthonormal. Component j of e_i is given by the Kronecker delta:

${\displaystyle (\mathbf {e} _{i})_{j}=\delta _{ij}}$

A vector in ℝⁿ takes the form:

${\displaystyle \mathbf {a} =a_{i}\mathbf {e} _{i}\equiv \sum _{i}a_{i}\mathbf {e} _{i}\,.}$

Similarly for the order 2 tensor above, for each vector a and b in ℝⁿ:

${\displaystyle \mathbf {T} =a_{i}b_{j}\mathbf {e} _{ij}\equiv \sum _{ij}a_{i}b_{j}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,,}$

or more generally:

${\displaystyle \mathbf {T} =T_{ij}\mathbf {e} _{ij}\equiv \sum _{ij}T_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,.}$

Transformations of Cartesian vectors (any number of dimensions)

The same position vector x represented in two 3d rectangular coordinate systems each with an orthonormal basis, the cuboids illustrate the parallelogram law for adding vector components.

Meaning of "invariance" under coordinate transformations

The position vector x in ℝⁿ is a simple and common example of a vector, and can be represented in any coordinate system. Consider the case of rectangular coordinate systems with orthonormal bases only. It is possible to have a coordinate system with rectangular geometry if the basis vectors are all mutually perpendicular and not normalized, in which case the basis is orthogonal but not orthonormal. However, orthonormal bases are easier to manipulate and are often used in practice. The following results are true for orthonormal bases, not orthogonal ones.

In one rectangular coordinate system, x as a contravector has coordinates xⁱ and basis vectors e_i, while as a covector it has coordinates x_i and basis covectors eⁱ, and we have:

${\displaystyle \mathbf {x} =x^{i}\mathbf {e} _{i}\,,\quad \mathbf {x} =x_{i}\mathbf {e} ^{i}}$

In another rectangular coordinate system, x as a contravector has coordinates xⁱ and bases e_i, while as a covector it has coordinates x_i and bases eⁱ, and we have:

${\displaystyle \mathbf {x} ={\bar {x}}^{i}{\bar {\mathbf {e} }}_{i}\,,\quad \mathbf {x} ={\bar {x}}_{i}{\bar {\mathbf {e} }}^{i}}$

Each new coordinate is a function of all the old ones, and vice versa for the inverse function:

${\displaystyle {\bar {x}}{}^{i}={\bar {x}}{}^{i}\left(x^{1},x^{2},\cdots \right)\quad \rightleftharpoons \quad x{}^{i}=x{}^{i}\left({\bar {x}}^{1},{\bar {x}}^{2},\cdots \right)}$

${\displaystyle {\bar {x}}{}_{i}={\bar {x}}{}_{i}\left(x_{1},x_{2},\cdots \right)\quad \rightleftharpoons \quad x{}_{i}=x{}_{i}\left({\bar {x}}_{1},{\bar {x}}_{2},\cdots \right)}$

and similarly each new basis vector is a function of all the old ones, and vice versa for the inverse function:

${\displaystyle {\bar {\mathbf {e} }}{}_{j}={\bar {\mathbf {e} }}{}_{j}\left(\mathbf {e} _{1},\mathbf {e} _{2}\cdots \right)\quad \rightleftharpoons \quad \mathbf {e} {}_{j}=\mathbf {e} {}_{j}\left({\bar {\mathbf {e} }}_{1},{\bar {\mathbf {e} }}_{2}\cdots \right)}$

${\displaystyle {\bar {\mathbf {e} }}{}^{j}={\bar {\mathbf {e} }}{}^{j}\left(\mathbf {e} ^{1},\mathbf {e} ^{2}\cdots \right)\quad \rightleftharpoons \quad \mathbf {e} {}^{j}=\mathbf {e} {}^{j}\left({\bar {\mathbf {e} }}^{1},{\bar {\mathbf {e} }}^{2}\cdots \right)}$

for all i, j.
A vector is invariant under any change of basis, so if coordinates transform according to a transformation matrix L, the bases transform according to the matrix inverse L⁻¹, and conversely if the coordinates transform according to inverse L⁻¹, the bases transform according to the matrix L.

If L is an orthogonal transformation (orthogonal matrix), the objects transforming by it are defined as Cartesian tensors. This geometrically has the interpretation that a rectangular coordinate system is mapped to another rectangular coordinate system, in which the norm of the vector x is preserved (and distances are preserved).

The determinant of L is det(L) = ±1, which corresponds to two types of orthogonal transformation: (+1) for rotations and (−1) for improper rotations (including reflections).

There are considerable algebraic simplifications, the matrix transpose is the inverse from the definition of an orthogonal transformation:

${\displaystyle {\boldsymbol {\mathsf {L}}}^{\mathrm {T} }={\boldsymbol {\mathsf {L}}}^{-1}\Rightarrow ({\boldsymbol {\mathsf {L}}}^{-1})_{i}{}^{j}=({\boldsymbol {\mathsf {L}}}^{\mathrm {T} })_{i}{}^{j}=({\boldsymbol {\mathsf {L}}})^{j}{}_{i}={\mathsf {L}}^{j}{}_{i}}$

From the previous table, orthogonal transformations of covectors and contravectors are identical. There is no need to differ between raising and lowering indices, and in this context and applications to physics and engineering the indices are usually all subscripted to remove confusion for exponents. All indices will be lowered in the remainder of this article. One can determine the actual raised and lowered indices by considering which quantities are covectors or contravectors, and the relevant transformation rules.

Exactly the same transformation rules apply to any vector a, not only the position vector. If its components a_i do not transform according to the rules, a is not a vector.

Despite the similarity between the expressions above, for the change of coordinates such as x^j = L_i^jxⁱ, and the action of a tensor on a vector like b_i = T_ija_j, L is not a tensor, but T is. In the change of coordinates, L is a matrix, used to relate two rectangular coordinate systems with orthonormal bases together. For the tensor relating a vector to a vector, the vectors and tensors throughout the equation all belong to the same coordinate system and basis.

Derivatives and Jacobian matrix elements

The entries of L are partial derivatives of the new or old coordinates with respect to the old or new coordinates, respectively.

Differentiating x_i with respect to x_k:

${\displaystyle {\frac {\partial {\bar {x}}_{i}}{\partial x_{k}}}={\frac {\partial }{\partial x_{k}}}(x_{j}{\mathsf {L}}_{ji})={\mathsf {L}}_{ji}{\frac {\partial x_{j}}{\partial x_{k}}}=\delta _{kj}{\mathsf {L}}_{ji}={\mathsf {L}}_{ki}}$

so

${\displaystyle {\mathsf {L}}_{i}{}^{j}\equiv {\mathsf {L}}_{ij}={\frac {\partial {\bar {x}}_{j}}{\partial x_{i}}}}$

is an element of the Jacobian matrix. There is a (partially mnemonical) correspondence between index positions attached to L and in the partial derivative: i at the top and j at the bottom, in each case, although for Cartesian tensors the indices can be lowered.
Conversely, differentiating x_j with respect to x_i:

${\displaystyle {\frac {\partial x_{j}}{\partial {\bar {x}}_{k}}}={\frac {\partial }{\partial {\bar {x}}_{k}}}({\bar {x}}_{i}({\boldsymbol {\mathsf {L}}}^{-1})_{ij})={\frac {\partial {\bar {x}}_{i}}{\partial {\bar {x}}_{k}}}({\boldsymbol {\mathsf {L}}}^{-1})_{ij}=\delta _{ki}({\boldsymbol {\mathsf {L}}}^{-1})_{ij}=({\boldsymbol {\mathsf {L}}}^{-1})_{kj}}$

so

${\displaystyle ({\boldsymbol {\mathsf {L}}}^{-1})_{i}{}^{j}\equiv ({\boldsymbol {\mathsf {L}}}^{-1})_{ij}={\frac {\partial x_{j}}{\partial {\bar {x}}_{i}}}}$

is an element of the inverse Jacobian matrix, with a similar index correspondence.

Many sources state transformations in terms of the partial derivatives:

${\displaystyle {\begin{array}{c}{\bar {x}}_{j}=x_{i}{\frac {\partial {\bar {x}}_{j}}{\partial x_{i}}}\\\upharpoonleft \downharpoonright \\x_{j}={\bar {x}}_{i}{\frac {\partial x_{j}}{\partial {\bar {x}}_{i}}}\end{array}}}$

and the explicit matrix equations in 3d are:

${\displaystyle {\bar {\mathbf {x} }}={\boldsymbol {\mathsf {L}}}\mathbf {x} }$

${\displaystyle {\begin{pmatrix}{\bar {x}}_{1}\\{\bar {x}}_{2}\\{\bar {x}}_{3}\end{pmatrix}}={\begin{pmatrix}{\frac {\partial {\bar {x}}_{1}}{\partial x_{1}}}&{\frac {\partial {\bar {x}}_{1}}{\partial x_{2}}}&{\frac {\partial {\bar {x}}_{1}}{\partial x_{3}}}\\{\frac {\partial {\bar {x}}_{2}}{\partial x_{1}}}&{\frac {\partial {\bar {x}}_{2}}{\partial x_{2}}}&{\frac {\partial {\bar {x}}_{2}}{\partial x_{3}}}\\{\frac {\partial {\bar {x}}_{3}}{\partial x_{1}}}&{\frac {\partial {\bar {x}}_{3}}{\partial x_{2}}}&{\frac {\partial {\bar {x}}_{3}}{\partial x_{3}}}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}}$

similarly for

${\displaystyle \mathbf {x} ={\boldsymbol {\mathsf {L}}}^{-1}{\bar {\mathbf {x} }}={\boldsymbol {\mathsf {L}}}^{\mathrm {T} }{\bar {\mathbf {x} }}}$

Projections along coordinate axes

Top: Angles from the x_i axes to the x_i axes. Bottom: Vice versa.

As with all linear transformations, L depends on the basis chosen. For two orthonormal bases

${\displaystyle {\bar {\mathbf {e} }}_{i}\cdot {\bar {\mathbf {e} }}_{j}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}=\delta _{ij}\,,\quad \left|\mathbf {e} _{i}\right|=\left|{\bar {\mathbf {e} }}_{i}\right|=1\,,}$

• projecting x to the x axes: ${\displaystyle {\bar {x}}_{i}={\bar {\mathbf {e} }}_{i}\cdot \mathbf {x} ={\bar {\mathbf {e} }}_{i}\cdot x_{j}\mathbf {e} _{j}=x_{i}{\mathsf {L}}_{ij}\,,}$
• projecting x to the x axes: ${\displaystyle x_{i}=\mathbf {e} _{i}\cdot \mathbf {x} =\mathbf {e} _{i}\cdot {\bar {x}}_{j}{\bar {\mathbf {e} }}_{j}={\bar {x}}_{j}({\boldsymbol {\mathsf {L}}}^{-1})_{ji}\,.}$

Hence the components reduce to direction cosines between the x_i and x_j axes:

${\displaystyle {\mathsf {L}}_{ij}={\bar {\mathbf {e} }}_{i}\cdot \mathbf {e} _{j}=\cos \theta _{ij}}$

${\displaystyle ({\boldsymbol {\mathsf {L}}}^{-1})_{ij}=\mathbf {e} _{i}\cdot {\bar {\mathbf {e} }}_{j}=\cos \theta _{ji}}$

where θ_ij and θ_ji are the angles between the x_i and x_j axes. In general, θ_ij is not equal to θ_ji, because for example θ₁₂ and θ₂₁ are two different angles.

The transformation of coordinates can be written:

${\displaystyle {\begin{array}{c}{\bar {x}}_{j}=x_{i}\left({\bar {\mathbf {e} }}_{i}\cdot \mathbf {e} _{j}\right)=x_{i}\cos \theta _{ij}\\\upharpoonleft \downharpoonright \\x_{j}={\bar {x}}_{i}\left(\mathbf {e} _{i}\cdot {\bar {\mathbf {e} }}_{j}\right)={\bar {x}}_{i}\cos \theta _{ji}\end{array}}}$

and the explicit matrix equations in 3d are:

${\displaystyle {\bar {\mathbf {x} }}={\boldsymbol {\mathsf {L}}}\mathbf {x} }$

${\displaystyle {\begin{pmatrix}{\bar {x}}_{1}\\{\bar {x}}_{2}\\{\bar {x}}_{3}\end{pmatrix}}={\begin{pmatrix}{\bar {\mathbf {e} }}_{1}\cdot \mathbf {e} _{1}&{\bar {\mathbf {e} }}_{1}\cdot \mathbf {e} _{2}&{\bar {\mathbf {e} }}_{1}\cdot \mathbf {e} _{3}\\{\bar {\mathbf {e} }}_{2}\cdot \mathbf {e} _{1}&{\bar {\mathbf {e} }}_{2}\cdot \mathbf {e} _{2}&{\bar {\mathbf {e} }}_{2}\cdot \mathbf {e} _{3}\\{\bar {\mathbf {e} }}_{3}\cdot \mathbf {e} _{1}&{\bar {\mathbf {e} }}_{3}\cdot \mathbf {e} _{2}&{\bar {\mathbf {e} }}_{3}\cdot \mathbf {e} _{3}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}\cos \theta _{11}&\cos \theta _{12}&\cos \theta _{13}\\\cos \theta _{21}&\cos \theta _{22}&\cos \theta _{23}\\\cos \theta _{31}&\cos \theta _{32}&\cos \theta _{33}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}}$

similarly for

${\displaystyle \mathbf {x} ={\boldsymbol {\mathsf {L}}}^{-1}{\bar {\mathbf {x} }}={\boldsymbol {\mathsf {L}}}^{\mathrm {T} }{\bar {\mathbf {x} }}}$

The geometric interpretation is the x_i components equal to the sum of projecting the x_j components onto the x_j axes.

The numbers e_i⋅e_j arranged into a matrix would form a symmetric matrix (a matrix equal to its own transpose) due to the symmetry in the dot products, in fact it is the metric tensor g. By contrast e_i⋅e_j or e_i⋅e_j do not form symmetric matrices in general, as displayed above. Therefore, while the L matrices are still orthogonal, they are not symmetric.

Apart from a rotation about any one axis, in which the x_i and x_i for some i coincide, the angles are not the same as Euler angles, and so the L matrices are not the same as the rotation matrices.

Transformation of the dot and cross products (three dimensions only)

The dot product and cross product occur very frequently, in applications of vector analysis to physics and engineering, examples include:

• power transferred P by an object exerting a force F with velocity v along a straight-line path:

${\displaystyle P=\mathbf {v} \cdot \mathbf {F} }$

• tangential velocity v at a point x of a rotating rigid body with angular velocity ω:

${\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {x} }$

• potential energy U of a magnetic dipole of magnetic moment m in a uniform external magnetic field B:

${\displaystyle U=-\mathbf {m} \cdot \mathbf {B} }$

• angular momentum J for a particle with position vector r and momentum p:

${\displaystyle \mathbf {J} =\mathbf {r} \times \mathbf {p} }$

• torque τ acting on an electric dipole of electric dipole moment p in a uniform external electric field E:

${\displaystyle {\boldsymbol {\tau }}=\mathbf {p} \times \mathbf {E} }$

• induced surface current density j_S in a magnetic material of magnetization M on a surface with unit normal n:

${\displaystyle \mathbf {j} _{\mathrm {S} }=\mathbf {M} \times \mathbf {n} }$

How these products transform under orthogonal transformations is illustrated below.

Dot product, Kronecker delta, and metric tensor

The dot product ⋅ of each possible pairing of the basis vectors follows from the basis being orthonormal. For perpendicular pairs we have

${\displaystyle {\begin{array}{cccc}\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{y}}&=\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{z}}&=\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{x}}\\\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{x}}&=\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{y}}&=\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{z}}&=0\end{array}}}$

while for parallel pairs we have

${\displaystyle \mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{x}}=\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{y}}=\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{z}}=1.}$

Replacing Cartesian labels by index notation as shown above, these results can be summarized by

${\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=\delta _{ij}}$

where δ_ij are the components of the Kronecker delta. The Cartesian basis can be used to represent δ in this way.

In addition, each metric tensor component g_ij with respect to any basis is the dot product of a pairing of basis vectors:

${\displaystyle g_{ij}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}.}$

For the Cartesian basis the components arranged into a matrix are:

${\displaystyle \mathbf {g} ={\begin{pmatrix}g_{\text{xx}}&g_{\text{xy}}&g_{\text{xz}}\\g_{\text{yx}}&g_{\text{yy}}&g_{\text{zz}}\\g_{\text{zx}}&g_{\text{zy}}&g_{\text{zz}}\\\end{pmatrix}}={\begin{pmatrix}\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{x}}&\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{y}}&\mathbf {e} _{\text{x}}\cdot \mathbf {e} _{\text{z}}\\\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{x}}&\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{y}}&\mathbf {e} _{\text{y}}\cdot \mathbf {e} _{\text{z}}\\\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{x}}&\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{y}}&\mathbf {e} _{\text{z}}\cdot \mathbf {e} _{\text{z}}\\\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\\end{pmatrix}}}$

so are the simplest possible for the metric tensor, namely the δ:

${\displaystyle g_{ij}=\delta _{ij}}$

This is not true for general bases: orthogonal coordinates have diagonal metrics containing various scale factors (i.e. not necessarily 1), while general curvilinear coordinates could also have nonzero entries for off-diagonal components.

The dot product of two vectors a and b transforms according to

${\displaystyle \mathbf {a} \cdot \mathbf {b} ={\bar {a}}_{j}{\bar {b}}_{j}=a_{i}{\mathsf {L}}_{ij}b_{k}({\boldsymbol {\mathsf {L}}}^{-1})_{jk}=a_{i}\delta _{i}{}_{k}b_{k}=a_{i}b_{i}}$

which is intuitive, since the dot product of two vectors is a single scalar independent of any coordinates. This also applies more generally to any coordinate systems, not just rectangular ones; the dot product in one coordinate system is the same in any other.

Cross and product, Levi-Civita symbol, and pseudovectors

Cyclic permutations of index values and positively oriented cubic volume.

Anticyclic permutations of index values and negatively oriented cubic volume.

Non-zero values of the Levi-Civita symbol ε_ijk as the volume e_i · e_j × e_k of a cube spanned by the 3d orthonormal basis.

For the cross product × of two vectors, the results are (almost) the other way round. Again, assuming a right-handed 3d Cartesian coordinate system, cyclic permutations in perpendicular directions yield the next vector in the cyclic collection of vectors:

${\displaystyle \mathbf {e} _{\text{x}}\times \mathbf {e} _{\text{y}}=\mathbf {e} _{\text{z}}\,\quad \mathbf {e} _{\text{y}}\times \mathbf {e} _{\text{z}}=\mathbf {e} _{\text{x}}\,\quad \mathbf {e} _{\text{z}}\times \mathbf {e} _{\text{x}}=\mathbf {e} _{\text{y}}}$

${\displaystyle \mathbf {e} _{\text{y}}\times \mathbf {e} _{\text{x}}=-\mathbf {e} _{\text{z}}\,\quad \mathbf {e} _{\text{z}}\times \mathbf {e} _{\text{y}}=-\mathbf {e} _{\text{x}}\,\quad \mathbf {e} _{\text{x}}\times \mathbf {e} _{\text{z}}=-\mathbf {e} _{\text{y}}}$

while parallel vectors clearly vanish:

${\displaystyle \mathbf {e} _{\text{x}}\times \mathbf {e} _{\text{x}}=\mathbf {e} _{\text{y}}\times \mathbf {e} _{\text{y}}=\mathbf {e} _{\text{z}}\times \mathbf {e} _{\text{z}}={\boldsymbol {0}}}$

and replacing Cartesian labels by index notation as above, these can be summarized by:

${\displaystyle \mathbf {e} _{i}\times \mathbf {e} _{j}=\left[{\begin{array}{cc}+\mathbf {e} _{k}&{\text{cyclic permutations: }}(i,j,k)=(1,2,3),(2,3,1),(3,1,2)\\-\mathbf {e} _{k}&{\text{anticyclic permutations: }}(i,j,k)=(2,1,3),(3,2,1),(1,3,2)\\{\boldsymbol {0}}&i=j\end{array}}\right.}$

where i, j, k are indices which take values 1, 2, 3. It follows that:

${\displaystyle {\mathbf {e} _{k}\cdot \mathbf {e} _{i}\times \mathbf {e} _{j}}=\left[{\begin{array}{cc}+1&{\text{cyclic permutations: }}(i,j,k)=(1,2,3),(2,3,1),(3,1,2)\\-1&{\text{anticyclic permutations: }}(i,j,k)=(2,1,3),(3,2,1),(1,3,2)\\0&i=j{\text{ or }}j=k{\text{ or }}k=i\end{array}}\right.}$

These permutation relations and their corresponding values are important, and there is an object coinciding with this property: the Levi-Civita symbol, denoted by ε. The Levi-Civita symbol entries can be represented by the Cartesian basis:

${\displaystyle \varepsilon _{ijk}=\mathbf {e} _{i}\cdot \mathbf {e} _{j}\times \mathbf {e} _{k}}$

which geometrically corresponds to the volume of a cube spanned by the orthonormal basis vectors, with sign indicating orientation (and not a "positive or negative volume"). Here, the orientation is fixed by ε₁₂₃ = +1, for a right-handed system. A left-handed system would fix ε₁₂₃ = −1 or equivalently ε₃₂₁ = +1.

The scalar triple product can now be written:

${\displaystyle \mathbf {c} \cdot \mathbf {a} \times \mathbf {b} =c_{i}\mathbf {e} _{i}\cdot a_{j}\mathbf {e} _{j}\times b_{k}\mathbf {e} _{k}=\varepsilon _{ijk}c_{i}a_{j}b_{k}}$

with the geometric interpretation of volume (of the parallelepiped spanned by a, b, c) and algebraically is a determinant:^[3]

${\displaystyle \mathbf {c} \cdot \mathbf {a} \times \mathbf {b} ={\begin{vmatrix}c_{\text{x}}&a_{\text{x}}&b_{\text{x}}\\c_{\text{y}}&a_{\text{y}}&b_{\text{y}}\\c_{\text{z}}&a_{\text{z}}&b_{\text{z}}\end{vmatrix}}}$

This in turn can be used to rewrite the cross product of two vectors as follows:

${\displaystyle {\begin{array}{ll}&(\mathbf {a} \times \mathbf {b} )_{i}={\mathbf {e} _{i}\cdot \mathbf {a} \times \mathbf {b} }=\varepsilon _{\ell jk}{(\mathbf {e} _{i})}_{\ell }a_{j}b_{k}=\varepsilon _{\ell jk}\delta _{i\ell }a_{j}b_{k}=\varepsilon _{ijk}a_{j}b_{k}\\\Rightarrow &{\mathbf {a} \times \mathbf {b} }=(\mathbf {a} \times \mathbf {b} )_{i}\mathbf {e} _{i}=\varepsilon _{ijk}a_{j}b_{k}\mathbf {e} _{i}\end{array}}}$

Contrary to its appearance, the Levi-Civita symbol is not a tensor, but a pseudotensor, the components transform according to:

${\displaystyle {\bar {\varepsilon }}_{pqr}=\det({\boldsymbol {\mathsf {L}}})\varepsilon _{ijk}{\mathsf {L}}_{ip}{\mathsf {L}}_{jq}{\mathsf {L}}_{kr}\,.}$

Therefore, the transformation of the cross product of a and b is:

${\displaystyle {\begin{aligned}({\bar {\mathbf {a} }}\times {\bar {\mathbf {b} }})_{i}&={\bar {\varepsilon }}_{ijk}{\bar {a}}_{j}{\bar {b}}_{k}\\&=\det({\boldsymbol {\mathsf {L}}})\;\;\varepsilon _{pqr}{\mathsf {L}}_{pi}{\mathsf {L}}_{qj}{\mathsf {L}}_{rk}\;\;a_{m}{\mathsf {L}}_{mj}\;\;b_{n}{\mathsf {L}}_{nk}\\&=\det({\boldsymbol {\mathsf {L}}})\;\;\varepsilon _{pqr}\;\;{\mathsf {L}}_{pi}\;\;{\mathsf {L}}_{qj}({\boldsymbol {\mathsf {L}}}^{-1})_{jm}\;\;{\mathsf {L}}_{rk}({\boldsymbol {\mathsf {L}}}^{-1})_{kn}\;\;a_{m}\;\;b_{n}\\&=\det({\boldsymbol {\mathsf {L}}})\;\;\varepsilon _{pqr}\;\;{\mathsf {L}}_{pi}\;\;\delta _{qm}\;\;\delta _{rn}\;\;a_{m}\;\;b_{n}\\&=\det({\boldsymbol {\mathsf {L}}})\;\;{\mathsf {L}}_{pi}\;\;\varepsilon _{pqr}a_{q}b_{r}\\&=\det({\boldsymbol {\mathsf {L}}})\;\;(\mathbf {a} \times \mathbf {b} )_{p}{\mathsf {L}}_{pi}\end{aligned}}}$

and so a × b transforms as a pseudovector, because of the determinant factor.

The tensor index notation applies to any object which has entities that form multidimensional arrays – not everything with indices is a tensor by default. Instead, tensors are defined by how their coordinates and basis elements change under a transformation from one coordinate system to another.
Note the cross product of two vectors is a pseudovector, while the cross product of a pseudovector with a vector is another vector.

Applications of the δ tensor and ε pseudotensor

Other identities can be formed from the δ tensor and ε pseudotensor, a notable and very useful identity is one that converts two Levi-Civita symbols adjacently contracted over two indices into an antisymmetrized combination of Kronecker deltas:

${\displaystyle \varepsilon _{ijk}\varepsilon _{pqk}=\delta _{ip}\delta _{jq}-\delta _{iq}\delta _{jp}}$

The index forms of the dot and cross products, together with this identity, greatly facilitate the manipulation and derivation of other identities in vector calculus and algebra, which in turn are used extensively in physics and engineering. For instance, it is clear the dot and cross products are distributive over vector addition:

${\displaystyle \mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )=a_{i}(b_{i}+c_{i})=a_{i}b_{i}+a_{i}c_{i}=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} }$

${\displaystyle \mathbf {a} \times (\mathbf {b} +\mathbf {c} )=\mathbf {e} _{i}\varepsilon _{ijk}a_{j}(b_{k}+c_{k})=\mathbf {e} _{i}\varepsilon _{ijk}a_{j}b_{k}+\mathbf {e} _{i}\varepsilon _{ijk}a_{j}c_{k}=\mathbf {a} \times \mathbf {b} +\mathbf {a} \times \mathbf {c} }$

without resort to any geometric constructions - the derivation in each case is a quick line of algebra. Although the procedure is less obvious, the vector triple product can also be derived. Rewriting in index notation:

${\displaystyle \left[\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )\right]_{i}=\varepsilon _{ijk}a_{j}(\varepsilon _{k\ell m}b_{\ell }c_{m})=(\varepsilon _{ijk}\varepsilon _{k\ell m})a_{j}b_{\ell }c_{m}}$

and because cyclic permutations of indices in the ε symbol does not change its value, cyclically permuting indices in ε_kℓm to obtain ε_ℓmk allows us to use the above δ-ε identity to convert the ε symbols into δ tensors:

${\displaystyle {\begin{aligned}\left[\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )\right]_{i}&=(\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell })a_{j}b_{\ell }c_{m}\\&=\delta _{i\ell }\delta _{jm}a_{j}b_{\ell }c_{m}-\delta _{im}\delta _{j\ell }a_{j}b_{\ell }c_{m}\\&=a_{j}b_{i}c_{j}-a_{j}b_{j}c_{i}\\\end{aligned}}}$

thusly:

${\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b} -(\mathbf {a} \cdot \mathbf {b} )\mathbf {c} }$

Note this is antisymmetric in b and c, as expected from the left hand side. Similarly, via index notation or even just cyclically relabelling a, b, and c in the previous result and taking the negative:

${\displaystyle (\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =(\mathbf {c} \cdot \mathbf {a} )\mathbf {b} -(\mathbf {c} \cdot \mathbf {b} )\mathbf {a} }$

and the difference in results show that the cross product is not associative. More complex identities, like quadruple products;

${\displaystyle (\mathbf {a} \times \mathbf {b} )\cdot (\mathbf {c} \times \mathbf {d} ),\quad (\mathbf {a} \times \mathbf {b} )\times (\mathbf {c} \times \mathbf {d} ),\ldots }$

and so on, can be derived in a similar manner.

Transformations of Cartesian tensors (any number of dimensions)

Tensors are defined as quantities which transform in a certain way under linear transformations of coordinates.

Second order

Text below contradics introduced above contravariant and covariant vectors (tensors)
Let a = a_ie_i and b = b_ie_i be two vectors, so that they transform according to a_j = a_iL_ij, b_j = b_iL_ij.
Taking the tensor product gives:

${\displaystyle \mathbf {a} \otimes \mathbf {b} =a_{i}\mathbf {e} _{i}\otimes b_{j}\mathbf {e} _{j}=a_{i}b_{j}\mathbf {e} _{i}\otimes \mathbf {e} _{j}}$

then applying the transformation to the components

${\displaystyle {\bar {a}}_{p}{\bar {b}}_{q}=a_{i}{\mathsf {L}}_{i}{}_{p}b_{j}{\mathsf {L}}_{j}{}_{q}={\mathsf {L}}_{i}{}_{p}{\mathsf {L}}_{j}{}_{q}a_{i}b_{j}}$

and to the bases

${\displaystyle {\bar {\mathbf {e} }}_{p}\otimes {\bar {\mathbf {e} }}_{q}=({\boldsymbol {\mathsf {L}}}^{-1})_{pi}\mathbf {e} _{i}\otimes ({\boldsymbol {\mathsf {L}}}^{-1})_{qj}{\bar {\mathbf {e} }}_{j}=({\boldsymbol {\mathsf {L}}}^{-1})_{pi}({\boldsymbol {\mathsf {L}}}^{-1})_{qj}\mathbf {e} _{i}\otimes {\bar {\mathbf {e} }}_{j}={\mathsf {L}}_{ip}{\mathsf {L}}_{jq}\mathbf {e} _{i}\otimes {\bar {\mathbf {e} }}_{j}}$

gives the transformation law of an order-2 tensor. The tensor a⊗b is invariant under this transformation:

${\displaystyle {\begin{array}{cl}{\bar {a}}_{p}{\bar {b}}_{q}{\bar {\mathbf {e} }}_{p}\otimes {\bar {\mathbf {e} }}_{q}&={\mathsf {L}}_{kp}{\mathsf {L}}_{\ell q}a_{k}b_{\ell }\,({\boldsymbol {\mathsf {L}}}^{-1})_{pi}({\boldsymbol {\mathsf {L}}}^{-1})_{qj}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\&={\mathsf {L}}_{kp}({\boldsymbol {\mathsf {L}}}^{-1})_{pi}{\mathsf {L}}_{\ell q}({\boldsymbol {\mathsf {L}}}^{-1})_{qj}\,a_{k}b_{\ell }\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\&=\delta _{k}{}_{i}\delta _{\ell j}\,a_{k}b_{\ell }\mathbf {e} _{i}\otimes \mathbf {e} _{j}\\&=a_{i}b_{j}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\end{array}}}$

More generally, for any order-2 tensor

${\displaystyle \mathbf {R} =R_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}\,,}$

the components transform according to;

${\displaystyle {\bar {R}}_{pq}={\mathsf {L}}_{i}{}_{p}{\mathsf {L}}_{j}{}_{q}R_{ij}}$,

and the basis transforms by:

${\displaystyle {\bar {\mathbf {e} }}_{p}\otimes {\bar {\mathbf {e} }}_{q}=({\boldsymbol {\mathsf {L}}}^{-1})_{ip}\mathbf {e} _{i}\otimes ({\boldsymbol {\mathsf {L}}}^{-1})_{jq}\mathbf {e} _{j}}$

If R does not transform according to this rule - whatever quantity R may be, it's not an order 2 tensor.

Any order

More generally, for any order p tensor

${\displaystyle \mathbf {T} =T_{j_{1}j_{2}\cdots j_{p}}\mathbf {e} _{j_{1}}\otimes \mathbf {e} _{j_{2}}\otimes \cdots \mathbf {e} _{j_{p}}}$

the components transform according to;

${\displaystyle {\bar {T}}_{j_{1}j_{2}\cdots j_{p}}={\mathsf {L}}_{i_{1}j_{1}}{\mathsf {L}}_{i_{2}j_{2}}\cdots {\mathsf {L}}_{i_{p}j_{p}}T_{i_{1}i_{2}\cdots i_{p}}}$

and the basis transforms by:

${\displaystyle {\bar {\mathbf {e} }}_{j_{1}}\otimes {\bar {\mathbf {e} }}_{j_{2}}\cdots \otimes {\bar {\mathbf {e} }}_{j_{p}}=({\boldsymbol {\mathsf {L}}}^{-1})_{j_{1}i_{1}}\mathbf {e} _{i_{1}}\otimes ({\boldsymbol {\mathsf {L}}}^{-1})_{j_{2}i_{2}}\mathbf {e} _{i_{2}}\cdots \otimes ({\boldsymbol {\mathsf {L}}}^{-1})_{j_{p}i_{p}}\mathbf {e} _{i_{p}}}$

For a pseudotensor S of order p, the components transform according to;

${\displaystyle {\bar {S}}_{j_{1}j_{2}\cdots j_{p}}=\det({\boldsymbol {\mathsf {L}}}){\mathsf {L}}_{i_{1}j_{1}}{\mathsf {L}}_{i_{2}j_{2}}\cdots {\mathsf {L}}_{i_{p}j_{p}}S_{i_{1}i_{2}\cdots i_{p}}\,.}$

Pseudovectors as antisymmetric second order tensors

The antisymmetric nature of the cross product can be recast into a tensorial form as follows. Let c be a vector, a be a pseudovector, b be another vector, and T be a second order tensor such that:

${\displaystyle \mathbf {c} =\mathbf {a} \times \mathbf {b} =\mathbf {T} \cdot \mathbf {b} }$

As the cross product is linear in a and b, the components of T can be found by inspection, and they are:

${\displaystyle \mathbf {T} ={\begin{pmatrix}0&-a_{\text{z}}&a_{\text{y}}\\a_{\text{z}}&0&-a_{\text{x}}\\-a_{\text{y}}&a_{\text{x}}&0\\\end{pmatrix}}}$

so the pseudovector a can be written as an antisymmetric tensor. This transforms as a tensor, not a pseudotensor. For the mechanical example above for the tangential velocity of a rigid body, given by v = ω × x, this can be rewritten as v = Ω · x where Ω is the tensor corresponding to the pseudovector ω:

${\displaystyle {\boldsymbol {\Omega }}={\begin{pmatrix}0&-\omega _{\text{z}}&\omega _{\text{y}}\\\omega _{\text{z}}&0&-\omega _{\text{x}}\\-\omega _{\text{y}}&\omega _{\text{x}}&0\\\end{pmatrix}}}$

For an example in electromagnetism, while the electric field E is a vector field, the magnetic field B is a pseudovector field. These fields are defined from the Lorentz force for a particle of electric charge q traveling at velocity v:

${\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )=q(\mathbf {E} -\mathbf {B} \times \mathbf {v} )}$

and considering the second term containing the cross product of a pseudovector B and velocity vector v, it can be written in matrix form, with F, E, and v as column vectors and B as an antisymmetric matrix:

${\displaystyle {\begin{pmatrix}F_{\text{x}}\\F_{\text{y}}\\F_{\text{z}}\\\end{pmatrix}}=q{\begin{pmatrix}E_{\text{x}}\\E_{\text{y}}\\E_{\text{z}}\\\end{pmatrix}}-q{\begin{pmatrix}0&-B_{\text{z}}&B_{\text{y}}\\B_{\text{z}}&0&-B_{\text{x}}\\-B_{\text{y}}&B_{\text{x}}&0\\\end{pmatrix}}{\begin{pmatrix}v_{\text{x}}\\v_{\text{y}}\\v_{\text{z}}\\\end{pmatrix}}}$

If a pseudovector is explicitly given by a cross product of two vectors (as opposed to entering the cross product with another vector), then such pseudovectors can also be written as antisymmetric tensors of second order, with each entry a component of the cross product. The angular momentum of a classical pointlike particle orbiting about an axis, defined by J = x × p, is another example of a pseudovector, with corresponding antisymmetric tensor:

${\displaystyle \mathbf {J} ={\begin{pmatrix}0&-J_{\text{z}}&J_{\text{y}}\\J_{\text{z}}&0&-J_{\text{x}}\\-J_{\text{y}}&J_{\text{x}}&0\\\end{pmatrix}}={\begin{pmatrix}0&-(xp_{\text{y}}-yp_{\text{x}})&(zp_{\text{x}}-xp_{\text{z}})\\(xp_{\text{y}}-yp_{\text{x}})&0&-(yp_{\text{z}}-zp_{\text{y}})\\-(zp_{\text{x}}-xp_{\text{z}})&(yp_{\text{z}}-zp_{\text{y}})&0\\\end{pmatrix}}}$

Although Cartesian tensors do not occur in the theory of relativity; the tensor form of orbital angular momentum J enters the spacelike part of the relativistic angular momentum tensor, and the above tensor form of the magnetic field B enters the spacelike part of the electromagnetic tensor.

Vector and tensor calculus

It should be emphasized the following formulae are only so simple in Cartesian coordinates - in general curvilinear coordinates there are factors of the metric and its determinant - see tensors in curvilinear coordinates for more general analysis.

Vector calculus

Following are the differential operators of vector calculus. Throughout, left Φ(r, t) be a scalar field, and

${\displaystyle \mathbf {A} (\mathbf {r} ,t)=A_{\text{x}}(\mathbf {r} ,t)\mathbf {e} _{\text{x}}+A_{\text{y}}(\mathbf {r} ,t)\mathbf {e} _{\text{y}}+A_{\text{z}}(\mathbf {r} ,t)\mathbf {e} _{\text{z}}}$

${\displaystyle \mathbf {B} (\mathbf {r} ,t)=B_{\text{x}}(\mathbf {r} ,t)\mathbf {e} _{\text{x}}+B_{\text{y}}(\mathbf {r} ,t)\mathbf {e} _{\text{y}}+B_{\text{z}}(\mathbf {r} ,t)\mathbf {e} _{\text{z}}}$

be vector fields, in which all scalar and vector fields are functions of the position vector r and time t.
The gradient operator in Cartesian coordinates is given by:

${\displaystyle \nabla =\mathbf {e} _{\text{x}}{\frac {\partial }{\partial x}}+\mathbf {e} _{\text{y}}{\frac {\partial }{\partial y}}+\mathbf {e} _{\text{z}}{\frac {\partial }{\partial z}}}$

and in index notation, this is usually abbreviated in various ways:

${\displaystyle \nabla _{i}\equiv \partial _{i}\equiv {\frac {\partial }{\partial x_{i}}}}$

This operator acts on a scalar field Φ to obtain the vector field directed in the maximum rate of increase of Φ:

${\displaystyle \left(\nabla \Phi \right)_{i}=\nabla _{i}\Phi }$

The index notation for the dot and cross products carries over to the differential operators of vector calculus.

The directional derivative of a scalar field Φ is the rate of change of Φ along some direction vector a (not necessarily a unit vector), formed out of the components of a and the gradient:

${\displaystyle \mathbf {a} \cdot (\nabla \Phi )=a_{j}(\nabla \Phi )_{j}}$

The divergence of a vector field A is:

${\displaystyle \nabla \cdot \mathbf {A} =\nabla _{i}A_{i}}$

Note the interchange of the components of the gradient and vector field yields a different differential operator

${\displaystyle \mathbf {A} \cdot \nabla =A_{i}\nabla _{i}}$

which could act on scalar or vector fields. In fact, if A is replaced by the velocity field u(r, t) of a fluid, this is a term in the material derivative (with many other names) of continuum mechanics, with another term being the partial time derivative:

${\displaystyle {\frac {D}{Dt}}={\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla }$

which usually acts on the velocity field leading to the non-linearity in the Navier-Stokes equations.
As for the curl of a vector field A, this can be defined as a pseudovector field by means of the ε symbol:

${\displaystyle \left(\nabla \times \mathbf {A} \right)_{i}=\varepsilon _{ijk}\nabla _{j}A_{k}}$

which is only valid in three dimensions, or an antisymmetric tensor field of second order via antisymmetrization of indices, indicated by delimiting the antisymmetrized indices by square brackets (see Ricci calculus):

${\displaystyle \left(\nabla \times \mathbf {A} \right)_{ij}=\nabla _{i}A_{j}-\nabla _{j}A_{i}=2\nabla _{[i}A_{j]}}$

which is valid in any number of dimensions. In each case, the order of the gradient and vector field components should not be interchanged as this would result in a different differential operator:

${\displaystyle \varepsilon _{ijk}A_{j}\nabla _{k}}$

${\displaystyle A_{i}\nabla _{j}-A_{j}\nabla _{i}=2A_{[i}\nabla _{j]}}$

which could act on scalar or vector fields.

Finally, the Laplacian operator is defined in two ways, the divergence of the gradient of a scalar field Φ:

${\displaystyle \nabla \cdot (\nabla \Phi )=\nabla _{i}(\nabla _{i}\Phi )}$

or the square of the gradient operator, which acts on a scalar field Φ or a vector field A:

${\displaystyle (\nabla \cdot \nabla )\Phi =(\nabla _{i}\nabla _{i})\Phi }$

${\displaystyle (\nabla \cdot \nabla )\mathbf {A} =(\nabla _{i}\nabla _{i})\mathbf {A} }$

In physics and engineering, the gradient, divergence, curl, and Laplacian operator arise inevitably in fluid mechanics, Newtonian gravitation, electromagnetism, heat conduction, and even quantum mechanics.
Vector calculus identities can be derived in a similar way to those of vector dot and cross products and combinations. For example, in three dimensions, the curl of a cross product of two vector fields A and B:

${\displaystyle {\begin{aligned}\left[\nabla \times (\mathbf {A} \times \mathbf {B} )\right]_{i}&=\varepsilon _{ijk}\nabla _{j}(\varepsilon _{k\ell m}A_{\ell }B_{m})\\&=(\varepsilon _{ijk}\varepsilon _{\ell mk})\nabla _{j}(A_{\ell }B_{m})\\&=(\delta _{i\ell }\delta _{jm}-\delta _{im}\delta _{j\ell })(B_{m}\nabla _{j}A_{\ell }+A_{\ell }\nabla _{j}B_{m})\\&=(B_{j}\nabla _{j}A_{i}+A_{i}\nabla _{j}B_{j})-(B_{i}\nabla _{j}A_{j}+A_{j}\nabla _{j}B_{i})\\&=(B_{j}\nabla _{j})A_{i}+A_{i}(\nabla _{j}B_{j})-B_{i}(\nabla _{j}A_{j})-(A_{j}\nabla _{j})B_{i}\\&=\left[(\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {A} (\nabla \cdot \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )-(\mathbf {A} \cdot \nabla )\mathbf {B} \right]_{i}\\\end{aligned}}}$

where the product rule was used, and throughout the differential operator was not interchanged with A or B. Thus:

${\displaystyle \nabla \times (\mathbf {A} \times \mathbf {B} )=(\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {A} (\nabla \cdot \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )-(\mathbf {A} \cdot \nabla )\mathbf {B} }$

Tensor calculus

One can continue the operations on tensors of higher order. Let T = T(r, t) denote a second order tensor field, again dependent on the position vector r and time t.

For instance, the gradient of a vector field in two equivalent notations ("dyadic" and "tensor", respectively) is:

${\displaystyle (\nabla \mathbf {A} )_{ij}\equiv (\nabla \otimes \mathbf {A} )_{ij}=\nabla _{i}A_{j}}$

which is a tensor field of second order.
The divergence of a tensor is:

${\displaystyle (\nabla \cdot \mathbf {T} )_{j}=\nabla _{i}T_{ij}}$

which is a vector field. This arises in continuum mechanics in Cauchy's laws of motion - the divergence of the Cauchy stress tensor σ is a vector field, related to body forces acting on the fluid.

Difference from the standard tensor calculus

Cartesian tensors are as in tensor algebra, but Euclidean structure of and restriction of the basis brings some simplifications compared to the general theory.

The general tensor algebra consists of general mixed tensors of type (p, q):

${\displaystyle \mathbf {T} =T_{j_{1}j_{2}\cdots j_{q}}^{i_{1}i_{2}\cdots i_{p}}\mathbf {e} _{i_{1}i_{2}\cdots i_{p}}^{j_{1}j_{2}\cdots j_{q}}}$

with basis elements:

${\displaystyle \mathbf {e} _{i_{1}i_{2}\cdots i_{p}}^{j_{1}j_{2}\cdots j_{q}}=\mathbf {e} _{i_{1}}\otimes \mathbf {e} _{i_{2}}\otimes \cdots \mathbf {e} _{i_{p}}\otimes \mathbf {e} ^{j_{1}}\otimes \mathbf {e} ^{j_{2}}\otimes \cdots \mathbf {e} ^{j_{q}}}$

the components transform according to:

${\displaystyle {\bar {T}}_{\ell _{1}\ell _{2}\cdots \ell _{q}}^{k_{1}k_{2}\cdots k_{p}}={\mathsf {L}}_{i_{1}}{}^{k_{1}}{\mathsf {L}}_{i_{2}}{}^{k_{2}}\cdots {\mathsf {L}}_{i_{p}}{}^{k_{p}}({\boldsymbol {\mathsf {L}}}^{-1})_{\ell _{1}}{}^{j_{1}}({\boldsymbol {\mathsf {L}}}^{-1})_{\ell _{2}}{}^{j_{2}}\cdots ({\boldsymbol {\mathsf {L}}}^{-1})_{\ell _{q}}{}^{j_{q}}T_{j_{1}j_{2}\cdots j_{q}}^{i_{1}i_{2}\cdots i_{p}}}$

as for the bases:

${\displaystyle {\bar {\mathbf {e} }}_{k_{1}k_{2}\cdots k_{p}}^{\ell _{1}\ell _{2}\cdots \ell _{q}}=({\boldsymbol {\mathsf {L}}}^{-1})_{k_{1}}{}^{i_{1}}({\boldsymbol {\mathsf {L}}}^{-1})_{k_{2}}{}^{i_{2}}\cdots ({\boldsymbol {\mathsf {L}}}^{-1})_{k_{p}}{}^{i_{p}}{\mathsf {L}}_{j_{1}}{}^{\ell _{1}}{\mathsf {L}}_{j_{2}}{}^{\ell _{2}}\cdots {\mathsf {L}}_{j_{q}}{}^{\ell _{q}}\mathbf {e} _{i_{1}i_{2}\cdots i_{p}}^{j_{1}j_{2}\cdots j_{q}}}$

For Cartesian tensors, only the order p + q of the tensor matters in a Euclidean space with an orthonormal basis, and all p + q indices can be lowered. A Cartesian basis does not exist unless the vector space has a positive-definite metric, and thus cannot be used in relativistic contexts.

History

Dyadic tensors were historically the first approach to formulating second-order tensors, similarly triadic tensors for third-order tensors, and so on. Cartesian tensors use tensor index notation, in which the variance may be glossed over and is often ignored, since the components remain unchanged by raising and lowering indices.

Heresy

From Wikipedia, the free encyclopedia

The Gospel (allegory) triumphs over Heresia and the Serpent. Gustaf Vasa Church, Stockholm, Sweden, sculpture by Burchard Precht.

 

The burning of the pantheistic Amalrician heretics in 1210, in the presence of King Philip II Augustus. In the background is the Gibbet of Montfaucon and, anachronistically, the Grosse Tour of the Temple. Illumination from the Grandes Chroniques de France, c. 1255-1260.

Heresy (/ˈhɛrəsi/) is any belief or theory that is strongly at variance with established beliefs or customs, in particular the accepted beliefs of a church or religious organization. A heretic is a proponent of such claims or beliefs. Heresy is distinct from both apostasy, which is the explicit renunciation of one's religion, principles or cause, and blasphemy, which is an impious utterance or action concerning God or sacred things.

The term is usually used to refer to violations of important religious teachings, but is used also of views strongly opposed to any generally accepted ideas. It is used in particular in reference to Christianity, Judaism, and Islam.

In certain historical Christian, Islamic and Jewish cultures, among others, espousing ideas deemed heretical has been and in some cases still is subjected not merely to punishments such as excommunication, but even the death penalty.

Etymology

The term heresy is from Greek αἵρεσις originally meant "choice" or "thing chosen", but it came to mean the "party or school of a man's choice" and also referred to that process whereby a young person would examine various philosophies to determine how to live. The word "heresy" is usually used within a Christian, Jewish, or Islamic context, and implies slightly different meanings in each. The founder or leader of a heretical movement is called a heresiarch, while individuals who espouse heresy or commit heresy are known as heretics. Heresiology is the study of heresy.

Christianity

Former German Catholic friar Martin Luther was famously excommunicated as a heretic by Pope Leo X by his Papal bull Decet Romanum Pontificem in 1520. To this day, the Papal decree has not been rescinded.

According to Titus 3:10 a divisive person should be warned twice before separating from him. The Greek for the phrase "divisive person" became a technical term in the early Church for a type of "heretic" who promoted dissension. In contrast correct teaching is called sound not only because it builds up the faith, but because it protects it against the corrupting influence of false teachers.

The Church Fathers identified Jews and Judaism with heresy. They saw deviations from orthodox Christianity as heresies that were essentially Jewish in spirit. Tertullian implied that it was the Jews who most inspired heresy in Christianity: "From the Jew the heretic has accepted guidance in this discussion [that Jesus was not the Christ.]" Peter of Antioch referred to Christians that refused to venerate religious images as having "Jewish minds".

The use of the word "heresy" was given wide currency by Irenaeus in his 2nd century tract Contra Haereses (Against Heresies) to describe and discredit his opponents during the early centuries of the Christian community. He described the community's beliefs and doctrines as orthodox (from ὀρθός, orthos "straight" + δόξα, doxa "belief") and the Gnostics' teachings as heretical. He also pointed out the concept of apostolic succession to support his arguments.

Constantine the Great, who along with Licinius had decreed toleration of Christianity in the Roman Empire by what is commonly called the "Edict of Milan", and was the first Roman Emperor baptized, set precedents for later policy. By Roman law the Emperor was Pontifex Maximus, the high priest of the College of Pontiffs (Collegium Pontificum) of all recognized religions in ancient Rome. To put an end to the doctrinal debate initiated by Arius, Constantine called the first of what would afterwards be called the ecumenical councils and then enforced orthodoxy by Imperial authority.

The first known usage of the term in a legal context was in AD 380 by the Edict of Thessalonica of Theodosius I, which made Christianity the state church of the Roman Empire. Prior to the issuance of this edict, the Church had no state-sponsored support for any particular legal mechanism to counter what it perceived as "heresy". By this edict the state's authority and that of the Church became somewhat overlapping. One of the outcomes of this blurring of Church and state was the sharing of state powers of legal enforcement with church authorities. This reinforcement of the Church's authority gave church leaders the power to, in effect, pronounce the death sentence upon those whom the church considered heretical.

Within six years of the official criminalization of heresy by the Emperor, the first Christian heretic to be executed, Priscillian, was condemned in 386 by Roman secular officials for sorcery, and put to death with four or five followers. However, his accusers were excommunicated both by Ambrose of Milan and Pope Siricius, who opposed Priscillian's heresy, but "believed capital punishment to be inappropriate at best and usually unequivocally evil". The edict of Theodosius II (435) provided severe punishments for those who had or spread writings of Nestorius. Those who possessed writings of Arius were sentenced to death.

For some years after the Reformation, Protestant churches were also known to execute those they considered heretics, including Catholics. The last known heretic executed by sentence of the Catholic Church was Spanish schoolmaster Cayetano Ripoll in 1826. The number of people executed as heretics under the authority of the various "ecclesiastical authorities" is not known.

Catholicism

Massacre of the Waldensians of Mérindol in 1545.

In the Catholic Church, obstinate and willful manifest heresy is considered to spiritually cut one off from the Church, even before excommunication is incurred. The Codex Justinianus (1:5:12) defines "everyone who is not devoted to the Catholic Church and to our Orthodox holy Faith" a heretic. The Church had always dealt harshly with strands of Christianity that it considered heretical, but before the 11th century these tended to centre on individual preachers or small localised sects, like Arianism, Pelagianism, Donatism, Marcionism and Montanism. The diffusion of the almost Manichaean sect of Paulicians westwards gave birth to the famous 11th and 12th century heresies of Western Europe. The first one was that of Bogomils in modern-day Bosnia, a sort of sanctuary between Eastern and Western Christianity. By the 11th century, more organised groups such as the Patarini, the Dulcinians, the Waldensians and the Cathars were beginning to appear in the towns and cities of northern Italy, southern France and Flanders.

In France the Cathars grew to represent a popular mass movement and the belief was spreading to other areas. The Cathar Crusade was initiated by the Catholic Church to eliminate the Cathar heresy in Languedoc. Heresy was a major justification for the Inquisition (Inquisitio Haereticae Pravitatis, Inquiry on Heretical Perversity) and for the European wars of religion associated with the Protestant Reformation.

Cristiano Banti's 1857 painting Galileo facing the Roman Inquisition.

Galileo Galilei was brought before the Inquisition for heresy, but abjured his views and was sentenced to house arrest, under which he spent the rest of his life. Galileo was found "vehemently suspect of heresy", namely of having held the opinions that the Sun lies motionless at the centre of the universe, that the Earth is not at its centre and moves, and that one may hold and defend an opinion as probable after it has been declared contrary to Holy Scripture. He was required to "abjure, curse and detest" those opinions.

Pope St. Gregory stigmatized Judaism and the Jewish people in many of his writings. He described Jews as enemies of Christ: "The more the Holy Spirit fills the world, the more perverse hatred dominates the souls of the Jews." He labeled all heresy as "Jewish", claiming that Judaism would "pollute [Catholics and] deceive them with sacrilegious seduction." The identification of Jews and heretics in particular occurred several times in Roman-Christian law.

Between 1420 and 1431 the Hussite heretics defeated five anti-Hussite Crusades ordered by the Pope.

Eastern Orthodox Church

In Eastern Orthodox Christianity heresy most commonly refers to those beliefs declared heretical by the first seven Ecumenical Councils. Since the Great Schism and the Protestant Reformation, various Christian churches have also used the concept in proceedings against individuals and groups those churches deemed heretical. The Orthodox Church also rejects the early Christian heresies such as Arianism, Gnosticism, Origenism, Montanism, Judaizers, Marcionism, Docetism, Adoptionism, Nestorianism, Monophysitism, Monothelitism and Iconoclasm.

Protestantism

In his work "On the Jews and Their Lies" (1543), German Reformation leader Martin Luther claims that Jewish history was "assailed by much heresy", and that Christ the logos swept away the Jewish heresy and goes on to do so, "as it still does daily before our eyes." He stigmatizes Jewish prayer as being "blasphemous" and a lie, and vilifies Jews in general as being spiritually "blind" and "surely possessed by all devils." Luther calls the members of the Catholic Church "papists" and heretics, and has a special spiritual problem with Jewish circumcision.

In England, the 16th-century European Reformation resulted in a number of executions on charges of heresy. During the thirty-eight years of Henry VIII's reign, about sixty heretics, mainly Protestants, were executed and a rather greater number of Catholics lost their lives on grounds of political offences such as treason, notably Sir Thomas More and Cardinal John Fisher, for refusing to accept the king's supremacy over the Church in England. Under Edward VI, the heresy laws were repealed in 1547 only to be reintroduced in 1554 by Mary I; even so two radicals were executed in Edward's reign (one for denying the reality of the incarnation, the other for denying Christ's divinity). Under Mary, around two hundred and ninety people were burned at the stake between 1555 and 1558 after the restoration of papal jurisdiction. When Elizabeth I came to the throne, the concept of heresy was retained in theory but severely restricted by the 1559 Act of Supremacy and the one hundred and eighty or so Catholics who were executed in the forty-five years of her reign were put to death because they were considered members of "...a subversive fifth column." The last execution of a "heretic" in England occurred under James VI and I in 1612. Although the charge was technically one of "blasphemy" there was one later execution in Scotland (still at that date an entirely independent kingdom) when in 1697 Thomas Aikenhead was accused, among other things, of denying the doctrine of the Trinity.

Another example of the persecution of heretics under Protestant rule was the execution of the Boston martyrs in 1659, 1660, and 1661. These executions resulted from the actions of the Anglican Puritans, who at that time wielded political as well as ecclesiastic control in the Massachusetts Bay Colony. At the time, the colony leaders were apparently hoping to achieve their vision of a "purer absolute theocracy" within their colony . As such, they perceived the teachings and practices of the rival Quaker sect as heretical, even to the point where laws were passed and executions were performed with the aim of ridding their colony of such perceived "heresies". It should be noticed that the Eastern Orthodox and Oriental Orthodox communions generally regard the Puritans themselves as having been heterodox or heretical.

Modern era

The era of mass persecution and execution of heretics under the banner of Christianity came to an end in 1826 with the last execution of a "heretic", Cayetano Ripoll, by the Spanish Inquisition. Although less common than in earlier periods, in modern times, formal charges of heresy within Christian churches still occur. Issues in the Protestant churches have included modern biblical criticism and the nature of God. In the Catholic Church, the Congregation for the Doctrine of the Faith criticizes writings for "ambiguities and errors" without using the word "heresy".

Perhaps due to the many modern negative connotations associated with the term heretic, such as the Spanish inquisition, the term is used less often today. The subject of Christian heresy opens up broader questions as to who has a monopoly on spiritual truth, as explored by Jorge Luis Borges in the short story "The Theologians" within the compilation Labyrinths.

On 11 July 2007, Pope Benedict XVI stated that all non-Catholic churches are "ecclesial communities." The members of these churches accuses Vatican of effectively calling them heretics.

Islam

Mehdiana Sahib: the Killing of Bhai Dayala, a Sikh, by Indian Muslims at Chandni Chowk, India in 1675

The Baha'i Faith is considered an Islamic heresy in Iran. To Mughal Emperor Aurangzeb, Sikhs were heretics.

Ottoman Sultan Selim the Grim, regarded the Shia Qizilbash as heretics, reportedly proclaimed that "the killing of one Shiite had as much otherworldly reward as killing 70 Christians." Shia, in general, have often been accused by Sunnis of being heretics.

Starting in medieval times, Muslims began to refer to heretics and those who antagonized Islam as zindiqs, the charge being punishable by death.

In some modern day nations and regions, heresy remains an offense punishable by death. One example is the 1989 fatwa issued by the government of Iran, offering a substantial bounty for anyone who succeeds in the assassination of author Salman Rushdie, whose writings were declared as heretical.

Judaism

Orthodox Judaism considers views on the part of Jews who depart from traditional Jewish principles of faith heretical. In addition, the more right-wing groups within Orthodox Judaism hold that all Jews who reject the simple meaning of Maimonides's 13 principles of Jewish faith are heretics. As such, most of Orthodox Judaism considers Reform and Reconstructionist Judaism heretical movements, and regards most of Conservative Judaism as heretical. The liberal wing of Modern Orthodoxy is more tolerant of Conservative Judaism, particularly its right wing, as there is some theological and practical overlap between these groups.

Other religions

The act of using Church of Scientology techniques in a form different than originally described by Hubbard is referred to within Scientology as "squirreling" and is said by Scientologists to be high treason. The Religious Technology Center has prosecuted breakaway groups that have practiced Scientology outside the official Church without authorization.

Although Zoroastrianism has had an historical tolerance for other religions, it also held sects like Zurvanism and Mazdakism heretical to its main dogma and has violently persecuted them, such as burying Mazdakians with their feet upright as "human gardens". In later periods Zoroastrians cooperated with Muslims to kill other Zoroastrians deemed as heretical.

Non-religious usage

The term "heresy" is used not only with regard to religion but also in the context of political theory.

In other contexts the term does not necessarily have pejorative overtones and may even be complimentary when used, in areas where innovation is welcome, of ideas that are in fundamental disagreement with the status quo in any practice and branch of knowledge. Scientist/author Isaac Asimov considered heresy as an abstraction, Asimov's views are in Forward: The Role of the Heretic. mentioning religious, political, socioeconomic and scientific heresies. He divided scientific heretics into endoheretics (those from within the scientific community) and exoheretics (those from without). Characteristics were ascribed to both and examples of both kinds were offered. Asimov concluded that science orthodoxy defends itself well against endoheretics (by control of science education, grants and publication as examples), but is nearly powerless against exoheretics. He acknowledged by examples that heresy has repeatedly become orthodoxy.

The revisionist paleontologist Robert T. Bakker, who published his findings as The Dinosaur Heresies, treated the mainstream view of dinosaurs as dogma. "I have enormous respect for dinosaur paleontologists past and present. But on average, for the last fifty years, the field hasn't tested dinosaur orthodoxy severely enough." page 27 "Most taxonomists, however, have viewed such new terminology as dangerously destabilizing to the traditional and well-known scheme..." page 462. This book apparently influenced Jurassic Park. The illustrations by the author show dinosaurs in very active poses, in contrast to the traditional perception of lethargy. He is an example of a recent scientific endoheretic.

Immanuel Velikovsky is an example of a recent scientific exoheretic; he did not have appropriate scientific credentials or did not publish in scientific journals. While the details of his work are in scientific disrepute, the concept of catastrophic change (extinction event and punctuated equilibrium) has gained acceptance in recent decades.

The term heresy is also used as an ideological pigeonhole for contemporary writers because, by definition, heresy depends on contrasts with an established orthodoxy. For example, the tongue-in-cheek contemporary usage of heresy, such as to categorize a "Wall Street heresy" a "Democratic heresy" or a "Republican heresy," are metaphors that invariably retain a subtext that links orthodoxies in geology or biology or any other field to religion. These expanded metaphoric senses allude to both the difference between the person's views and the mainstream and the boldness of such a person in propounding these views.

Selected quotations

• Thomas Aquinas: "Wherefore if forgers of money and other evil-doers are forthwith condemned to death by the secular authority, much more reason is there for heretics, as soon as they are convicted of heresy, to be not only excommunicated but even put to death." (Summa Theologica, c. 1270)
• Isaac Asimov: "Science is in a far greater danger from the absence of challenge than from the coming of any number of even absurd challenges."
• Gerald Brenan: "Religions are kept alive by heresies, which are really sudden explosions of faith. Dead religions do not produce them." (Thoughts in a Dry Season, 1978)
• Geoffrey Chaucer: "Thu hast translated the Romance of the Rose, That is a heresy against my law, And maketh wise folk from me withdraw." (The Prologue to The Legend of Good Women, c. 1386)
• G. K. Chesterton: "Truths turn into dogmas the instant that they are disputed. Thus every man who utters a doubt defines a religion." (Heretics, 12th Edition, 1919)
• G. K. Chesterton: "But to have avoided [all heresies] has been one whirling adventure; and in my vision the heavenly chariot flies thundering through the ages, the dull heresies sprawling and prostrate, the wild truth reeling but erect." (Orthodoxy, 1908)
• Benjamin Franklin: "Many a long dispute among divines may be thus abridged: It is so. It is not. It is so. It is not." (Poor Richard's Almanack, 1879)
• Helen Keller: "The heresy of one age becomes the orthodoxy of the next." (Optimism, 1903)
• Lao Tzu: "Those who are intelligent are not ideologues. Those who are ideologues are not intelligent." (Tao Te Ching, Verse 81, 6th century BCE)
• James G. March on the relations among madness, heresy, and genius: "... we sometimes find that such heresies have been the foundation for bold and necessary change, but heresy is usually just new ideas that are foolish or dangerous and appropriately rejected or ignored. So while it may be true that great geniuses are usually heretics, heretics are rarely great geniuses."
• Montesquieu: "No kingdom has ever had as many civil wars as the kingdom of Christ." (Persian Letters, 1721)
• Friedrich Nietzsche: "Whoever has overthrown an existing law of custom has hitherto always first been accounted a bad man: but when, as did happen, the law could not afterwards be reinstated and this fact was accepted, the predicate gradually changed; - history treats almost exclusively of these bad men who subsequently became good men!" (Daybreak, § 20)

Inquisition

From Wikipedia, the free encyclopedia

A 19th-century depiction of Galileo before the Holy Office, by Joseph-Nicolas Robert-Fleury

The Inquisition was a group of institutions within the government system of the Catholic Church whose aim was to combat public heresy committed by baptized Christians. It started in 12th-century France to combat religious dissent, in particular the Cathars and the Waldensians. Other groups investigated later included the Spiritual Franciscans, the Hussites (followers of Jan Hus) and the Beguines. Beginning in the 1250s, inquisitors were generally chosen from members of the Dominican Order, replacing the earlier practice of using local clergy as judges. The term Medieval Inquisition covers these courts up to mid-15th century.

During the Late Middle Ages and early Renaissance, the concept and scope of the Inquisition significantly expanded in response to the Protestant Reformation and the Catholic Counter-Reformation. It expanded to other European countries, resulting in the Spanish Inquisition and Portuguese Inquisition. The Spanish and Portuguese operated inquisitorial courts throughout their empires in Africa, Asia, and the Americas (resulting in the Peruvian Inquisition and Mexican Inquisition). The Spanish and Portuguese inquisitions focused particularly on the issue of Jewish anusim and Muslim converts to Catholicism, partly because these minority groups were more numerous in Spain and Portugal than in many other parts of Europe, and partly because they were often considered suspect due to the assumption that they had secretly reverted to their previous religions.

Except within the Papal States, the institution of the Inquisition was abolished in the early 19th century, after the Napoleonic Wars in Europe and after the Spanish American wars of independence in the Americas. The institution survived as part of the Roman Curia, but in 1908 was given the new name of "Supreme Sacred Congregation of the Holy Office". In 1965 it became the Congregation for the Doctrine of the Faith.

Definition and purpose

Tribunal at the Inquisitor's Palace in Birgu, Malta

The term Inquisition comes from Medieval Latin "inquisitio", which referred to any court process that was based on Roman law, which had gradually come back into usage in the late medieval period. Today, the English term "Inquisition" can apply to any one of several institutions that worked against heretics (or other offenders against canon law) within the judicial system of the Roman Catholic Church. Although the term Inquisition is usually applied to ecclesiastical courts of the Catholic Church, it has several different usages:

• an ecclesiastical tribunal,
• the institution of the Catholic Church for combating heresy,
• a number of historical expurgation movements against heresy (orchestrated by the Catholic Church or a Catholic state), or
• the trial of an individual accused of heresy.

"[T]he Inquisition, as a church-court, had no jurisdiction over Moors and Jews as such." Generally, the Inquisition was concerned only with the heretical behaviour of Catholic adherents or converts.

"The overwhelming majority of sentences seem to have consisted of penances like wearing a cross sewn on one's clothes, going on pilgrimage, etc." When a suspect was convicted of unrepentant heresy, the inquisitorial tribunal was required by law to hand the person over to the secular authorities for final sentencing, at which point a magistrate would determine the penalty, which was usually burning at the stake although the penalty varied based on local law. The laws were inclusive of proscriptions against certain religious crimes (heresy, etc.), and the punishments included death by burning, although usually the penalty was banishment or imprisonment for life, which was generally commuted after a few years. Thus the inquisitors generally knew what would be the fate of anyone so remanded, and cannot be considered to have divorced the means of determining guilt from its effects.

The 1578 edition of the Directorium Inquisitorum (a standard Inquisitorial manual) spelled out the purpose of inquisitorial penalties: ... quoniam punitio non refertur primo & per se in correctionem & bonum eius qui punitur, sed in bonum publicum ut alij terreantur, & a malis committendis avocentur (translation: "... for punishment does not take place primarily and per se for the correction and good of the person punished, but for the public good in order that others may become terrified and weaned away from the evils they would commit").

Origin

Before 1100, the Catholic Church suppressed what they believed to be heresy, usually through a system of ecclesiastical proscription or imprisonment, but without using torture, and seldom resorting to executions. Such punishments were opposed by a number of clergymen and theologians, although some countries punished heresy with the death penalty.

In the 12th century, to counter the spread of Catharism, prosecution of heretics became more frequent. The Church charged councils composed of bishops and archbishops with establishing inquisitions (the Episcopal Inquisition). The first Inquisition was temporarily established in Languedoc (south of France) in 1184. The murder of Pope Innocent's papal legate Pierre de Castelnau in 1208 sparked the Albigensian Crusade (1209–1229). The Inquisition was permanently established in 1229, run largely by the Dominicans in Rome and later at Carcassonne in Languedoc.

Medieval Inquisition

Historians use the term "Medieval Inquisition" to describe the various inquisitions that started around 1184, including the Episcopal Inquisition (1184–1230s) and later the Papal Inquisition (1230s). These inquisitions responded to large popular movements throughout Europe considered apostate or heretical to Christianity, in particular the Cathars in southern France and the Waldensians in both southern France and northern Italy. Other Inquisitions followed after these first inquisition movements. The legal basis for some inquisitorial activity came from Pope Innocent IV's papal bull Ad extirpanda of 1252, which explicitly authorized (and defined the appropriate circumstances for) the use of torture by the Inquisition for eliciting confessions from heretics. However, Nicholas Eymerich, the inquisitor who wrote the "Directorium Inquisitorum", stated: 'Quaestiones sunt fallaces et ineficaces' ("interrogations via torture are misleading and futile"). By 1256 inquisitors were given absolution if they used instruments of torture.

In the 13th century, Pope Gregory IX (reigned 1227–1241) assigned the duty of carrying out inquisitions to the Dominican Order and Franciscan Order. By the end of the Middle Ages, England and Castile were the only large western nations without a papal inquisition. Most inquisitors were friars who taught theology and/or law in the universities. They used inquisitorial procedures, a common legal practice adapted from the earlier Ancient Roman court procedures. They judged heresy along with bishops and groups of "assessors" (clergy serving in a role that was roughly analogous to a jury or legal advisers), using the local authorities to establish a tribunal and to prosecute heretics. After 1200, a Grand Inquisitor headed each Inquisition. Grand Inquisitions persisted until the mid 19th century.

Early Modern European history

With the sharpening of debate and of conflict between the Protestant Reformation and the Catholic Counter-Reformation, Protestant societies came to see/use the Inquisition as a terrifying "Other", while staunch Catholics regarded the Holy Office as a necessary bulwark against the spread of reprehensible heresies.

Witch-trials

Emblem of the Spanish Inquisition (1571)

While belief in witchcraft, and persecutions directed at or excused by it, were widespread in pre-Christian Europe, and reflected in Germanic law, the influence of the Church in the early medieval era resulted in the revocation of these laws in many places, bringing an end to traditional pagan witch hunts. Throughout the medieval era mainstream Christian teaching had denied the existence of witches and witchcraft, condemning it as pagan superstition. However, Christian influence on popular beliefs in witches and maleficium (harm committed by magic) failed to entirely eradicate folk belief in witches.

The fierce denunciation and persecution of supposed sorceresses that characterized the cruel witchhunts of a later age were not generally found in the first thirteen hundred years of the Christian era. The medieval Church distinguished between "white" and "black" magic. Local folk practice often mixed chants, incantations, and prayers to the appropriate patron saint to ward off storms, to protect cattle, or ensure a good harvest. Bonfires on Midsummer's Eve were intended to deflect natural catastrophes or the influence of fairies, ghosts, and witches. Plants, often harvested under particular conditions, were deemed effective in healing.

Black magic was that which was used for a malevolent purpose. This was generally dealt with through confession, repentance, and charitable work assigned as penance. Early Irish canons treated sorcery as a crime to be visited with excommunication until adequate penance had been performed. In 1258 Pope Alexander IV ruled that inquisitors should limit their involvement to those cases in which there was some clear presumption of heretical belief.

The prosecution of witchcraft generally became more prominent throughout the late medieval and Renaissance era, perhaps driven partly by the upheavals of the era – the Black Death, Hundred Years' War, and a gradual cooling of the climate that modern scientists call the Little Ice Age (between about the 15th and 19th centuries). Witches were sometimes blamed. Since the years of most intense witch-hunting largely coincide with the age of the Reformation, some historians point to the influence of the Reformation on the European witch-hunt.

Dominican priest Heinrich Kramer was assistant to the Archbishop of Salzburg. In 1484 Kramer requested that Pope Innocent VIII clarify his authority to prosecute witchcraft in Germany, where he had been refused assistance by the local ecclesiastical authorities. They maintained that Kramer could not legally function in their areas.

The bull Summis desiderantes affectibus sought to remedy this jurisdictional dispute by specifically identifying the dioceses of Mainz, Köln, Trier, Salzburg, and Bremen. Some scholars view the bull as "clearly political". The bull failed to ensure that Kramer obtained the support he had hoped for, in fact he was subsequently expelled from the city of Innsbruck by the local bishop, George Golzer, who ordered Kramer to stop making false accusations. Golzer described Kramer as senile in letters written shortly after the incident. This rebuke led Kramer to write a justification of his views on witchcraft in his book Malleus Maleficarum, written in 1486. In the book, Kramer stated his view that witchcraft was to blame for bad weather. The book is also noted for its animus against women. Despite Kramer's claim that the book gained acceptance from the clergy at the university of Cologne, it was in fact condemned by the clergy at Cologne for advocating views that violated Catholic doctrine and standard inquisitorial procedure. In 1538 the Spanish Inquisition cautioned its members not to believe everything the Malleus said.

Spanish Inquisition

Pedro Berruguete, Saint Dominic Guzmán presiding over an Auto da fe (c. 1495). Many artistic representations depict torture and burning at the stake during the auto-da-fé (Portuguese for "Act of Faith").

Portugal and Spain in the late Middle Ages consisted largely of multicultural territories of Muslim and Jewish influence, reconquered from Islamic control, and the new Christian authorities could not assume that all their subjects would suddenly become and remain orthodox Roman Catholics. So the Inquisition in Iberia, in the lands of the Reconquista counties and kingdoms like León, Castile and Aragon, had a special socio-political basis as well as more fundamental religious motives.

In some parts of Spain towards the end of the 14th century, there was a wave of violent anti-Judaism, encouraged by the preaching of Ferrand Martinez, Archdeacon of Ecija. In the pogroms of June 1391 in Seville, hundreds of Jews were killed, and the synagogue was completely destroyed. The number of people killed was also high in other cities, such as Córdoba, Valencia and Barcelona.

One of the consequences of these pogroms was the mass conversion of thousands of surviving Jews. Forced baptism was contrary to the law of the Catholic Church, and theoretically anybody who had been forcibly baptized could legally return to Judaism. However, this was very narrowly interpreted. Legal definitions of the time theoretically acknowledged that a forced baptism was not a valid sacrament, but confined this to cases where it was literally administered by physical force. A person who had consented to baptism under threat of death or serious injury was still regarded as a voluntary convert, and accordingly forbidden to revert to Judaism. After the public violence, many of the converted "felt it safer to remain in their new religion." Thus, after 1391, a new social group appeared and were referred to as conversos or New Christians.

King Ferdinand II of Aragon and Queen Isabella I of Castile established the Spanish Inquisition in 1478. In contrast to the previous inquisitions, it operated completely under royal Christian authority, though staffed by clergy and orders, and independently of the Holy See. It operated in Spain and in all Spanish colonies and territories, which included the Canary Islands, the Spanish Netherlands, the Kingdom of Naples, and all Spanish possessions in North, Central, and South America. It primarily targeted forced converts from Islam (Moriscos, Conversos and secret Moors) and from Judaism (Conversos, Crypto-Jews and Marranos) — both groups still resided in Spain after the end of the Islamic control of Spain — who came under suspicion of either continuing to adhere to their old religion or of having fallen back into it.

In 1492 all Jews who had not converted were expelled from Spain; those who converted became nominal Catholics and thus subject to the Inquisition.

Inquisition in the Spanish overseas empire

In the Americas, King Philip II set up three tribunals (each formally titled Tribunal del Santo Oficio de la Inquisición) in 1569, one in Mexico, Cartagena de Indias (in modern-day Colombia) and Peru. The Mexican office administered Mexico (central and southeastern Mexico), Nueva Galicia (northern and western Mexico), the Audiencias of Guatemala (Guatemala, Chiapas, El Salvador, Honduras, Nicaragua, Costa Rica), and the Spanish East Indies. The Peruvian Inquisition, based in Lima, administered all the Spanish territories in South America and Panama.

Portuguese Inquisition

A copper engraving from 1685: "Die Inquisition in Portugall"

The Portuguese Inquisition formally started in Portugal in 1536 at the request of King João III. Manuel I had asked Pope Leo X for the installation of the Inquisition in 1515, but only after his death in 1521 did Pope Paul III acquiesce. At its head stood a Grande Inquisidor, or General Inquisitor, named by the Pope but selected by the Crown, and always from within the royal family. The Portuguese Inquisition principally targeted the Sephardic Jews, whom the state forced to convert to Christianity. Spain had expelled its Sephardic population in 1492; many of these Spanish Jews left Spain for Portugal but eventually were targeted there as well.

The Portuguese Inquisition held its first auto-da-fé in 1540. The Portuguese inquisitors mostly targeted the Jewish New Christians (i.e. conversos or marranos). The Portuguese Inquisition expanded its scope of operations from Portugal to its colonial possessions, including Brazil, Cape Verde, and Goa. In the colonies, it continued as a religious court, investigating and trying cases of breaches of the tenets of orthodox Roman Catholicism until 1821. King João III (reigned 1521–57) extended the activity of the courts to cover censorship, divination, witchcraft, and bigamy. Originally oriented for a religious action, the Inquisition exerted an influence over almost every aspect of Portuguese society: political, cultural, and social.

The Goa Inquisition, an inquisition largely aimed at Catholic converts from Hinduism or Islam who were thought to have returned to their original ways, started in 1560. In addition, the Inquisition prosecuted non-converts who broke prohibitions against the observance of Hindu or Muslim rites or interfered with Portuguese attempts to convert non-Christians to Catholicism. Aleixo Dias Falcão and Francisco Marques set it up in the palace of the Sabaio Adil Khan.

According to Henry Charles Lea, between 1540 and 1794, tribunals in Lisbon, Porto, Coimbra, and Évora resulted in the burning of 1,175 persons, the burning of another 633 in effigy, and the penancing of 29,590. But documentation of 15 out of 689 autos-da-fé has disappeared, so these numbers may slightly understate the activity.

Roman Inquisition

With the Protestant Reformation, Catholic authorities became much more ready to suspect heresy in any new ideas, including those of Renaissance humanism, previously strongly supported by many at the top of the Church hierarchy. The extirpation of heretics became a much broader and more complex enterprise, complicated by the politics of territorial Protestant powers, especially in northern Europe. The Catholic Church could no longer exercise direct influence in the politics and justice-systems of lands that officially adopted Protestantism. Thus war (the French Wars of Religion, the Thirty Years' War), massacre (the St. Bartholomew's Day massacre) and the missional and propaganda work (by the Sacra congregatio de propaganda fide) of the Counter-Reformation came to play larger roles in these circumstances, and the Roman law type of a "judicial" approach to heresy represented by the Inquisition became less important overall. In 1542 Pope Paul III established the Congregation of the Holy Office of the Inquisition as a permanent congregation staffed with cardinals and other officials. It had the tasks of maintaining and defending the integrity of the faith and of examining and proscribing errors and false doctrines; it thus became the supervisory body of local Inquisitions. Arguably the most famous case tried by the Roman Inquisition was that of Galileo Galilei in 1633.

The penances and sentences for those who confessed or were found guilty were pronounced together in a public ceremony at the end of all the processes. This was the sermo generalis or auto-da-fé. Penances (not matters for the civil authorities) might consist of a pilgrimage, a public scourging, a fine, or the wearing of a cross. The wearing of two tongues of red or other brightly colored cloth, sewn onto an outer garment in an "X" pattern, marked those who were under investigation. The penalties in serious cases were confiscation of property by the Inquisition or imprisonment. This led to the possibility of false charges to enable confiscation being made against those over a certain income, particularly rich marranos. Following the French invasion of 1798, the new authorities sent 3,000 chests containing over 100,000 Inquisition documents to France from Rome.

Ending of the Inquisition in the 19th and 20th centuries

The wars of independence of the former Spanish colonies in the Americas concluded with the abolition of the Inquisition in every quarter of Hispanic America between 1813 and 1825.

By decree of Napoleon's government in 1797, the Inquisition in Venice was abolished in 1806
In Portugal, in the wake of the Liberal Revolution of 1820, the "General Extraordinary and Constituent Courts of the Portuguese Nation" abolished the Portuguese inquisition in 1821.

The last execution of the Inquisition was in Spain in 1826. This was the execution by garroting of the school teacher Cayetano Ripoll for purportedly teaching Deism in his school. In Spain the practices of the Inquisition were finally outlawed in 1834.

In Italy, after the restoration of the Pope as the ruler of the Papal States in 1814, the activity of the Papal States Inquisition continued on until the mid-19th century, notably in the well-publicised Mortara Affair (1858–1870). In 1908 the name of the Congregation became "The Sacred Congregation of the Holy Office", which in 1965 further changed to "Congregation for the Doctrine of the Faith", as retained to the present day.

Statistics

Beginning in the 19th century, historians have gradually compiled statistics drawn from the surviving court records, from which estimates have been calculated by adjusting the recorded number of convictions by the average rate of document loss for each time period. Gustav Henningsen and Jaime Contreras studied the records of the Spanish Inquisition, which list 44,674 cases of which 826 resulted in executions in person and 778 in effigy (i.e. a straw dummy was burned in place of the person). William Monter estimated there were 1000 executions between 1530–1630 and 250 between 1630–1730. Jean-Pierre Dedieu studied the records of Toledo's tribunal, which put 12,000 people on trial. For the period prior to 1530, Henry Kamen estimated there were about 2,000 executions in all of Spain's tribunals. Italian Renaissance history professor and Inquisition expert Carlo Ginzburg had his doubts about using statistics to reach a judgment about the period. "In many cases, we don’t have the evidence, the evidence has been lost," said Ginzburg.

Appearance in popular media

Series 2 Episode 2 of Monty Python's Flying Circus is entitled "The Spanish Inquisition", and features Michael Palin, Terry Jones, and Terry Gilliam as an inept—not to mention anachronistic—team of Inquisitors attempting to menace 20th-Century Britons, who seem unfazed by their inane torture implements (such as "the rack", which was taken out of a dishwasher, or even the dreaded "Comfy Chair") and woefully unpolished theatrics (Palin's character discovers to his horror that having more "chief weapons" rather than fewer seems to take some of the punch out of his diabolical monologue). Their signature gimmick is bursting into the room whenever someone says they "didn't expect the Spanish Inquisition", and declaring that "nobody expects the Spanish Inquisition", a line which has itself inspired numerous homages and parodies since.

The 1982 novel Baltasar and Blimunda by José Saramago, portrays how the Portuguese Inquisition impacts the fortunes of the title characters as well as several others from history, including the priest and aviation pioneer Bartolomeu de Gusmão.

The 1981 comedy film History of the World, Part I, produced and directed by Mel Brooks, features a segment on the Spanish Inquisition.

Inquisitio is a French television series set in the Middle Ages.

In the novel Name of the Rose by Umberto Eco, there is some discussion about various sects of Christianity and inquisition, a small discussion about the ethics and purpose of inquisition, and a scene of Inquisition. In the movie by the same name, The Inquisition plays a prominent role including torture and a burning at the stake.

In the novel La Catedral del Mar by Ildefonso Falcones, there are scenes of inquisition investigations in small towns and a great scene in Barcelona.