Home Work Set # 2, Physics 217, Due: September 19,
Check the fundamental theorem for gradients, using the function
and the points a
= (0, 0, 0) and b
= (1, 1, 1), and the following
three paths (see Figure 1).
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 parabolic path
1. Problem 1.
Evaluate the following
One consequence of the fundamental theorem for curls is that the surface
integral of the curl of a vector function
depends only on the boundary line, not on the particular surface used. Consider
the following vector function:
a) Calculate the line integral of
along the boundary of the square shown in Figure 2a.
b) Calculate the surface
over the surface of the square shown in Figure 2a.
c) Calculate the surface
by integrating over the five sides of the cube shown in Figure 2b.
the result of part a) with the result of part b) and c). What do you
2. Problem 3.
a) Show that
: Use integration by parts.
be the "step function":
Calculate the divergence and curl of
Which one can be written as the gradient of a scalar? Find a scalar potential
that does the job. Which one can be written as the curl of a vector function?
Find a suitable vector potential.
b) Show that
can be written both as the gradient of a scalar and as the curl of a vector.
Find scalar and vector potentials for this function.
a) Compute the divergence of the function
b) Check the divergence theorem for this function, using as you volume the
inverted hemisphere of radius R, resting on the x-y plane
and centered at the origin (see Figure 3).
3. Problem 6.
Compute the line integral of
along the triangular path shown in Figure 4. Check your answer using
4. Problem 6.