Computing the Laplacian in different coordinate systems.

The Laplacian of a function ta(x.y.z) in 3D Cartesian coordinates in defined by

∇²u = (∂²u/∂x²) + (∂²u/∂y²) + (∂²u/∂z²)

Any function in 3 variables can also be expressed in terms of either cylindrical polar coordinates (r, θ, Φ) or spherical polar coordinates (r, θ, Φ). In these coordinate systems the Laplacian has its own form. These are

∇²u =(1/r) (∂/∂r) (r ∂u/∂r) + (1/r²) (∂²u/∂θ²) + (∂²u/∂z²) (cylindrical polar)

∇²u = (1/r²) (∂/∂r) (r² ∂u/∂r) + (1/r² sin θ) (∂/∂θ) (sin (θ) ∂u/∂θ) + (1/r² sin² θ)(∂²u/∂Φ²)

a. Take the function u(x. y,z) = -x³ - 3xy²+ z². Compute the Laplacian in 3D Cartesian coordinates.

b. Convert the formula for u into a formula for u involving cylindrical polar coordinates. Then compute the Laplacian ∇²u using the cylindrical polar form. Show that your answer here is the same as your answer from (a)

c. For this section take u(x, y, z) = x² Convert the formula for u into a formula for u involving spherical polar coordinates. Then compute the Laplacian ∇²u using the spherical polar form. Check that your answer is 2 .