NationMaster - Encyclopedia: 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. See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ... Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

1 Terminology
- 1.1 Example
2 Line, surface, and volume integrals
- 2.1 Line integrals
- 2.2 Surface integrals
3 Div, curl, grad
- 3.1 Primary and secondary sources
4 See also

Terminology

In R³, for example, if we have some transformation

$H(x_1', x_2', x_3')=(x_1, x_2, x_3)=mathbf{x}$

giving curvilinear coordinates x₁′, x₂′,x₃′, for x₁, x₂, x₃, if this transformation is locally invertible everywhere, the Jacobian determinant In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, 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

${ partial mathbf{x} over partial x_i' }$

must form a basis for R³.

From these basis vectors, we define scale factors

$h_{x_i'}=h_i=left|{partial mathbf{x} over partial{x_i'}}right|$

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

$mathbf{e}_{x_i'}=1/h_i {partial mathbf{x} over partial{x_i'}}$

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 â†” â‡” â‰¡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

$mathbf{e}_{x_i'}cdotmathbf{e}_{x_j'} = delta_{ij}$

where δ_ij is the Kronecker delta. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

Example

If we consider polar coordinates for R², note that

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

(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.

Line, surface, and volume 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. Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ...

Line integrals

Normally in the calculation of line integrals we are interested in calculating This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...

$int_C f ,ds = int_a^b f(mathbf{x}(t))left|{partial mathbf{x} over partial t}right|; dt$

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

$left|{partial mathbf{x} over partial t}right| = left|{partial mathbf{x} over partial x_i'}{partial x_i' over partial t}right|$

by the chain rule. But from the definition of the curvilinear coordinates, In calculus, the chain rule is a formula for the derivative of the composition of two functions. ...

${partial mathbf{x} over partial x_i'} = h_i mathbf{e}_{x_i'}$

and thus

$left|{partial mathbf{x} over partial t}right| = sqrt{sum h_i mathbf{e}_{x_i'} {partial x_i' over partial t}}$

and we can proceed normally.

Surface integrals

Likewise, if we are interested in a surface integral, the relevant calculation, with the parametrisation of the surface in Cartesian coordinates is: In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ...

$int_S f ,ds = iint_T f(mathbf{x}(s, t)) left|{partial mathbf{x} over partial s}times {partial mathbf{x} over partial t}right| ds dt$

Again, in curvilinear coordinates, the term

$left|{partial mathbf{x} over partial s}times {partial mathbf{x} over partial t}right| = left|{partial mathbf{x} over partial x_i'}{partial x_i' over partial s} times {partial mathbf{x} over partial x_i'}{partial x_i' over partial t}right|$

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

${partial mathbf{x} over partial x_i'}{partial x_i' over partial s} = sum {partial x_i' over partial s} h_{x_i'} mathbf{e}_{x_i'}$

and

${partial mathbf{x} over partial x_i'}{partial x_i' over partial t} = sum {partial x_i' over partial t} h_{x_i'} mathbf{e}_{x_i'}$

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

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

Div, curl, grad

In orthogonal curvilinear coordinates, we can express the divergence, curl and gradient of a function or vector field are as follows: In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

$nabla f = sum {1 over h_i} mathbf{e}_{x_i'} {partial f over partial {x_i'}}$

$nablatimes F = {1 over {Pi h_i}} begin{pmatrix} h_1 partial_1 vdots h_n partial_n end{pmatrix}times F$

$nablacdot F = sum {1 over Pi} left ({{Pi over h_i} cdot {partial F_{x_i'} over partial {x_i'}}}right)$

Where $Π$ is the product of all $h i$

Primary and secondary sources

Arfken, George (1995). Mathematical Methods for Physicists, Academic Press.

Curvilinear coordinates - definition of Curvilinear coordinates in Encyclopedia (316 words)
	Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system, such that the transformation is locally invertible at each point.
	(r, θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation (r,θ) → (r cos θ, r sin θ) is r.
	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.

Coordinates (mathematics) - Wikipedia, the free encyclopedia (1186 words)
	The coordinates of a point are the components of a tuple of numbers used to represent the location of the point in the plane or space.
	The polar coordinate systems are coordinate systems in which a point is identified by a distance from some fixed feature in space and one or more subtended angles.
	In terms of the Cartesian coordinate system, one usually picks O to be the origin (0,0) and L to be the positive x-axis (the right half of the x-axis).

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COMMENTARY

dinesh dubey (mp, india)
22nd January 2009

hello,

there is some problem to understand the derivation of transformation of parabolic cylindrical coordinate. please solve my problem.
thank you

e mail: dinesh_dubey2004@yahoo.com

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