PHZ 3113 Fall 2011 – Homework 5
Due at the start of class on Wednesday, October 5. Half credit will be available
for homework submitted after the deadline but no later than the start of class on Friday,
October 7.
Answer all questions. Please write neatly and include your name on the front page of your
answers. You must also clearly identify all your collaborators on this assignment. To gain
maximum credit you should explain your reasoning and show all working.
Throughout the questions below, x̂, ŷ, and ẑ are unit vectors along the Cartesian x, y, and
z axes, respectively.
R
1. Calculate the outward flux A F · da of the vector field F = 2xyz 2 x̂ + y 2 z 3 ŷ + x2 zẑ
through the surface A of the cuboid spanning 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 3.
2. Find the outward flux of the vector field F = (y − x)x̂ + x2 z ŷ + (x2 + z)ẑ through the
region z > 0 of the surface of a sphere of radius R centered on the origin.
Hint: Convert the open surface to a closed one and apply the divergence theorem.
H
3. Calculate the circulation C F · dr of the vector field F = x3 x̂ + 3xy 2 ŷ − z 3 ẑ around the
perimeter of the ellipse x2 + 4y 2 = 1, z = 0, traveling counterclockwise when viewed
from positive values of z.
Hint: It may simplify the integral that you must perform to change variables using
x = 2r cos φ and y = r sin φ.
4. Use Stokes’ theorem to calculate the circulation of the vector field F = (z − x)x̂+
(x−z)ŷ+(y −z)ẑ around the perimeter P QR of the triangle with vertices P = (2, 0, 0),
Q = (0, 1, 0), and R = (0, 0, 3).
−→ −→
Hint: You can use a vector cross product such as P Q × QR to construct a vector
normal to the plane of the triangle and also to calculate the area of the triangle.
5. Green’s theorem states that if p and q are functions of x and y having continuous first
partial derivatives within some region A of the x-y plane with boundary C, then
Z I
∂q
∂p
−
dxdy,
(1)
(pdx + qdy) =
∂x ∂y
C
A
where the integral on the left-hand side is taken counterclockwise around the boundary.
Equation (1) can be regarded as a special case of Stokes’ theorem for a vector field
with Cartesian components (p, q, 0) where A is confined to lie in a plane of constant z.
It can also be derived from the two-dimensional version of the divergence theorem,
Z
I
∇ · F da =
F · (dr × ẑ)
A
C
(where dr is an infinitesimal vector along the boundary C and dr×ẑ is therefore a vector
of length |dr| pointing along the outward normal direction) by setting F = (q, −p).
As one application
H of Green’s theorem, the area of a region A can be found by evaluating
the line integral C (pdx + qdy) for any p and q such that ∂q/∂x − ∂p/∂y = 1, e.g.,
I
I
I
1
area =
xdy = − ydx =
(xdy − ydx).
2 C
C
C
H
(a) Use Green’s theorem to evaluate the line integral C v · dr for v = (2xy 2 , 4x) and
a path C taken counterclockwise around the edge of a triangle with vertices at
(0, 0), (3, 0), and (0, 2).
(b) Use Green’s theorem to find the area of the ellipse parametrized by x = a cos φ,
y = b sin φ for 0 ≤ φ < 2π.
(c) Use Green’s theorem to find the area of the shape enclosed by the parametric
equations x = a cos3 φ, y = a sin3 φ for 0 ≤ φ < 2π. Sketch the shape.