Curvilinear coordinates

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https://en.wikipedia.org/wiki/Curvilinear_coordinates

Curvilinear (top), affine (right), and Cartesian (left) coordinates in two-dimensional space

In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.

Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R³) are cylindrical and spherical coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example z = 0 defines the x-y plane. In the same space, the coordinate surface r = 1 in spherical coordinates is the surface of a unit sphere, which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems.

Curvilinear coordinates are often used to define the location or distribution of physical quantities which may be, for example, scalars, vectors, or tensors. Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient, divergence, curl, and Laplacian) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. Such expressions then become valid for any curvilinear coordinate system.

A curvilinear coordinate system may be simpler to use than the Cartesian coordinate system for some applications. The motion of particles under the influence of central forces is usually easier to solve in spherical coordinates than in Cartesian coordinates; this is true of many physical problems with spherical symmetry defined in R³. Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system. While one might describe the motion of a particle in a rectangular box using Cartesian coordinates, it is easier to describe the motion in a sphere with spherical coordinates. Spherical coordinates are the most common curvilinear coordinate systems and are used in Earth sciences, cartography, quantum mechanics, relativity, and engineering.

Orthogonal curvilinear coordinates in 3 dimensions

Coordinates, basis, and vectors

Fig. 1 - Coordinate surfaces, coordinate lines, and coordinate axes of general curvilinear coordinates.

Fig. 2 - Coordinate surfaces, coordinate lines, and coordinate axes of spherical coordinates. Surfaces: r - spheres, θ - cones, Φ - half-planes; Lines: r - straight beams, θ - vertical semicircles, Φ - horizontal circles; Axes: r - straight beams, θ - tangents to vertical semicircles, Φ - tangents to horizontal circles

For now, consider 3-D space. A point P in 3-D space (or its position vector r) can be defined using Cartesian coordinates (x, y, z) [equivalently written (x¹, x², x³)], by ${\displaystyle \mathbf{r}=x{\mathbf{e}}_x+y{\mathbf{e}}_y+z{\mathbf{e}}_z}$, where e_x, e_y, e_z are the standard basis vectors.

It can also be defined by its curvilinear coordinates (q¹, q², q³) if this triplet of numbers defines a single point in an unambiguous way. The relation between the coordinates is then given by the invertible transformation functions:

${\displaystyle x=f^1(q^1,q^2,q^3),y=f^2(q^1,q^2,q^3),z=f^3(q^1,q^2,q^3)}$

${\displaystyle q^1=g^1(x,y,z),q^2=g^2(x,y,z),q^3=g^3(x,y,z)}$

The surfaces q¹ = constant, q² = constant, q³ = constant are called the coordinate surfaces; and the space curves formed by their intersection in pairs are called the coordinate curves. The coordinate axes are determined by the tangents to the coordinate curves at the intersection of three surfaces. They are not in general fixed directions in space, which happens to be the case for simple Cartesian coordinates, and thus there is generally no natural global basis for curvilinear coordinates.

In the Cartesian system, the standard basis vectors can be derived from the derivative of the location of point P with respect to the local coordinate

${\displaystyle {\mathbf{e}}_x={\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }x}};{\mathbf{e}}_y={\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }y}};{\mathbf{e}}_z={\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }z}}.}$

Applying the same derivatives to the curvilinear system locally at point P defines the natural basis vectors:

${\displaystyle {\mathbf{h}}_1={\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^1}};{\mathbf{h}}_2={\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^2}};{\mathbf{h}}_3={\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^3}}.}$

Such a basis, whose vectors change their direction and/or magnitude from point to point is called a local basis. All bases associated with curvilinear coordinates are necessarily local. Basis vectors that are the same at all points are global bases, and can be associated only with linear or affine coordinate systems.

For this article e is reserved for the standard basis (Cartesian) and h or b is for the curvilinear basis.

These may not have unit length, and may also not be orthogonal. In the case that they are orthogonal at all points where the derivatives are well-defined, we define the Lamé coefficients (after Gabriel Lamé) by

${\displaystyle h_1=|{\mathbf{h}}_1|;h_2=|{\mathbf{h}}_2|;h_3=|{\mathbf{h}}_3|}$

and the curvilinear orthonormal basis vectors by

${\displaystyle {\mathbf{b}}_1={\displaystyle \frac{{\mathbf{h}}_1}{h_1}};{\mathbf{b}}_2={\displaystyle \frac{{\mathbf{h}}_2}{h_2}};{\mathbf{b}}_3={\displaystyle \frac{{\mathbf{h}}_3}{h_3}}.}$

These basis vectors may well depend upon the position of P; it is therefore necessary that they are not assumed to be constant over a region. (They technically form a basis for the tangent bundle of ${\displaystyle {\mathbb{R}}^3}$ at P, and so are local to P.)

In general, curvilinear coordinates allow the natural basis vectors h_i not all mutually perpendicular to each other, and not required to be of unit length: they can be of arbitrary magnitude and direction. The use of an orthogonal basis makes vector manipulations simpler than for non-orthogonal. However, some areas of physics and engineering, particularly fluid mechanics and continuum mechanics, require non-orthogonal bases to describe deformations and fluid transport to account for complicated directional dependences of physical quantities. A discussion of the general case appears later on this page.

Vector calculus

See also: Differential geometry

Differential elements

In orthogonal curvilinear coordinates, since the total differential change in r is

${\displaystyle d\mathbf{r}={\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^1}}d{q}^1+{\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^2}}d{q}^2+{\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^3}}d{q}^3=h_{1}d{q}^1{\mathbf{b}}_1+h_{2}d{q}^2{\mathbf{b}}_2+h_{3}d{q}^3{\mathbf{b}}_3}$

so scale factors are ${\displaystyle h_i=\left|\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^i}\right|}$

In non-orthogonal coordinates the length of ${\displaystyle d\mathbf{r}=d{q}^1{\mathbf{h}}_1+d{q}^2{\mathbf{h}}_2+d{q}^3{\mathbf{h}}_3}$ is the positive square root of ${\displaystyle d\mathbf{r}\cdot d\mathbf{r}=d{q}^{i}d{q}^j{\mathbf{h}}_i\cdot {\mathbf{h}}_j}$ (with Einstein summation convention). The six independent scalar products g_ij=h_i.h_j of the natural basis vectors generalize the three scale factors defined above for orthogonal coordinates. The nine g_ij are the components of the metric tensor, which has only three non zero components in orthogonal coordinates: g₁₁=h₁h₁, g₂₂=h₂h₂, g₃₃=h₃h₃.

Covariant and contravariant bases

Main articles: Covariance and contravariance of vectors and Raising and lowering indices

Further information: Orthogonal coordinates § Covariant and contravariant bases

A vector v (red) represented by • a vector basis (yellow, left: e₁, e₂, e₃), tangent vectors to coordinate curves (black) and • a covector basis or cobasis (blue, right: e¹, e², e³), normal vectors to coordinate surfaces (grey) in general (not necessarily orthogonal) curvilinear coordinates (q¹, q², q³). The basis and cobasis do not coincide unless the coordinate system is orthogonal.

Spatial gradients, distances, time derivatives and scale factors are interrelated within a coordinate system by two groups of basis vectors:

1. basis vectors that are locally tangent to their associated coordinate pathline: $${\displaystyle {\mathbf{b}}_i={\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^i}}}$$ are contravariant vectors (denoted by lowered indices), and
2. basis vectors that are locally normal to the isosurface created by the other coordinates: $${\displaystyle {\mathbf{b}}^i=\mathrm{\nabla }{q}^i}$$ are covariant vectors (denoted by raised indices), ∇ is the del operator.

Note that, because of Einstein's summation convention, the position of the indices of the vectors is the opposite of that of the coordinates.

Consequently, a general curvilinear coordinate system has two sets of basis vectors for every point: {b₁, b₂, b₃} is the contravariant basis, and {b¹, b², b³} is the covariant (a.k.a. reciprocal) basis. The covariant and contravariant basis vectors types have identical direction for orthogonal curvilinear coordinate systems, but as usual have inverted units with respect to each other.

Note the following important equality: $${\displaystyle {\mathbf{b}}^i\cdot {\mathbf{b}}_j={\delta }_j^i}$$ wherein ${\displaystyle {\delta }_j^i}$ denotes the generalized Kronecker delta.

Proof

In the Cartesian coordinate system ${\displaystyle ({\mathbf{e}}_x,{\mathbf{e}}_y,{\mathbf{e}}_z)}$, we can write the dot product as:

${\displaystyle {\mathbf{b}}_i\cdot {\mathbf{b}}^j=\left({\displaystyle \frac{\mathrm{\partial }x}{\mathrm{\partial }{q}_i}},{\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{q}_i}},{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{q}_i}}\right)\cdot \left({\displaystyle \frac{\mathrm{\partial }{q}_j}{\mathrm{\partial }x}},{\displaystyle \frac{\mathrm{\partial }{q}_j}{\mathrm{\partial }y}},{\displaystyle \frac{\mathrm{\partial }{q}_j}{\mathrm{\partial }z}}\right)={\displaystyle \frac{\mathrm{\partial }x}{\mathrm{\partial }{q}_i}}{\displaystyle \frac{\mathrm{\partial }{q}_j}{\mathrm{\partial }x}}+{\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{q}_i}}{\displaystyle \frac{\mathrm{\partial }{q}_j}{\mathrm{\partial }y}}+{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{q}_i}}{\displaystyle \frac{\mathrm{\partial }{q}_j}{\mathrm{\partial }z}}}$

Consider an infinitesimal displacement ${\displaystyle d\mathbf{r}=dx\cdot {\mathbf{e}}_x+dy\cdot {\mathbf{e}}_y+dz\cdot {\mathbf{e}}_z}$. Let dq₁, dq₂ and dq₃ denote the corresponding infinitesimal changes in curvilinear coordinates q₁, q₂ and q₃ respectively.

By the chain rule, dq₁ can be expressed as:

${\displaystyle d{q}_1={\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }x}}dx+{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }y}}dy+{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }z}}dz={\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }x}}dx+{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }y}}\left({\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{q}_1}}d{q}_1+{\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{q}_2}}d{q}_2+{\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{q}_3}}d{q}_3\right)+{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }z}}\left({\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{q}_1}}d{q}_1+{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{q}_2}}d{q}_2+{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{q}_3}}d{q}_3\right)}$

If the displacement dr is such that dq₂ = dq₃ = 0, i.e. the position vector r moves by an infinitesimal amount along the coordinate axis q₂=const and q₃=const, then:

${\displaystyle d{q}_1={\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }x}}dx+{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }y}}{\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{q}_1}}d{q}_1+{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }z}}{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{q}_1}}d{q}_1}$

Dividing by dq₁, and taking the limit dq₁ → 0:

${\displaystyle 1={\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }x}}{\displaystyle \frac{\mathrm{\partial }x}{\mathrm{\partial }{q}_1}}+{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }y}}{\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{q}_1}}+{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }z}}{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{q}_1}}={\displaystyle \frac{\mathrm{\partial }x}{\mathrm{\partial }{q}_1}}{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }x}}+{\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{q}_1}}{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }y}}+{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{q}_1}}{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }z}}}$

or equivalently:

${\displaystyle {\mathbf{b}}_1\cdot {\mathbf{b}}^1=1}$

Now if the displacement dr is such that dq₁=dq₃=0, i.e. the position vector r moves by an infinitesimal amount along the coordinate axis q₁=const and q₃=const, then:

${\displaystyle 0={\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }x}}dx+{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }y}}{\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{q}_2}}d{q}_2+{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }z}}{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{q}_2}}d{q}_2}$

Dividing by dq₂, and taking the limit dq₂ → 0:

${\displaystyle 0={\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }x}}{\displaystyle \frac{\mathrm{\partial }x}{\mathrm{\partial }{q}_2}}+{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }y}}{\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{q}_2}}+{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }z}}{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{q}_2}}={\displaystyle \frac{\mathrm{\partial }x}{\mathrm{\partial }{q}_2}}{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }x}}+{\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{q}_2}}{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }y}}+{\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{q}_2}}{\displaystyle \frac{\mathrm{\partial }{q}_1}{\mathrm{\partial }z}}}$

or equivalently:

${\displaystyle {\mathbf{b}}_2\cdot {\mathbf{b}}^1=0}$

And so forth for the other dot products.

Alternative Proof:

${\displaystyle {\delta }_j^{i}d{q}^j=d{q}^i=\mathrm{\nabla }{q}^i\cdot d\mathbf{r}={\mathbf{b}}^i\cdot {\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }{q}^j}}d{q}^j={\mathbf{b}}^i\cdot {\mathbf{b}}_{j}d{q}^j}$

and the Einstein summation convention is implied.

A vector v can be specified in terms of either basis, i.e.,

${\displaystyle \mathbf{v}=v^1{\mathbf{b}}_1+v^2{\mathbf{b}}_2+v^3{\mathbf{b}}_3=v_1{\mathbf{b}}^1+v_2{\mathbf{b}}^2+v_3{\mathbf{b}}^3}$

Using the Einstein summation convention, the basis vectors relate to the components by^{[2]: 30–32}

${\displaystyle \mathbf{v}\cdot {\mathbf{b}}^i=v^k{\mathbf{b}}_k\cdot {\mathbf{b}}^i=v^k{\delta }_k^i=v^i}$

${\displaystyle \mathbf{v}\cdot {\mathbf{b}}_i=v_k{\mathbf{b}}^k\cdot {\mathbf{b}}_i=v_k{\delta }_i^k=v_i}$

and

${\displaystyle \mathbf{v}\cdot {\mathbf{b}}_i=v^k{\mathbf{b}}_k\cdot {\mathbf{b}}_i=g_{ki}{v}^k}$

${\displaystyle \mathbf{v}\cdot {\mathbf{b}}^i=v_k{\mathbf{b}}^k\cdot {\mathbf{b}}^i=g^{ki}{v}_k}$

where g is the metric tensor (see below).

A vector can be specified with covariant coordinates (lowered indices, written v_k) or contravariant coordinates (raised indices, written v^k). From the above vector sums, it can be seen that contravariant coordinates are associated with covariant basis vectors, and covariant coordinates are associated with contravariant basis vectors.

A key feature of the representation of vectors and tensors in terms of indexed components and basis vectors is invariance in the sense that vector components which transform in a covariant manner (or contravariant manner) are paired with basis vectors that transform in a contravariant manner (or covariant manner).

Integration

Main article: Covariant transformation

Constructing a covariant basis in one dimension

Fig. 3 – Transformation of local covariant basis in the case of general curvilinear coordinates

Consider the one-dimensional curve shown in Fig. 3. At point P, taken as an origin, x is one of the Cartesian coordinates, and q¹ is one of the curvilinear coordinates. The local (non-unit) basis vector is b₁ (notated h₁ above, with b reserved for unit vectors) and it is built on the q¹ axis which is a tangent to that coordinate line at the point P. The axis q¹ and thus the vector b₁ form an angle ${\displaystyle \alpha }$ with the Cartesian x axis and the Cartesian basis vector e₁.

It can be seen from triangle PAB that

${\displaystyle \cos \alpha =\frac{|{\mathbf{e}}_1|}{|{\mathbf{b}}_1|}\Rightarrow |{\mathbf{e}}_1|=|{\mathbf{b}}_1|\cos \alpha }$

where |e₁|, |b₁| are the magnitudes of the two basis vectors, i.e., the scalar intercepts PB and PA. PA is also the projection of b₁ on the x axis.

However, this method for basis vector transformations using directional cosines is inapplicable to curvilinear coordinates for the following reasons:

1. By increasing the distance from P, the angle between the curved line q¹ and Cartesian axis x increasingly deviates from ${\displaystyle \alpha }$.
2. At the distance PB the true angle is that which the tangent at point C forms with the x axis and the latter angle is clearly different from ${\displaystyle \alpha }$.

The angles that the q¹ line and that axis form with the x axis become closer in value the closer one moves towards point P and become exactly equal at P.

Let point E be located very close to P, so close that the distance PE is infinitesimally small. Then PE measured on the q¹ axis almost coincides with PE measured on the q¹ line. At the same time, the ratio PD/PE (PD being the projection of PE on the x axis) becomes almost exactly equal to ${\displaystyle \cos \alpha }$.

Let the infinitesimally small intercepts PD and PE be labelled, respectively, as dx and dq¹. Then

${\displaystyle \cos \alpha =\frac{dx}{d{q}^1}=\frac{|{\mathbf{e}}_1|}{|{\mathbf{b}}_1|}}$.

Thus, the directional cosines can be substituted in transformations with the more exact ratios between infinitesimally small coordinate intercepts. It follows that the component (projection) of b₁ on the x axis is

${\displaystyle p^1={\mathbf{b}}_1\cdot \frac{{\mathbf{e}}_1}{|{\mathbf{e}}_1|}=|{\mathbf{b}}_1|\frac{|{\mathbf{e}}_1|}{|{\mathbf{e}}_1|}\cos \alpha =|{\mathbf{b}}_1|\frac{dx}{d{q}^1}\Rightarrow \frac{p^1}{|{\mathbf{b}}_1|}=\frac{dx}{d{q}^1}}$.

If qⁱ = qⁱ(x₁, x₂, x₃) and x_i = x_i(q¹, q², q³) are smooth (continuously differentiable) functions the transformation ratios can be written as ${\displaystyle \frac{\mathrm{\partial }{q}^i}{\mathrm{\partial }{x}_j}}$ and ${\displaystyle \frac{\mathrm{\partial }{x}_i}{\mathrm{\partial }{q}^j}}$. That is, those ratios are partial derivatives of coordinates belonging to one system with respect to coordinates belonging to the other system.

Constructing a covariant basis in three dimensions

Doing the same for the coordinates in the other 2 dimensions, b₁ can be expressed as:

${\displaystyle {\mathbf{b}}_1=p^1{\mathbf{e}}_1+p^2{\mathbf{e}}_2+p^3{\mathbf{e}}_3=\frac{\mathrm{\partial }{x}_1}{\mathrm{\partial }{q}^1}{\mathbf{e}}_1+\frac{\mathrm{\partial }{x}_2}{\mathrm{\partial }{q}^1}{\mathbf{e}}_2+\frac{\mathrm{\partial }{x}_3}{\mathrm{\partial }{q}^1}{\mathbf{e}}_3}$

Similar equations hold for b₂ and b₃ so that the standard basis {e₁, e₂, e₃} is transformed to a local (ordered and normalised) basis {b₁, b₂, b₃} by the following system of equations:

${\displaystyle \begin{array}{rl}{\mathbf{b}}_1& =\frac{\mathrm{\partial }{x}_1}{\mathrm{\partial }{q}^1}{\mathbf{e}}_1+\frac{\mathrm{\partial }{x}_2}{\mathrm{\partial }{q}^1}{\mathbf{e}}_2+\frac{\mathrm{\partial }{x}_3}{\mathrm{\partial }{q}^1}{\mathbf{e}}_3\\ {\mathbf{b}}_2& =\frac{\mathrm{\partial }{x}_1}{\mathrm{\partial }{q}^2}{\mathbf{e}}_1+\frac{\mathrm{\partial }{x}_2}{\mathrm{\partial }{q}^2}{\mathbf{e}}_2+\frac{\mathrm{\partial }{x}_3}{\mathrm{\partial }{q}^2}{\mathbf{e}}_3\\ {\mathbf{b}}_3& =\frac{\mathrm{\partial }{x}_1}{\mathrm{\partial }{q}^3}{\mathbf{e}}_1+\frac{\mathrm{\partial }{x}_2}{\mathrm{\partial }{q}^3}{\mathbf{e}}_2+\frac{\mathrm{\partial }{x}_3}{\mathrm{\partial }{q}^3}{\mathbf{e}}_3\end{array}}$

By analogous reasoning, one can obtain the inverse transformation from local basis to standard basis:

${\displaystyle \begin{array}{rl}{\mathbf{e}}_1& =\frac{\mathrm{\partial }{q}^1}{\mathrm{\partial }{x}_1}{\mathbf{b}}_1+\frac{\mathrm{\partial }{q}^2}{\mathrm{\partial }{x}_1}{\mathbf{b}}_2+\frac{\mathrm{\partial }{q}^3}{\mathrm{\partial }{x}_1}{\mathbf{b}}_3\\ {\mathbf{e}}_2& =\frac{\mathrm{\partial }{q}^1}{\mathrm{\partial }{x}_2}{\mathbf{b}}_1+\frac{\mathrm{\partial }{q}^2}{\mathrm{\partial }{x}_2}{\mathbf{b}}_2+\frac{\mathrm{\partial }{q}^3}{\mathrm{\partial }{x}_2}{\mathbf{b}}_3\\ {\mathbf{e}}_3& =\frac{\mathrm{\partial }{q}^1}{\mathrm{\partial }{x}_3}{\mathbf{b}}_1+\frac{\mathrm{\partial }{q}^2}{\mathrm{\partial }{x}_3}{\mathbf{b}}_2+\frac{\mathrm{\partial }{q}^3}{\mathrm{\partial }{x}_3}{\mathbf{b}}_3\end{array}}$

Jacobian of the transformation

The above systems of linear equations can be written in matrix form using the Einstein summation convention as

${\displaystyle \frac{\mathrm{\partial }{x}_i}{\mathrm{\partial }{q}^k}{\mathbf{e}}_i={\mathbf{b}}_k,\frac{\mathrm{\partial }{q}^i}{\mathrm{\partial }{x}_k}{\mathbf{b}}_i={\mathbf{e}}_k}$.

This coefficient matrix of the linear system is the Jacobian matrix (and its inverse) of the transformation. These are the equations that can be used to transform a Cartesian basis into a curvilinear basis, and vice versa.

In three dimensions, the expanded forms of these matrices are

${\displaystyle \mathbf{J}=\left[\begin{array}{ccc}\frac{\mathrm{\partial }{x}_1}{\mathrm{\partial }{q}^1}& \frac{\mathrm{\partial }{x}_1}{\mathrm{\partial }{q}^2}& \frac{\mathrm{\partial }{x}_1}{\mathrm{\partial }{q}^3}\\ \frac{\mathrm{\partial }{x}_2}{\mathrm{\partial }{q}^1}& \frac{\mathrm{\partial }{x}_2}{\mathrm{\partial }{q}^2}& \frac{\mathrm{\partial }{x}_2}{\mathrm{\partial }{q}^3}\\ \frac{\mathrm{\partial }{x}_3}{\mathrm{\partial }{q}^1}& \frac{\mathrm{\partial }{x}_3}{\mathrm{\partial }{q}^2}& \frac{\mathrm{\partial }{x}_3}{\mathrm{\partial }{q}^3}\end{array}\right],{\mathbf{J}}^{-1}=\left[\begin{array}{ccc}\frac{\mathrm{\partial }{q}^1}{\mathrm{\partial }{x}_1}& \frac{\mathrm{\partial }{q}^1}{\mathrm{\partial }{x}_2}& \frac{\mathrm{\partial }{q}^1}{\mathrm{\partial }{x}_3}\\ \frac{\mathrm{\partial }{q}^2}{\mathrm{\partial }{x}_1}& \frac{\mathrm{\partial }{q}^2}{\mathrm{\partial }{x}_2}& \frac{\mathrm{\partial }{q}^2}{\mathrm{\partial }{x}_3}\\ \frac{\mathrm{\partial }{q}^3}{\mathrm{\partial }{x}_1}& \frac{\mathrm{\partial }{q}^3}{\mathrm{\partial }{x}_2}& \frac{\mathrm{\partial }{q}^3}{\mathrm{\partial }{x}_3}\end{array}\right]}$

In the inverse transformation (second equation system), the unknowns are the curvilinear basis vectors. For any specific location there can only exist one and only one set of basis vectors (else the basis is not well defined at that point). This condition is satisfied if and only if the equation system has a single solution. In linear algebra, a linear equation system has a single solution (non-trivial) only if the determinant of its system matrix is non-zero:

${\displaystyle det({\mathbf{J}}^{-1})\ne 0}$

which shows the rationale behind the above requirement concerning the inverse Jacobian determinant.

Generalization to n dimensions

The formalism extends to any finite dimension as follows.

Consider the real Euclidean n-dimensional space, that is Rⁿ = R × R × ... × R (n times) where R is the set of real numbers and × denotes the Cartesian product, which is a vector space.

The coordinates of this space can be denoted by: x = (x₁, x₂,...,x_n). Since this is a vector (an element of the vector space), it can be written as:

${\displaystyle \mathbf{x}=\sum _{i=1}^n{x}_i{\mathbf{e}}^i}$

where e¹ = (1,0,0...,0), e² = (0,1,0...,0), e³ = (0,0,1...,0),...,eⁿ = (0,0,0...,1) is the standard basis set of vectors for the space Rⁿ, and i = 1, 2,...n is an index labelling components. Each vector has exactly one component in each dimension (or "axis") and they are mutually orthogonal (perpendicular) and normalized (has unit magnitude).

More generally, we can define basis vectors b_i so that they depend on q = (q₁, q₂,...,q_n), i.e. they change from point to point: b_i = b_i(q). In which case to define the same point x in terms of this alternative basis: the coordinates with respect to this basis v_i also necessarily depend on x also, that is v_i = v_i(x). Then a vector v in this space, with respect to these alternative coordinates and basis vectors, can be expanded as a linear combination in this basis (which simply means to multiply each basis vector e_i by a number v_i – scalar multiplication):

${\displaystyle \mathbf{v}=\sum _{j=1}^n{\overline{v}}^j{\mathbf{b}}_j=\sum _{j=1}^n{\overline{v}}^j(\mathbf{q}){\mathbf{b}}_j(\mathbf{q})}$

The vector sum that describes v in the new basis is composed of different vectors, although the sum itself remains the same.

Transformation of coordinates

From a more general and abstract perspective, a curvilinear coordinate system is simply a coordinate patch on the differentiable manifold Eⁿ (n-dimensional Euclidean space) that is diffeomorphic to the Cartesian coordinate patch on the manifold. Two diffeomorphic coordinate patches on a differential manifold need not overlap differentiably. With this simple definition of a curvilinear coordinate system, all the results that follow below are simply applications of standard theorems in differential topology.

The transformation functions are such that there's a one-to-one relationship between points in the "old" and "new" coordinates, that is, those functions are bijections, and fulfil the following requirements within their domains:

1. They are smooth functions: qⁱ = qⁱ(x)
2. The inverse Jacobian determinant

   ${\displaystyle J^{-1}=\left|\begin{array}{cccc}{\displaystyle \frac{\mathrm{\partial }{q}^1}{\mathrm{\partial }{x}_1}}& {\displaystyle \frac{\mathrm{\partial }{q}^1}{\mathrm{\partial }{x}_2}}& \cdots & {\displaystyle \frac{\mathrm{\partial }{q}^1}{\mathrm{\partial }{x}_n}}\\ {\displaystyle \frac{\mathrm{\partial }{q}^2}{\mathrm{\partial }{x}_1}}& {\displaystyle \frac{\mathrm{\partial }{q}^2}{\mathrm{\partial }{x}_2}}& \cdots & {\displaystyle \frac{\mathrm{\partial }{q}^2}{\mathrm{\partial }{x}_n}}\\ ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \frac{\mathrm{\partial }{q}^n}{\mathrm{\partial }{x}_1}}& {\displaystyle \frac{\mathrm{\partial }{q}^n}{\mathrm{\partial }{x}_2}}& \cdots & {\displaystyle \frac{\mathrm{\partial }{q}^n}{\mathrm{\partial }{x}_n}}\end{array}\right|\ne 0}$

   is not zero; meaning the transformation is invertible: x_i(q) according to the inverse function theorem. The condition that the Jacobian determinant is not zero reflects the fact that three surfaces from different families intersect in one and only one point and thus determine the position of this point in a unique way.

Vector and tensor algebra in three-dimensional curvilinear coordinates

Note: the Einstein summation convention of summing on repeated indices is used below.

Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in mechanics and physics and can be indispensable to understanding work from the early and mid-1900s, for example the text by Green and Zerna. Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, Naghdi, Simmonds, Green and Zerna, Basar and Weichert, and Ciarlet.

Tensors in curvilinear coordinates

Main article: Tensors in curvilinear coordinates

A second-order tensor can be expressed as

${\displaystyle \mathit{S}=S^{ij}{\mathbf{b}}_i\otimes {\mathbf{b}}_j=S^i{}_j{\mathbf{b}}_i\otimes {\mathbf{b}}^j=S_i{}^j{\mathbf{b}}^i\otimes {\mathbf{b}}_j=S_{ij}{\mathbf{b}}^i\otimes {\mathbf{b}}^j}$

where ${\displaystyle \otimes }$ denotes the tensor product. The components S^ij are called the contravariant components, Sⁱ _j the mixed right-covariant components, S_i ^j the mixed left-covariant components, and S_ij the covariant components of the second-order tensor. The components of the second-order tensor are related by

${\displaystyle S^{ij}=g^{ik}{S}_k{}^j=g^{jk}{S}^i{}_k=g^{ik}{g}^{j\ell }{S}_{k\ell }}$

The metric tensor in orthogonal curvilinear coordinates

Main article: Metric tensor

At each point, one can construct a small line element dx, so the square of the length of the line element is the scalar product dx • dx and is called the metric of the space, given by:

${\displaystyle d\mathbf{x}\cdot d\mathbf{x}=\frac{\mathrm{\partial }{x}_i}{\mathrm{\partial }{q}^j}\frac{\mathrm{\partial }{x}_i}{\mathrm{\partial }{q}^k}d{q}^{j}d{q}^k}$.

The following portion of the above equation

${\displaystyle \frac{\mathrm{\partial }{x}_k}{\mathrm{\partial }{q}^i}\frac{\mathrm{\partial }{x}_k}{\mathrm{\partial }{q}^j}=g_{ij}(q^i,q^j)={\mathbf{b}}_i\cdot {\mathbf{b}}_j}$

is a symmetric tensor called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates.

Indices can be raised and lowered by the metric:

${\displaystyle v^i=g^{ik}{v}_k}$

Relation to Lamé coefficients

Defining the scale factors h_i by

${\displaystyle h_i{h}_j=g_{ij}={\mathbf{b}}_i\cdot {\mathbf{b}}_j\Rightarrow h_i=\sqrt{g_{ii}}=\left|{\mathbf{b}}_i\right|=\left|\frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{q}^i}\right|}$

gives a relation between the metric tensor and the Lamé coefficients, and

${\displaystyle g_{ij}=\frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{q}^i}\cdot \frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{q}^j}=\left(h_{ki}{\mathbf{e}}_k\right)\cdot \left(h_{mj}{\mathbf{e}}_m\right)=h_{ki}{h}_{kj}}$

where h_ij are the Lamé coefficients. For an orthogonal basis we also have:

${\displaystyle g=g_{11}{g}_{22}{g}_{33}=h_1^2{h}_2^2{h}_3^2\Rightarrow \sqrt{g}=h_1{h}_2{h}_3=J}$

Example: Polar coordinates

If we consider polar coordinates for R²,

${\displaystyle (x,y)=(r\cos \theta ,r\sin \theta )}$

(r, θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation (r,θ) → (r cos θ, r sin θ) is r.

The orthogonal basis vectors are b_r = (cos θ, sin θ), b_θ = (−r sin θ, r cos θ). The scale factors are h_r = 1 and h_θ= r. The fundamental tensor is g₁₁ =1, g₂₂ =r², g₁₂ = g₂₁ =0.

The alternating tensor

In an orthonormal right-handed basis, the third-order alternating tensor is defined as

${\displaystyle \mathcal{E}={\epsilon }_{ijk}{\mathbf{e}}^i\otimes {\mathbf{e}}^j\otimes {\mathbf{e}}^k}$

In a general curvilinear basis the same tensor may be expressed as

${\displaystyle \mathcal{E}={\mathcal{E}}_{ijk}{\mathbf{b}}^i\otimes {\mathbf{b}}^j\otimes {\mathbf{b}}^k={\mathcal{E}}^{ijk}{\mathbf{b}}_i\otimes {\mathbf{b}}_j\otimes {\mathbf{b}}_k}$

It can also be shown that

${\displaystyle {\mathcal{E}}^{ijk}=\frac{1}{J}{\epsilon }_{ijk}=\frac{1}{+\sqrt{g}}{\epsilon }_{ijk}}$

Christoffel symbols

Christoffel symbols of the first kind ${\displaystyle {\mathrm{\Gamma }}_{kij}}$

${\displaystyle {\mathbf{b}}_{i,j}=\frac{\mathrm{\partial }{\mathbf{b}}_i}{\mathrm{\partial }{q}^j}={\mathbf{b}}^k{\mathrm{\Gamma }}_{kij}\Rightarrow {\mathbf{b}}_k\cdot {\mathbf{b}}_{i,j}={\mathrm{\Gamma }}_{kij}}$

where the comma denotes a partial derivative (see Ricci calculus). To express Γ_kij in terms of g_ij,

${\displaystyle \begin{array}{rl}{g}_{ij,k}& =({\mathbf{b}}_i\cdot {\mathbf{b}}_j{)}_{,k}={\mathbf{b}}_{i,k}\cdot {\mathbf{b}}_j+{\mathbf{b}}_i\cdot {\mathbf{b}}_{j,k}={\mathrm{\Gamma }}_{jik}+{\mathrm{\Gamma }}_{ijk}\\ g_{ik,j}& =({\mathbf{b}}_i\cdot {\mathbf{b}}_k{)}_{,j}={\mathbf{b}}_{i,j}\cdot {\mathbf{b}}_k+{\mathbf{b}}_i\cdot {\mathbf{b}}_{k,j}={\mathrm{\Gamma }}_{kij}+{\mathrm{\Gamma }}_{ikj}\\ g_{jk,i}& =({\mathbf{b}}_j\cdot {\mathbf{b}}_k{)}_{,i}={\mathbf{b}}_{j,i}\cdot {\mathbf{b}}_k+{\mathbf{b}}_j\cdot {\mathbf{b}}_{k,i}={\mathrm{\Gamma }}_{kji}+{\mathrm{\Gamma }}_{jki}\end{array}}$

Since

${\displaystyle {\mathbf{b}}_{i,j}={\mathbf{b}}_{j,i}\Rightarrow {\mathrm{\Gamma }}_{kij}={\mathrm{\Gamma }}_{kji}}$

using these to rearrange the above relations gives

${\displaystyle {\mathrm{\Gamma }}_{kij}=\frac{1}{2}(g_{ik,j}+g_{jk,i}-g_{ij,k})=\frac{1}{2}[({\mathbf{b}}_i\cdot {\mathbf{b}}_k{)}_{,j}+({\mathbf{b}}_j\cdot {\mathbf{b}}_k{)}_{,i}-({\mathbf{b}}_i\cdot {\mathbf{b}}_j{)}_{,k}]}$

Christoffel symbols of the second kind ${\displaystyle {\mathrm{\Gamma }}^k{}_{ji}}$

${\displaystyle {\mathrm{\Gamma }}^k{}_{ij}=g^{kl}{\mathrm{\Gamma }}_{lij}={\mathrm{\Gamma }}^k{}_{ji},\frac{\mathrm{\partial }{\mathbf{b}}_i}{\mathrm{\partial }{q}^j}={\mathbf{b}}_k{\mathrm{\Gamma }}^k{}_{ij}}$

This implies that

${\displaystyle {\mathrm{\Gamma }}^k{}_{ij}=\frac{\mathrm{\partial }{\mathbf{b}}_i}{\mathrm{\partial }{q}^j}\cdot {\mathbf{b}}^k=-{\mathbf{b}}_i\cdot \frac{\mathrm{\partial }{\mathbf{b}}^k}{\mathrm{\partial }{q}^j}}$ since ${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{q}^j}({\mathbf{b}}_i\cdot {\mathbf{b}}^k)=0}$.

Other relations that follow are

${\displaystyle \frac{\mathrm{\partial }{\mathbf{b}}^i}{\mathrm{\partial }{q}^j}=-{\mathrm{\Gamma }}^i{}_{jk}{\mathbf{b}}^k,\mathbf{\nabla }{\mathbf{b}}_i={\mathrm{\Gamma }}^k{}_{ij}{\mathbf{b}}_k\otimes {\mathbf{b}}^j,\mathbf{\nabla }{\mathbf{b}}^i=-{\mathrm{\Gamma }}^i{}_{jk}{\mathbf{b}}^k\otimes {\mathbf{b}}^j}$

Vector operations

1. Dot product:

   The scalar product of two vectors in curvilinear coordinates is

   ${\displaystyle \mathbf{u}\cdot \mathbf{v}=u^i{v}_i=u_i{v}^i=g_{ij}{u}^i{v}^j=g^{ij}{u}_i{v}_j}$

2. Cross product:

   The cross product of two vectors is given by

   ${\displaystyle \mathbf{u}\times \mathbf{v}={ϵ}_{ijk}{u}_j{v}_k{\mathbf{e}}_i}$

   where ${\displaystyle {ϵ}_{ijk}}$ is the permutation symbol and ${\displaystyle {\mathbf{e}}_i}$ is a Cartesian basis vector. In curvilinear coordinates, the equivalent expression is

   ${\displaystyle \mathbf{u}\times \mathbf{v}=[({\mathbf{b}}_m\times {\mathbf{b}}_n)\cdot {\mathbf{b}}_s]u^m{v}^n{\mathbf{b}}^s={\mathcal{E}}_{smn}{u}^m{v}^n{\mathbf{b}}^s}$

   where ${\displaystyle {\mathcal{E}}_{ijk}}$ is the third-order alternating tensor.

Vector and tensor calculus in three-dimensional curvilinear coordinates

Note: the Einstein summation convention of summing on repeated indices is used below.

Adjustments need to be made in the calculation of line, surface and volume integrals. For simplicity, the following restricts to three dimensions and orthogonal curvilinear coordinates. However, the same arguments apply for n-dimensional spaces. When the coordinate system is not orthogonal, there are some additional terms in the expressions.

Simmonds, in his book on tensor analysis, quotes Albert Einstein saying

The magic of this theory will hardly fail to impose itself on anybody who has truly understood it; it represents a genuine triumph of the method of absolute differential calculus, founded by Gauss, Riemann, Ricci, and Levi-Civita.

Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity, in the mechanics of curved shells, in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials and in many other fields.

Some useful relations in the calculus of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, Simmonds, Green and Zerna, Basar and Weichert, and Ciarlet.

Let φ = φ(x) be a well defined scalar field and v = v(x) a well-defined vector field, and λ₁, λ₂... be parameters of the coordinates

Geometric elements

1. Tangent vector: If x(λ) parametrizes a curve C in Cartesian coordinates, then

   ${\displaystyle \frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }\lambda }=\frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{q}^i}\frac{\mathrm{\partial }{q}^i}{\mathrm{\partial }\lambda }=\left(h_{ki}\frac{\mathrm{\partial }{q}^i}{\mathrm{\partial }\lambda }\right){\mathbf{b}}_k}$

   is a tangent vector to C in curvilinear coordinates (using the chain rule). Using the definition of the Lamé coefficients, and that for the metric g_ij = 0 when i ≠ j, the magnitude is:

   ${\displaystyle \left|\frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }\lambda }\right|=\sqrt{h_{ki}{h}_{kj}\frac{\mathrm{\partial }{q}^i}{\mathrm{\partial }\lambda }\frac{\mathrm{\partial }{q}^j}{\mathrm{\partial }\lambda }}=\sqrt{g_{ij}\frac{\mathrm{\partial }{q}^i}{\mathrm{\partial }\lambda }\frac{\mathrm{\partial }{q}^j}{\mathrm{\partial }\lambda }}=\sqrt{h_i^2{\left(\frac{\mathrm{\partial }{q}^i}{\mathrm{\partial }\lambda }\right)}^2}}$

2. Tangent plane element: If x(λ₁, λ₂) parametrizes a surface S in Cartesian coordinates, then the following cross product of tangent vectors is a normal vector to S with the magnitude of infinitesimal plane element, in curvilinear coordinates. Using the above result,

   ${\displaystyle \frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{\lambda }_1}\times \frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{\lambda }_2}=\left(\frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{q}^i}\frac{\mathrm{\partial }{q}^i}{\mathrm{\partial }{\lambda }_1}\right)\times \left(\frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{q}^j}\frac{\mathrm{\partial }{q}^j}{\mathrm{\partial }{\lambda }_2}\right)={\mathcal{E}}_{kmp}\left(h_{ki}\frac{\mathrm{\partial }{q}^i}{\mathrm{\partial }{\lambda }_1}\right)\left(h_{mj}\frac{\mathrm{\partial }{q}^j}{\mathrm{\partial }{\lambda }_2}\right){\mathbf{b}}_p}$

   where ${\displaystyle \mathcal{E}}$ is the permutation symbol. In determinant form:

   ${\displaystyle \frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{\lambda }_1}\times \frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{\lambda }_2}=\left|\begin{array}{ccc}{\mathbf{e}}_1& {\mathbf{e}}_2& {\mathbf{e}}_3\\ h_{1i}{\displaystyle \frac{\mathrm{\partial }{q}^i}{\mathrm{\partial }{\lambda }_1}}& h_{2i}{\displaystyle \frac{\mathrm{\partial }{q}^i}{\mathrm{\partial }{\lambda }_1}}& h_{3i}{\displaystyle \frac{\mathrm{\partial }{q}^i}{\mathrm{\partial }{\lambda }_1}}\\ h_{1j}{\displaystyle \frac{\mathrm{\partial }{q}^j}{\mathrm{\partial }{\lambda }_2}}& h_{2j}{\displaystyle \frac{\mathrm{\partial }{q}^j}{\mathrm{\partial }{\lambda }_2}}& h_{3j}{\displaystyle \frac{\mathrm{\partial }{q}^j}{\mathrm{\partial }{\lambda }_2}}\end{array}\right|}$

Integration

Operator	Scalar field	Vector field
Line integral	${\displaystyle {\int }_C\phi (\mathbf{x})ds={\int }_a^b\phi (\mathbf{x}(\lambda ))\left|\frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }\lambda }\right|d\lambda }$	${\displaystyle {\int }_C\mathbf{v}(\mathbf{x})\cdot d\mathbf{s}={\int }_a^b\mathbf{v}(\mathbf{x}(\lambda ))\cdot \left(\frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }\lambda }\right)d\lambda }$
Surface integral	${\displaystyle {\int }_S\phi (\mathbf{x})dS={\iint }_T\phi (\mathbf{x}({\lambda }_1,{\lambda }_2))\left|\frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{\lambda }_1}\times \frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{\lambda }_2}\right|d{\lambda }_{1}d{\lambda }_2}$	${\displaystyle {\int }_S\mathbf{v}(\mathbf{x})\cdot dS={\iint }_T\mathbf{v}(\mathbf{x}({\lambda }_1,{\lambda }_2))\cdot \left(\frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{\lambda }_1}\times \frac{\mathrm{\partial }\mathbf{x}}{\mathrm{\partial }{\lambda }_2}\right)d{\lambda }_{1}d{\lambda }_2}$
Volume integral	${\displaystyle {\iiint }_V\phi (x,y,z)dV={\iiint }_V\chi (q_1,q_2,q_3)Jd{q}_{1}d{q}_{2}d{q}_3}$	${\displaystyle {\iiint }_V\mathbf{u}(x,y,z)dV={\iiint }_V\mathbf{v}(q_1,q_2,q_3)Jd{q}_{1}d{q}_{2}d{q}_3}$

Differentiation

The expressions for the gradient, divergence, and Laplacian can be directly extended to n-dimensions, however the curl is only defined in 3D.

The vector field b_i is tangent to the qⁱ coordinate curve and forms a natural basis at each point on the curve. This basis, as discussed at the beginning of this article, is also called the covariant curvilinear basis. We can also define a reciprocal basis, or contravariant curvilinear basis, bⁱ. All the algebraic relations between the basis vectors, as discussed in the section on tensor algebra, apply for the natural basis and its reciprocal at each point x.

Operator	Scalar field	Vector field	2nd order tensor field
Gradient	${\displaystyle \mathrm{\nabla }\phi =\frac{1}{h_i}\frac{\mathrm{\partial }\phi }{\mathrm{\partial }{q}^i}{\mathbf{b}}^i}$	${\displaystyle \mathrm{\nabla }\mathbf{v}=\frac{1}{h_i^2}\frac{\mathrm{\partial }\mathbf{v}}{\mathrm{\partial }{q}^i}\otimes {\mathbf{b}}_i}$	${\displaystyle \mathbf{\nabla }\mathit{S}=\frac{\mathrm{\partial }\mathit{S}}{\mathrm{\partial }{q}^i}\otimes {\mathbf{b}}^i}$
Divergence	N/A	${\displaystyle \mathrm{\nabla }\cdot \mathbf{v}=\frac{1}{\prod _j{h}_j}\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^i}(v^i\prod _{j\ne i}{h}_j)}$	${\displaystyle (\mathbf{\nabla }\cdot \mathit{S})\cdot \mathbf{a}=\mathbf{\nabla }\cdot (\mathit{S}\cdot \mathbf{a})}$ where a is an arbitrary constant vector. In curvilinear coordinates, ${\displaystyle \mathbf{\nabla }\cdot \mathit{S}=\left[\frac{\mathrm{\partial }{S}_{ij}}{\mathrm{\partial }{q}^k}-{\mathrm{\Gamma }}_{ki}^l{S}_{lj}-{\mathrm{\Gamma }}_{kj}^l{S}_{il}\right]g^{ik}{\mathbf{b}}^j}$
Laplacian	${\displaystyle {\mathrm{\nabla }}^2\phi =\frac{1}{\prod _j{h}_j}\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^i}\left(\frac{\prod _j{h}_j}{h_i^2}\frac{\mathrm{\partial }\phi }{\mathrm{\partial }{q}^i}\right)}$	${\displaystyle {\mathrm{\nabla }}^2\mathbf{v}\equiv \mathrm{\nabla }\mathrm{\nabla }\cdot \mathbf{v}-\mathrm{\nabla }\times \mathrm{\nabla }\times \mathbf{v}}$ ${\displaystyle =\hat{\mathbf{x}}{\mathrm{\nabla }}^2{v}_x+\hat{\mathbf{y}}{\mathrm{\nabla }}^2{v}_y+\hat{\mathbf{z}}{\mathrm{\nabla }}^2{v}_z}$ (First equality in 3D only; second equality in Cartesian components only)
Curl	N/A	For vector fields in 3D only, ${\displaystyle \mathrm{\nabla }\times \mathbf{v}=\frac{1}{h_1{h}_2{h}_3}{\mathbf{e}}_i{ϵ}_{ijk}{h}_i\frac{\mathrm{\partial }(h_k{v}_k)}{\mathrm{\partial }{q}^j}}$ where ${\displaystyle {ϵ}_{ijk}}$ is the Levi-Civita symbol.	See Curl of a tensor field

Fictitious forces in general curvilinear coordinates

By definition, if a particle with no forces acting on it has its position expressed in an inertial coordinate system, (x₁, x₂, x₃, t), then there it will have no acceleration (d²x_j/dt² = 0). In this context, a coordinate system can fail to be "inertial" either due to non-straight time axis or non-straight space axes (or both). In other words, the basis vectors of the coordinates may vary in time at fixed positions, or they may vary with position at fixed times, or both. When equations of motion are expressed in terms of any non-inertial coordinate system (in this sense), extra terms appear, called Christoffel symbols. Strictly speaking, these terms represent components of the absolute acceleration (in classical mechanics), but we may also choose to continue to regard d²x_j/dt² as the acceleration (as if the coordinates were inertial) and treat the extra terms as if they were forces, in which case they are called fictitious forces. The component of any such fictitious force normal to the path of the particle and in the plane of the path's curvature is then called centrifugal force.

This more general context makes clear the correspondence between the concepts of centrifugal force in rotating coordinate systems and in stationary curvilinear coordinate systems. (Both of these concepts appear frequently in the literature.) For a simple example, consider a particle of mass m moving in a circle of radius r with angular speed w relative to a system of polar coordinates rotating with angular speed W. The radial equation of motion is mr” = F_r + mr(w + W)². Thus the centrifugal force is mr times the square of the absolute rotational speed A = w + W of the particle. If we choose a coordinate system rotating at the speed of the particle, then W = A and w = 0, in which case the centrifugal force is mrA², whereas if we choose a stationary coordinate system we have W = 0 and w = A, in which case the centrifugal force is again mrA². The reason for this equality of results is that in both cases the basis vectors at the particle's location are changing in time in exactly the same way. Hence these are really just two different ways of describing exactly the same thing, one description being in terms of rotating coordinates and the other being in terms of stationary curvilinear coordinates, both of which are non-inertial according to the more abstract meaning of that term.

When describing general motion, the actual forces acting on a particle are often referred to the instantaneous osculating circle tangent to the path of motion, and this circle in the general case is not centered at a fixed location, and so the decomposition into centrifugal and Coriolis components is constantly changing. This is true regardless of whether the motion is described in terms of stationary or rotating coordinates.

Sex differences in cognition

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Sex_differences_in_cognition

Sex differences in cognition are widely studied in the current scientific literature. Biological and genetic differences in combination with environment and culture have resulted in the cognitive differences among males and females. Among biological factors, hormones such as testosterone and estrogen may play some role mediating these differences. Among differences of diverse mental and cognitive abilities, the largest or most well known are those relating to spatial abilities, social cognition and verbal skills and abilities.

Cognitive abilities

Cognitive abilities are mental abilities that a person uses in everyday life, as well as specific demand tasks. The most basic of these abilities are memory, executive function, processing speed and perception, which combine to form a larger perceptual umbrella relating to different social, affective, verbal and spatial information. Memory, which is one of the primary core of cognitive abilities can be broken down into short-term memory, working memory and long-term memory. There are also other abilities relating to perceptual information such as mental rotation, spatial visualization ability, verbal fluency and reading comprehension. Other larger perceptual umbrellas include social cognition, empathy, spatial perception and verbal abilities.

Sex differences in memory

Main article: Sex differences in memory

Short term memory

Various researchers have conducted studies to determine the differences between males and females and their abilities within their short-term memory. For example, a study conducted by Lowe, Mayfield, and Reynolds (2003) examined sex differences among children and adolescents on various short-term memory measures. This study included 1,279 children and adolescents, 637 males and 642 females, between the ages of 5 and 19. They found that females scored higher on two verbal subtests: Word Selective Reminding and Object Recall, and males scored higher on the Memory for Location and Abstract Visual Memory subtests, the key spatial memory tasks. In two different studies researchers have found that women perform higher on verbal tasks and men perform higher on spatial tasks (Voyer, Voyer, & Saint-Aubin, 2016). These findings are consistent with studies of intelligence with regards to pattern, females performing higher on certain verbal tasks and males performing higher on certain spatial tasks (Voyer, Voyer, & Saint-Aubin, 2016). Same results have been also found cross culturally. Sex differences in verbal short-term memory have been found regardless of age even among adults, for example a review published in the journal Neuropsychologia which evaluated studies from 1990 to 2013 found greater female verbal memory from ages 11–89 years old.

Working memory

There are usually no sex differences in overall working memory except those involving spatial information such as space and object. A 2004 study published in the journal of Applied Cognitive Psychology found significantly higher male performance on four visuo-spatial working memory. Another 2010 study published in the journal Brain and Cognition found a male advantage in spatial and object working memory on an n-back test but not for verbal working memory. Similarly another study published in the journal Human Brain Mapping found no sex differences in a verbal n-back working memory task among adults from ages 18–58 years old. There was also no sex differences in verbal working memory among a study of university students published in the Journal of Dental and Medical Sciences. However, they still found greater male spatial working memory in studies published in the journals Brain Cognition and Intelligence. Also, even though they found no sex differences in verbal working memory, researchers have found lower brain activity or thermodynamics in the prefrontal cortex of females which suggested greater neural efficiency and less effort for the same performance. Researchers indicate females might have greater working memory on tasks that only relies on the prefrontal cortex. However, in another study of working memory, where the goal was to detect sex differences under high loads of working memory, males outperformed females under high loads of working memory. The authors of the study state: "Results indicated sex effects at high loads across tasks and within each task, such that males had higher accuracy, even among groups that were matched for performance at lower loads". A 2006 review and study on working memory published in the journal European Journal of Cognitive Psychology also found no sex differences in working memory processes except in a double-span task where females outperformed males. There have also been no sex differences found in a popular working memory task known as n-back among a large number of studies.

Long term memory

Studies have found a greater female ability in episodic memory involving verbal or both verbal and visual-spatial tasks while a higher male ability that only involves complex visual-spatial episodic memory. For example, a study published in the journal Neuropsychology found that women perform at a higher level on most verbal episodic tasks and tasks involving some or little visual-spatial episodic memory. Another study published the following year found that women perform at a higher level in verbal and non-verbal (non-spatial visual) episodic memory but men performed at a higher level in complex visual-spatial episodic memory. A review published in the journal Current Directions in Psychological Science by researcher Agneta Herlitz also conclude that higher ability in women on episodic-memory tasks requiring both verbal and visuospatial episodic memory and on face-recognition tasks, while men have higher abilities for episodic memory, where visual-spatial skills of high complexity are required.

Sex differences in semantic memory have also been found with a higher female ability which can be explained by a female advantage in verbal fluency. One other study also found greater female free-recall among the ages 5–17.

In another study, when using multiple tests for episodic memory, there were no differences between men and women. A similar result was also found among children from 3 to 6 years old. As for semantic memory related to general knowledge and knowledge of facts from the world. That is, in most areas of cognition, men show higher results on semantic memory.

Sex differences in executive functions

There has not been enough literature or studies assessing sex difference in executive functioning, especially since executive functions are not a unitary concept. However, in the ones that have been done, there have been differences found in attention and inhibition.

Attention

A 2002 study published in the Journal of Vision found that males were faster at shifting attention from one object to another as well as shifting attention within objects. 2012–2014 studies published in the Journal of Neuropsychology with a sample size ranging from 3500 to 9138 participants by researcher Ruben C Gur found higher female attention accuracy in a neurocognitive battery assessing individuals from ages 8–21. A 2013 study published in the Chinese Medical Journal found no sex differences in executive and alerting of attention networks but faster orientation of attention among females. A 2010 study published in Neuropsychologia also found greater female responsiveness in attention to processing overall sensory stimulation.

Inhibition and self-regulation

A 2008 study published in the journal Psychophysiology found faster reaction time to deviant stimuli in women. The study also analyzed past literature and found higher female performance in withholding social behavior such as aggressive responses and improper sexual arousal. Furthermore, they found evidence that women were better at resisting temptation in tasks, delaying gratification and controlling emotional expressions. They also found lower female effort in response inhibition in equal performance for the same tasks implying an advantage for females in response inhibition based on neural efficiency. In another study published in 2011 in the journal Brain and Cognition, it was found that females outperformed males on the Sustained Attention to Response Task which is a test that measures inhibitory control. Researchers have hypothesized that any female advantage in inhibition or self-regulation may have evolved as a response to greater parenting responsibilities in ancestral settings.

Sex differences in processing speed

Sex differences in processing speed has been largely noted in literature. Studies published in the journal Intelligence have found faster processing speed in women. For example, a 2006 study published in Intelligence by researcher Stephen Camarata and Richard Woodcock found faster processing speed in females across all age groups in a sample of 4,213 participants. This was followed by another study published in 2008 by researchers Timothy Z Keith and Matthew R. Reynolds who found faster processing speed in females from ages 6 to 89 years old. The sample also had a number of 8,818 participants. Other studies by Keith have also found faster processing speed in females from ages 5 to 17. In one recent study, groups of men and women were tested using the WAIS-IV and WAIS-R tests. According to the research results, there were no differences in processing speed between men and women.

Sex differences in semantic perception

Studies of sex differences in semantic perception (attribution of meaning) of words reported that males conceptualize items in terms of physical or observable attributes whereas females use more evaluative concepts. Another study of young adults in three cultures showed significant sex differences in semantic perception (attribution of meaning) of most common and abstract words. Contrary to common beliefs, women gave more negative scores to the concepts describing sensational objects, social and physical attractors but more positive estimations to work- and reality-related words, in comparison to men  This suggests that men favour concepts related to extreme experience and women favour concepts related to predictable and controllable routines. In a light of the higher rates of sensation seeking and deviancy in males, in comparison to females, these sex differences in meaning attribution were interpreted as support for the evolutionary theory of sex.

Sex differences in spatial abilities

Rubik's cube puzzle involving mental rotation

Sex differences in spatial abilities are widely established in literature. Males have much higher level of performance in three major spatial tasks which include spatial visualization, spatial perception and mental rotation. Spatial visualization elicits the smallest difference with a deviation of 0.13, perception a deviation of 0.44 and mental rotation the largest with a deviation of 0.73. Another 2013 meta-analysis published in the journal Educational Review found greater male mental rotation in a deviation of 0.57 which only grew larger as time limits were added. These male advantages manifests themselves in math and mechanical tasks for example significantly higher male performance on tests of geometry, measurement, probability, statistics and especially mechanical reasoning. It also manifests and largely mediates higher male performance in arithmetic and computational fluency All of these math and technical fields involve spatial abilities such as rotation and manipulation of imagined space, symbols and objects. Mental rotation has also been linked to higher success in fields of engineering, physics and chemistry regardless of gender. Spatial visualization on the other hand also correlate with higher math achievement in a range of 0.30 to 0.60. Furthermore, male advantage in spatial abilities can be accounted for by their greater ability in spatial working memory. Sex differences in mental rotation also reaches almost a single deviation (1.0) when the tasks require navigation, as found in one study with participants who used Oculus Rift in a virtual environment. A 2009 study using data from the BBC of over 200,000 people in 53 nations showed that in all nations examined, men outperformed women in both mental rotation and in angle judgment, and that these differences increased with measures of gender equality. A 2019 meta-analysis of the literature from 1988 to 2018 likewise found the same results at both the behavioral and neural levels, though the effect sizes were larger for large-scale spatial ability than small-scale spatial ability.

Even though most spatial abilities are higher in men, object location memory or the ability to memorize spatial cues involving categorical relations are higher in women. But it depends on the type of stimulus (object) and the task. In some conditions, men's productivity is higher (for example, when "male" objects are used), in other conditions, women's productivity may be higher or there are no differences between the sexes. Higher female ability in visual recognition of objects and shapes have also been found.

Sex differences in verbal abilities

Like spatial ability, sex differences in verbal abilities have been widely established in literature. There is a clear higher female performance on a number of verbal tasks prominently a higher level of performance in speech production which reaches a deviation of 0.33 and also a higher performance in writing. Studies have also found greater female performance in phonological processing, identifying alphabetical sequences, and word fluency tasks. Studies have also found that females outperform males in verbal learning especially on tests such as Rey Auditory Verbal Learning Test and Verbal Paired Associates. It has also been found that the hormone estrogen increases ability of speech production and phonological processing in women, which could be tied to their advantages in these areas. Overall better female performance have also been found in verbal fluency which include a trivial advantage in reading comprehension while a significantly higher performance in speech production and essay writing. This manifests in higher female international PISA scores in reading and higher female Grade 12 scores in national reading, writing and study skills. Researchers Joseph M. Andreano and Larry Cahill have also found that the female verbal advantage extends into numerous tasks, including tests of spatial and autobiographical abilities.

In a fairly large meta-analysis that analyzed 165 different studies, a very small difference of 0.11 standard deviations was found. The authors of this study postulate: "The difference is so small that we argue that gender differences in verbal ability no longer exist."

A recent meta-analysis of 168 studies, 496 effect sizes, and 355,173 participants found a small but robust female advantage in both verbal fluency and verbal episodic memory.

Sex differences in social cognition

Main article: Sex differences in emotional intelligence

Current literature suggests women have higher level of social cognition. A 2012 review published in the journal Neuropsychologia found that women are better at recognizing facial effects, expression processing and emotions in general. Men were only better at recognizing specific behaviour which includes anger, aggression and threatening cues. A 2012 study published in the journal Neuropsychology with a sample of 3,500 individuals from ages 8–21, found that females outperformed males on face memory and all social cognition tests. In 2014, another study published in the journal Cerebral Cortex found that females had larger activity in the right temporal cortex, an essential core of the social brain connected to perception and understanding the social behaviour of others such as intentions, emotions, and expectations. In 2014, a meta-analysis of 215 study sample by researcher A.E. Johnson and D Voyeur in the journal Cognition and Emotion found overall female advantage in emotional recognition. Other studies have also indicated greater female superiority to discriminate vocal and facial expression regardless of valence, and also being able to accurately process emotional speech. Studies have also found males to be slower in making social judgments than females. Structural studies with MRI neuroimaging has also shown that women have bigger regional grey matter volumes in a number of regions related to social information processing including the Inferior frontal cortex and bigger cortical folding in the Inferior frontal cortex and parietal cortex  Researchers suppose that these sex differences in social cognition predisposes males to high rates of autism spectrum disorders which is characterized by lower social cognition.

A recent study that aimed to identify gender differences in social cognition did not show significant differences, with few exceptions. The study authors state: "The presence of sex differences in social cognition is controversial". Results showed no significant sex differences in affective and cognitive ToM, in the recognition of emotional facial expressions, or in the ability to identify and regulate one's own emotions.

Empathy

Main article: Sex differences in emotional intelligence § Empathy

Empathy is a large part of social cognition and facilitates its cognitive components known as theory of mind. Current literature suggests a higher level of empathy in women compared to men.

A 2014 analysis from the journal of Neuroscience & Biobehavioral Reviews reported that there is evidence that "sex differences in empathy have phylogenetic and ontogenetic roots in biology and are not merely cultural byproducts driven by socialization." Other research has found no differences in empathy between women and men, and suggest that perceived gender differences are the result of motivational differences.