Cartesian Coordinates

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CartesianCoordinatesAxes

Cartesian coordinates are rectilinear two- or three-dimensional coordinates (and therefore a special case of curvilinear coordinates) which are also called rectangular coordinates. The two axes of two-dimensional Cartesian coordinates, conventionally denoted the x- and y-axes (a notation due to Descartes), are chosen to be linear and mutually perpendicular. Typically, the x-axis is thought of as the "left and right" or horizontal axis while the y-axis is thought of as the "up and down" or vertical axis. In two dimensions, the coordinates x and y may lie anywhere in the interval (-infty,infty), and an ordered pair (x,y) in two-dimensional Cartesian coordinates is often called a point or a 2-vector.

The three-dimensional Cartesian coordinate system is a natural extension of the two-dimensional version formed by the addition of a third "in and out" axis mutually perpendicular to the x- and y-axes defined above. This new axis is conventionally referred to as the z-axis and the coordinate z may lie anywhere in the interval (-infty,infty). An ordered triple (x,y,z) in three-dimensional Cartesian coordinates is often called a point or a 3-vector.

CartesianCoordinatesEllipse

In René Descartes' original treatise (1637), which introduced the use of coordinates for describing plane curves, the axes were omitted, and only positive values of the x- and the y-coordinates were considered, since they were defined as distances between points. For an ellipse this meant that, instead of the full picture which we would plot nowadays (left figure), Descartes drew only the upper half (right figure).

The inversion of three-dimensional Cartesian coordinates is called 6-sphere coordinates.

The scale factors of Cartesian coordinates are all unity, h_i=1. The line element is given by

 ds=dxx^^+dyy^^+dzz^^,
(1)

and the volume element by

 dV=dxdydz.
(2)

The gradient has a particularly simple form,

 del =x^^partial/(partialx)+y^^partial/(partialy)+z^^partial/(partialz),
(3)

as does the Laplacian

 del ^2=(partial^2)/(partialx^2)+(partial^2)/(partialy^2)+(partial^2)/(partialz^2).
(4)

The vector Laplacian is

del ^2F=(partial^2F)/(partialx^2)+(partial^2F)/(partialy^2)+(partial^2F)/(partialz^2)
(5)
=x^^((partial^2F_x)/(partialx^2)+(partial^2F_x)/(partialy^2)+(partial^2F_x)/(partialz^2))+y^^((partial^2F_y)/(partialx^2)+(partial^2F_y)/(partialy^2)+(partial^2F_y)/(partialz^2))+z^^((partial^2F_z)/(partialx^2)+(partial^2F_z)/(partialy^2)+(partial^2F_z)/(partialz^2)).
(6)

The divergence is

 del ·F=(partialF_x)/(partialx)+(partialF_y)/(partialy)+(partialF_z)/(partialz),
(7)

and the curl is

del xF=|x^^ y^^ z^^; partial/(partialx) partial/(partialy) partial/(partialz); F_x F_y F_z|
(8)
=((partialF_z)/(partialy)-(partialF_y)/(partialz))x^^+((partialF_x)/(partialz)-(partialF_z)/(partialx))y^^+((partialF_y)/(partialx)-(partialF_x)/(partialy))z^^.
(9)

The gradient of the divergence is

del (del ·u)=[partial/(partialx)((partialu_x)/(partialx)+(partialu_y)/(partialy)+(partialu_z)/(partialz)); partial/(partialy)((partialu_x)/(partialx)+(partialu_y)/(partialy)+(partialu_z)/(partialz)); partial/(partialz)((partialu_x)/(partialx)+(partialu_y)/(partialy)+(partialu_z)/(partialz))]
(10)
=[partial/(partialx); partial/(partialy); partial/(partialz)]((partialu_x)/(partialx)+(partialu_y)/(partialy)+(partialu_z)/(partialz)).
(11)

Laplace's equation is separable in Cartesian coordinates.

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