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The change-of-variables formula with 3 (or more) variables is just like the formula for two variables. If we do a change-of-variables Φ from coordinates (u,v,w) to coordinates (x,y,z), then the Jacobian is the determinant (x,y,z)(u,v,w) = |xuxvxwyuyvywzuzvzw|, and the volume element is dV = dxdydz = |(x,y,z)(u,v,w)|dudvdw.


After rectangular (aka Cartesian) coordinates, the two most common an useful coordinate systems in 3 dimensions are cylindrical coordinates (sometimes called cylindrical polar coordinates) and spherical coordinates (sometimes called spherical polar coordinates).



Cylindrical Coordinates: When there's symmetry about an axis, it's convenient to take the z-axis as the axis of symmetry and use polar coordinates (r,θ) in the xy-plane to measure rotation around the z-axis. Check the interactive figure to the right. A point P is specified by coordinates (r,θ,z) where z is the height of P above the xy-plane.

(i) What happens to P as z changes?

(ii) What's the relation between r, P and the axis of symmetry?

(iii) What are the natural restrictions on θ?

(iv) The relation between Cartesian coordinates (x,y,z) and Cylindrical coordinates (r,θ,z) for each point P in 3-space is x = rcosθ,y = rsinθ,z = z.

Problem: Find the Jacobian of the transformation (r,θ,z)(x,y,z) of cylindrical coordinates.

Solution: This calculation is almost identical to finding the Jacobian for polar coordinates. Our partial derivatives are: xr=cos(θ),xθ=rsin(θ),xz=0,yr=sin(θ),yθ=rcos(θ),yz=0,zr=0,zθ=0,zz=1.
Our Jacobian is then the 3×3 determinant (x,y,z)(r,θ,z) = |cos(θ)rsin(θ)0sin(θ)rcos(θ)0001| = r, and our volume element is dV=dxdydz=rdrdθdz.

Spherical Coordinates: A sphere is symmetric in all directions about its center, so it's convenient to take the center of the sphere as the origin. Then we let ρ be the distance from the origin to P and ϕ the angle this line from the origin to P makes with the z-axis. Finally, as before, we use θ from polar coordinates in the xy-plane to measure rotation around the z-axis. Investigate the interactive figure to the right. A point P is specified by 3 coordinates (ρ,θ,ϕ).
[Warning: Most physics texts swap the roles of θ and ϕ.]
(i) The relation between Cartesian coordinates (x,y,z) and Spherical Polar coordinates (ρ,θ,ϕ) for each point P in 3-space is  x = ρcosθsinϕ,y = ρsinθsinϕ,z = ρcosϕ. (ii) The natural restrictions on ρ,θ, and ϕ are 0ρ<,0θ<2π,0ϕπ. (iii) Points on the earth are frequently specified by Latitude and Longitude. How do these relate to θ and ϕ?

Problem: Find the Jacobian of the transformation (ρ,θ,ϕ)(x,y,z) of spherical coordinates.

Solution: Now our partial derivatives are: xρ=cos(θ)sin(ϕ),xθ=ρsin(θ)sin(ϕ),xϕ=ρcos(θ)cos(ϕ),yρ=sin(θ)sin(ϕ),yθ=ρcos(θ)sin(ϕ),yϕ=ρsin(θ)cos(ϕ),zρ=cos(ϕ),zθ=0,zϕ=ρsin(ϕ).
Our Jacobian (x,y,z)(ρ,θ,ϕ) is then the 3×3 determinant |cos(θ)sin(ϕ)ρsin(θ)sin(ϕ)ρcos(θ)cos(ϕ)sin(θ)sin(ϕ)ρcos(θ)sin(ϕ)ρsin(θ)cos(ϕ)cos(ϕ)0ρsin(ϕ).| which works out to ρ2sin(ϕ), and our volume element is dV=dxdydz=ρ2sin(ϕ)dρdθdϕ.


Problem: Compute the volume of the ball ρR or radius R.

Solution: If B is the unit ball, then its volume is . We convert to spherical coordinates to get
\begin{eqnarray*} \hbox{Vol}(B) & = & \int_0^{\pi}\int_0^{2\pi} \int_0^R \rho^2 \sin(\phi) d\rho d\theta d\phi \\ & = & \int_0^\pi \int_0^{2\pi} \frac{R^3\sin(\phi)}{3} d\theta d\phi \\ & = & \int_0^\pi \frac{2 \pi R^3 \sin(\phi)}{3} d\phi \\ & = & \frac{4 \pi R^3}{3}. \end{eqnarray*}