Curvilinear coordinates

Encyclopedia : C : CU : CUR : Curvilinear coordinates

---

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

[ \mathbf x_i=x_i(x_1', x_2', x_3'); i=1,2,3 ]

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

[ \partial(x_1, x_2, x_3) \over \partial(x_1', x_2', x_3')]

is nonzero, and for this to happen, the vectors

[ \over \partial x_i' }]

must form a basis for R³.

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

[ h_=h_i=\left| \over \partial} \right| =\sqrt^3 }})^2}} ]

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

[ \mathbf_=1/h_i \over \partial}]

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

[ \mathbf_\cdot\mathbf_ = \delta_]

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

[ =\sum_ x_ y_ ].

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

[ = dx_1^2+dx_2^2+dx_3^2 ].

The same Euclidean metric in curvilinear coordinates is

[ = \sum_^3 \frac}} \frac}} dx_i' dx_j' ].

The symmetric tensor

[ g_(x_i',x_j')= \sum_^3 \frac}} \frac}}]

are called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates. Connection between fundamental tensor and Lamé coefficients is [ g_(x_i',x_j')= h_i^2].

Example

If we consider polar coordinates for R², note that

[ (x, y)=(r \mathrm\theta, r \mathrm \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 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

[ \int_C f \,ds = \int_a^b f(\mathbf(t))\left| \over \partial t}\right|\; dt]

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

[ \left| \over \partial t}\right| = \left| \sum \over \partial x_i'}\right|]

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

[ \over \partial x_i'} = h_i \mathbf_ ]

and thus

[ \left| \over \partial t}\right| = \sqrt_ }]

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:

[\int_S f \,ds = \iint_T f(\mathbf(s, t)) \left| \over \partial s}\times \over \partial t}\right| ds dt]

Again, in curvilinear coordinates, the term

[\left| \over \partial s}\times \over \partial t}\right| = \left| \over \partial x_i'} \times \over \partial x_i'}\right|]

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

[ \over \partial x_i'} = \sum h_ \mathbf_]

and

[ \over \partial x_i'} = \sum h_ \mathbf_]

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

[\begin \mathbf_ & \mathbf_ & \mathbf_ \\ && \\h_1 & h_2 & h_3 \\&& \\h_1 & h_2 & h_3 \end]

Grad, curl, div

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

[ \nabla f = \sum } \hat e_ ]

[ \nabla\times = } \begin h_1 \partial_1 \\ \vdots \\ h_n \partial_n \end\times ]

[ \nabla\cdot = \sum }} \left ( \right ),]

where [\Pi] is the product of all [h_i.]

Reference

See also

• Covariance and contravariance

 

---

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.

Search Titles

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

?