Cartesian Coordinates

$\begin{figure}\begin{center}\BoxedEPSF{Axes.epsf}\end{center}\end{figure}$

Cartesian coordinates are rectilinear 2-D or 3-D coordinates (and therefore a special case of Curvilinear Coordinates) which are also called Rectangular Coordinates. The three axes of 3-D Cartesian coordinates, conventionally denoted the x-Axis-, y-Axis-, and z-Axis (a Notation due to Descartes ) are chosen to be linear and mutually Perpendicular. In 3-D, the coordinates , , and may lie anywhere in the Interval $(-\infty,\infty)$ .

The Scale Factors of Cartesian coordinates are all unity, . The Line Element is given by

$\begin{displaymath} d{\bf s} = dx\, \hat {\bf x} + dy \,\hat {\bf y} + dz\, \hat {\bf z}, \end{displaymath}$

(1)

and the Volume Element by

$\begin{displaymath} dV = dx\,dy\,dz. \end{displaymath}$

(2)

The Gradient has a particularly simple form,

$\begin{displaymath} \nabla \equiv \hat{\bf x}{\partial\over\partial x} + \hat{\b... ...rtial\over \partial y} + \hat{\bf z}{\partial\over\partial z}, \end{displaymath}$

(3)

as does the Laplacian

$\begin{displaymath} \nabla^2 \equiv {\partial^2\over\partial x^2}+{\partial^2\over\partial y^2}+{\partial^2\over\partial z^2}. \end{displaymath}$

(4)

The Laplacian is

$\displaystyle \nabla^2{\bf F}$	$\textstyle \equiv$	$\displaystyle \nabla \cdot (\nabla {\bf F}) = {\partial^2 {\bf F}\over\partial ... ... + {\partial^2{\bf F}\over\partial y^2} + {\partial^2 {\bf F}\over\partial z^2}$
	$\textstyle =$	$\displaystyle \hat {\bf x} \left({{\partial^2 F_x\over\partial x^2} + {\partial^2 F_x\over\partial y^2} + {\partial^2 F_x\over\partial z^2}}\right)$
	$\textstyle \phantom{=}$	$\displaystyle + \hat {\bf y} \left({{\partial^2 F_y\over\partial x^2} + {\partial^2 F_y\over\partial y^2} + {\partial^2 F_y\over\partial z^2}}\right)$
	$\textstyle \phantom{=}$	$\displaystyle + \hat {\bf z} \left({{\partial^2 F_z\over\partial x^2} + {\partial^2 F_z\over\partial y^2} + {\partial^2 F_z\over\partial z^2}}\right).$	(5)

The Divergence is

$\begin{displaymath} \nabla \cdot {\bf F} = {\partial F_x\over\partial x} + {\partial F_y\over\partial y} + {\partial F_z\over\partial z}, \end{displaymath}$

(6)

and the Curl is

$\begin{displaymath} \nabla \times {\bf F} \equiv \left\vert\matrix{\hat {\bf x} ... ...artial x} - {\partial F_x\over\partial y}}\right)\hat {\bf z}. \end{displaymath}$

(7)

The Gradient of the Divergence is

$\displaystyle \nabla(\nabla\cdot{\bf u})$	$\textstyle =$	$\displaystyle \left[\begin{array}{c} {\partial\over\partial x}\left({{\partial ... ...ial u_y\over\partial y}+{\partial u_z\over\partial z}}\right)\end{array}\right]$
	$\textstyle =$	$\displaystyle \left[\begin{array}{c}{\partial\over\partial x}\\ {\partial\over... ...partial x}+{\partial u_y\over\partial y}+{\partial u_z\over\partial z}}\right).$	(8)

Laplace's Equation is separable in Cartesian coordinates.

References

Arfken, G. ``Special Coordinate Systems--Rectangular Cartesian Coordinates.'' §2.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 94-95, 1985.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 656, 1953.