Physics 311 Homework Set 2 solved

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1. Calculate the divergence of the following vector functions:
~va = x4 xˆ + 4×2
z2 yˆ 2y3
z zˆ
~vb = x2
y xˆ + 3×2
z yˆ + 2yz2 zˆ
~vc = 4y3 xˆ + (2×2
z + z3
) ˆy + 2xy2 zˆ (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 xˆ + z yˆ + 2x zˆ
B~ = 2x xˆ 3z yˆ (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 xˆ + 2zy2 yˆ zy zˆ

6. Consider the generic vector function
B~ = Bx xˆ + By yˆ + Bz zˆ (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 xˆ + 2zy2 yˆ zy zˆ from the origin to the
point (1, 1, 1) by three di↵erent 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.