Please do the following exercises from the text "Foundations of Analysis" by Joseph L. Taylor.
- 9.4[1,4,8*,11,12*,13*] (Problem 7 is not due until next week.)
Please do the following additional exercises.
- A*. (See 9.3) Suppose that (x,y,z) are the Cartesian coordinates of a point in R3 and the spherical coordinates of the same point is given by
x = r cos θ sin φ, y = r sin θ sin φ, z = r cos φ.
Let u = f(x,y,z) be a C2 function on R3. Find a formula for the partial derivatives of u with respect to x,y,z in terms of partial derivatives with respect to r,θ,φ. Find a formula for the Laplacian of u in terms of partial derivatives with respect to r,θ,φ, where the Laplacian is given by
Δu = ∂2u/∂x2 + ∂2u/∂y2 + ∂2u/∂z2.