M3220-1 Eighth Homework Assignment

MATH 3220 § 1 EIGHTH HOMEWORK ASSIGNMENT Due Friday,

A. Treibergs October 26, 2007.

MATH 3220 § 1	EIGHTH HOMEWORK ASSIGNMENT	Due Friday,
A. Treibergs		October 26, 2007.

You are responsible for knowing how to solve the following exercises. Please hand in the starred "*" problems.

Please do the following exercises from the text "Foundations of Analysis" by Joseph L. Taylor.

9.3[1,3*,5,9*]

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[8]) 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.

You are responsible for knowing how to solve the following exercises. Please hand in the starred "*" problems.
Please do the following exercises from the text "Foundations of Analysis" by Joseph L. Taylor. 9.3[1,3,5,9] 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[8]) 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.