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Mathematics LibreTexts

3-D Coordinate Systems

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 Cartesian Coordinates

Here is a figure showing the definitions of the three Cartesian coordinates (x,y,z)

cart1.svg

and here are three figures showing a surface of constant x, a surface of constant x, and a surface of constant z.

cart3.svg        cart4.svg        cart2.svg

Finally here is a figure showing the volume element dV in cartesian coordinates.

cart5.svg

Cylindrical Coordinates

Here is a figure showing the definitions of the three cylindrical coordinates

r= distance from (0,0,0) to (x,y,0)θ= angle between the the x axis and the line joining (x,y,0) to (0,0,0)z= signed distance from (x,y,z) to the xy-plane

cyl1 (1).svg

The cartesian and cylindrical coordinates are related by

x=rcosθy=rsinθz=zr=x2+y2θ=arctanyxz=z

Here are three figures showing a surface of constant r, a surface of constant θ, and a surface of constant z.

cyl3.svg        cyl4.svg        cyl2.svg

Finally here is a figure showing the volume element dV in cylindrical coordinates.

cyl5.svg

 Spherical Coordinates

Here is a figure showing the definitions of the three spherical coordinates

ρ= distance from (0,0,0) to (x,y,z)φ= angle between the z axis and the line joining (x,y,z) to (0,0,0)θ= angle between the x axis and the line joining (x,y,0) to (0,0,0)

spherical.svg

and here are two more figures giving the side and top views of the previous figure.

sphericalSide.svg            sphericalTop.svg

The cartesian and spherical coordinates are related by

x=ρsinφcosθy=ρsinφsinθz=ρcosφρ=x2+y2+z2θ=arctanyxφ=arctanx2+y2z

Here are three figures showing a surface of constant ρ, a surface of constant θ, and a surface of constant φ.

spher2.svg            spher3.svg            spher4.svg

Finally, here is a figure showing the volume element dV in spherical coordinates

spher5.svg

and two extracts of the above figure to make it easier to see how the factors ρ dφ and ρsinφ dθ arise.

spher6.svg                    spher7.svg


This page titled 3-D Coordinate Systems is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform.

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