Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system. We need the same number of coordinates. If we consider the 2D case, then instead of Cartesian coordinates x and y we use e.g. p and q; the level curves of p and q in the xy-plane, as well as those of x and y in the pq-plane are in general curved. Required is that the transformation is locally invertible at each point. This means that we can convert a point given in one coordinate system to its curvilinear coordinates and back. Depending on the application, a curvilinear coordinate system may be simpler than the Cartesian coordinate system. This also has consequences that we can express many of the concepts in vector calculus which are given in Cartesian or spherical coordinates or any other arbitrary coordinate system, also in curvilinear coordinates.

Terminology

---

In R³, for example, we have some transformation

$\backslash mathbf\{\}\; x\_i=x\_i(x\_1\text{'},\; x\_2\text{'},\; x\_3\text{'});\; i=1,2,3$

giving curvilinear coordinates x₁′, x₂′,x₃′, for x₁, x₂, x₃. If this transformation is locally invertible everywhere, the Jacobian determinant

$\backslash partial(x\_1,\; x\_2,\; x\_3)\; \backslash over\; \backslash partial(x\_1\text{'},\; x\_2\text{'},\; x\_3\text{'})$

is nonzero, and for this to happen, the vectors

$\{\; \backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; x\_i\text{'}\; \}$

must form a basis for R³.

From these basis vectors, we define scale factors or Lamé coefficients, named after Gabriel Lamé,

$h\_\{x\_i\text{'}\}=h\_i=\backslash left|\{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\{x\_i\text{'}\}\}\; \backslash right|\; =\backslash sqrt\{\backslash sum\_\{k=0\}^3\; \{\; (\backslash frac\{\backslash partial\{x\_k\}\}\{\backslash partial\{x\_i\text{'}\}\})^2\}\}$

and thus arrive at the unit basis vectors for the curvilinear coordinates to be

$\backslash mathbf\{e\}\_\{x\_i\text{'}\}=1/h\_i\; \{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\{x\_i\text{'}\}\}$

Note that the coordinate system we choose need not be orthogonal, but for the purposes of this article, they are treated as being so. The system is orthogonal iff

$\backslash mathbf\{e\}\_\{x\_i\text{'}\}\backslash cdot\backslash mathbf\{e\}\_\{x\_j\text{'}\}\; =\; \backslash delta\_\{ij\}$

where δ_ij is the Kronecker delta.

Cartesian coordinates $x\_1,x\_2,x\_3$ which have the scalar product, are called Euclidean coordinates. It is often convenient to associate the points of Euclidean space with vectors, for example, with each point P we associate the vector (or arrow) with its tail at the origin of coordinates and its tip at P. This vector is called the radius vector with components $(x\_1,\; x\_2,\; x\_3)$. At any point P of Euclidean space we can construct the small line element

$d\; x\; =\; (dx\_1,dx\_2,dx\_3)$

which is vector too. Two vectors $h\; =\; (x\_1,\; x\_2,\; x\_3)$ and $f=(y\_l,\; y\_2,\; y\_3)$ from the same origin can be added and result is the vector with coordinates $(x\_l+\; y\_l,\; x\_2\; +\; y\_2,\; x\_3\; +\; y\_3)$. A vector can also be multiplied by any real number. The Euclidean scalar product of two (real) vectors is the number

.

The scalar product of the vector with itself give the square of the vector length. The square of the length of a line element in space with scalar product is called the metric of the space. The metric of Euclidean space is

.

The same Euclidean metric in curvilinear coordinates is

.

The symmetric tensor

$g\_\{i,j\}(x\_i\text{'},x\_j\text{'})=\; \backslash sum\_\{k=1\}^3\; \backslash frac\{\backslash partial\{x\_k\}\}\{\backslash partial\{x\_i\text{'}\}\}\; \backslash frac\{\backslash partial\{x\_k\}\}\{\backslash partial\{x\_j\text{'}\}\}$

are called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates. Connection between fundamental tensor and Lamé coefficients is $g\_\{i,i\}(x\_i\text{'},x\_j\text{'})=\; h\_i^2$.

Example

If we consider polar coordinates for R², note that

$(x,\; y)=(r\; \backslash mathrm\{cos\}\backslash theta,\; r\; \backslash mathrm\{sin\}\; \backslash theta)\; \backslash ,\backslash !$

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

The basis vectors are b_r = (cos θ, sin θ), b_θ = (-r sin θ, r cos θ), with unit basis vectors e_r = (cos θ, sin θ), e_θ = (- sin θ, cos θ) with scale factors h_r = 1 and h_θ= r. The fundamental tensor is g_1,1 =1, g_2,2 =r², g_1,2 = g_2,1 =0.

Line and surface integrals

---

Since we use curvilinear coordinates to aid in the calculation in vector calculus, there are adjustments we need to make in the calculation of line, surface and volume integrals.

Line integrals

Normally in the calculation of line integrals we are interested in calculating

$\backslash int\_C\; f\; \backslash ,ds\; =\; \backslash int\_a^b\; f(\backslash mathbf\{x\}(t))\backslash left|\{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; t\}\backslash right|\backslash ;\; dt$

where x(t) parametrizes C in Cartesian coordinates. In curvilinear coordinates, the term

$\backslash left|\{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; t\}\backslash right|\; =\; \backslash left|\; \backslash sum\; \{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; x\_i\text{'}\}\{\backslash partial\; x\_i\text{'}\; \backslash over\; \backslash partial\; t\}\backslash right|$

by the chain rule. But from the definition of the curvilinear coordinates,

$\{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; x\_i\text{'}\}\; =\; h\_i\; \backslash mathbf\{e\}\_\{x\_i\text{'}\}$

and thus

$\backslash left|\{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; t\}\backslash right|\; =\; \backslash sqrt\{\backslash sum\; h\_i\; \backslash mathbf\{e\}\_\{x\_i\text{'}\}\; \{\backslash partial\; x\_i\text{'}\; \backslash over\; \backslash partial\; t\}\}$

and we can proceed normally.

Surface integrals

Likewise, if we are interested in a surface integral, the relevant calculation, with the parameterization of the surface in Cartesian coordinates is:

$\backslash int\_S\; f\; \backslash ,ds\; =\; \backslash iint\_T\; f(\backslash mathbf\{x\}(s,\; t))\; \backslash left|\{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; s\}\backslash times\; \{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; t\}\backslash right|\; ds\; dt$

Again, in curvilinear coordinates, the term

$\backslash left|\{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; s\}\backslash times\; \{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; t\}\backslash right|\; =\; \backslash left|\{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; x\_i\text{'}\}\{\backslash partial\; x\_i\text{'}\; \backslash over\; \backslash partial\; s\}\; \backslash times\; \{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; x\_i\text{'}\}\{\backslash partial\; x\_i\text{'}\; \backslash over\; \backslash partial\; t\}\backslash right|$

and we make use of the definition of curvilinear coordinates again to yield

$\{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; x\_i\text{'}\}\{\backslash partial\; x\_i\text{'}\; \backslash over\; \backslash partial\; s\}\; =\; \backslash sum\; \{\backslash partial\; x\_i\text{'}\; \backslash over\; \backslash partial\; s\}\; h\_\{x\_i\text{'}\}\; \backslash mathbf\{e\}\_\{x\_i\text{'}\}$

and

$\{\backslash partial\; \backslash mathbf\{x\}\; \backslash over\; \backslash partial\; x\_i\text{'}\}\{\backslash partial\; x\_i\text{'}\; \backslash over\; \backslash partial\; t\}\; =\; \backslash sum\; \{\backslash partial\; x\_i\text{'}\; \backslash over\; \backslash partial\; t\}\; h\_\{x\_i\text{'}\}\; \backslash mathbf\{e\}\_\{x\_i\text{'}\}$

where the cross product, in terms of curvilinear coordinates, will be:

$\backslash begin\{vmatrix\}$

\mathbf{e}_{x_1'} & \mathbf{e}_{x_2'} & \mathbf{e}_{x_3'} \\ && \\ h_1 {\partial x_1' \over \partial s} & h_2 {\partial x_2' \over \partial s} & h_3 {\partial x_3' \over \partial s} \\ && \\ h_1 {\partial x_1' \over \partial t} & h_2 {\partial x_2' \over \partial t} & h_3 {\partial x_3' \over \partial t} \end{vmatrix}

Grad, curl, div

---

In orthogonal curvilinear coordinates, one can express the gradient, curl and divergence of a function or vector field as follows:

$\backslash nabla\; f\; =\; \backslash sum\; \{1\; \backslash over\; h\_i\}\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; \{x\_i\}\}\; \backslash hat\; e\_\{x\_i\}$

$\backslash nabla\backslash times\; \{\backslash vec\; v\}\; =\; \{1\; \backslash over\; \{\backslash Pi\; h\_i\}\}\; \backslash begin\{pmatrix\}\; h\_1\; \backslash partial\_1\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; h\_n\; \backslash partial\_n\; \backslash end\{pmatrix\}\backslash times\; \{\backslash vec\; v\}$

$\backslash nabla\backslash cdot\; \{\backslash vec\; v\}\; =\; \backslash sum\; \{1\; \backslash over\; \backslash Pi\}\; \{\backslash partial\; \backslash over\; \{\backslash partial\_\{x\_i\}\}\}\; \backslash left\; (\{\backslash Pi\; \backslash cdot\; v\_i\; \backslash over\; h\_i\}\; \backslash right\; ),$

where $\backslash Pi$ is the product of all $h\_i.$

Reference

---

See also

---

• Covariance and contravariance

Coordinate systems