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)
and here are three figures showing a surface of constant x, a surface of constant x, and a surface of constant z.
Finally here is a figure showing the volume element dV in cartesian coordinates.
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
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.
Finally here is a figure showing the volume element dV in cylindrical coordinates.
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)
and here are two more figures giving the side and top views of the previous figure.
The cartesian and spherical coordinates are related by
x=ρsinφcosθy=ρsinφsinθz=ρcosφρ=√x2+y2+z2θ=arctanyxφ=arctan√x2+y2z
Here are three figures showing a surface of constant ρ, a surface of constant θ, and a surface of constant φ.
Finally, here is a figure showing the volume element dV in spherical coordinates
and two extracts of the above figure to make it easier to see how the factors ρ dφ and ρsinφ dθ arise.