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Cartesian Coordinates


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 $x$, $y$, and $z$ may lie anywhere in the Interval $(-\infty,\infty)$.

The Scale Factors of Cartesian coordinates are all unity, $h_i = 1$. The Line Element is given by

d{\bf s} = dx\, \hat {\bf x} + dy \,\hat {\bf y} + dz\, \hat {\bf z},
\end{displaymath} (1)

and the Volume Element by
dV = dx\,dy\,dz.
\end{displaymath} (2)

The Gradient has a particularly simple form,
\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
\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
\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

\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.

See also Coordinates, Helmholtz Differential Equation--Cartesian Coordinates


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.

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© 1996-9 Eric W. Weisstein