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) = |∂x∂u∂x∂v∂x∂w∂y∂u∂y∂v∂y∂w∂z∂u∂z∂v∂z∂w|, 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. |
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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: ∂x∂r=cos(θ),∂x∂θ=−rsin(θ),∂x∂z=0,∂y∂r=sin(θ),∂y∂θ=rcos(θ),∂y∂z=0,∂z∂r=0,∂z∂θ=0,∂z∂z=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 0≤ρ<∞,0≤θ<2π,0≤ϕ≤π. (iii) Points on the earth are frequently specified by Latitude and Longitude. How do these relate to θ and ϕ? |
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Problem: Find the Jacobian of the transformation
(ρ,θ,ϕ)→(x,y,z) of spherical coordinates.
Solution: Now our partial derivatives are: ∂x∂ρ=cos(θ)sin(ϕ),∂x∂θ=−ρsin(θ)sin(ϕ),∂x∂ϕ=ρcos(θ)cos(ϕ),∂y∂ρ=sin(θ)sin(ϕ),∂y∂θ=ρcos(θ)sin(ϕ),∂y∂ϕ=ρsin(θ)cos(ϕ),∂z∂ρ=cos(ϕ),∂z∂θ=0,∂z∂ϕ=−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)0−ρsin(ϕ).| which works out to ρ2sin(ϕ), and our volume element is dV=dxdydz=ρ2sin(ϕ)dρdθdϕ. |
Problem: Compute the volume of the ball ρ≤R
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*} |