3-D Coordinate Systems

 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 d⁢𝑉 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 𝑥⁢𝑦-plane

The cartesian and cylindrical coordinates are related by

𝑥=𝑟⁢cos⁡𝜃𝑦=𝑟⁢sin⁡𝜃𝑧=𝑧𝑟=√𝑥2+𝑦2𝜃=arctan⁡𝑦𝑥𝑧=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 d⁢𝑉 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=𝜌⁢sin⁡𝜑⁢cos⁡𝜃𝑦=𝜌⁢sin⁡𝜑⁢sin⁡𝜃𝑧=𝜌⁢cos⁡𝜑𝜌=√𝑥2+𝑦2+𝑧2𝜃=arctan⁡𝑦𝑥𝜑=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 d⁢𝑉 in spherical coordinates

and two extracts of the above figure to make it easier to see how the factors 𝜌 d⁢𝜑 and 𝜌⁢sin⁡𝜑 d⁢𝜃 arise.