[SOLVED] Physics 311 Homework Set 2

1. Calculate the divergence of the following vector functions:
~va = x4 x + 42
z2 y 2y3
z z
~vb = x2
y x + 32
z y + 2yz2 z
~vc = 4y3 x + (22
z + z3
) y + 2xy2 z (1)2. Calculate the curl of the vector functions in the previous problem.3. Construct a vector function that has zero divergence and zero curl everywhere, a constant
vector function is too trivial!4. Check product rule (vi) (by calculating each term separately) for the functions
A~ = 3y x + z y + 2x z
B~ = 2x x 3z y (2)5. Calculate the Laplacian of the following functions:
a) Ta = z2 + 2zx + 3y + 5
b) Tb = sin x cos y sin z
c) Tc = e2z sin 3x cos 4y
d) ~v = z2 x + 2zy2 y zy z6. Consider the generic vector function
B~ = Bx x + By y + Bz z (3)
where Bx, By, Bz are the components.a) Show that the divergence of a curl of this generic vector function is always zero.
b) Show that the curl of a gradient of the scalar function, g = g(x, y, z), is always zero.7. Calculate the line integral of the function ~v = z2 x + 2zy2 y zy z from the origin to the
point (1, 1, 1) by three dierent routes:
a) (0, 0, 0) ! (1, 0, 0) ! (1, 1, 0) ! (1, 1, 1);
b) (0, 0, 0) ! (0, 0, 1) ! (0, 1, 1) ! (1, 1, 1);
c) The direct straight line.