MA330: Assignment 5
Required Reading.
• Read Chapter 5, § 44-46
To be turned in April 26th at the start of class.
1. Textbook, page 122, #1
2. Textbook, page 122, #2
3. Textbook, page 122, #4
4. Textbook, page 122, #6
5. In applications such as fluid mechanics and signal processing, it’s sometimes useful to be able to compute integrals
in the complex plane, such as
Z
zdz
Γ
where Γ is some path connecting the point z1 = a1 + ib1 to z2 = a2 + ib2 in the complex plane. There are ways
of computing this integral directly, using complex variable methods, but it’s also possible to compute it using ideas
from vector calculus.
(a) If we write the complex number z in terms of it’s real and imaginary parts, z = x + iy, we can use this to
compute dz, and then substitute into the integral. You should now be able to write the integral as
Z
Z
Z
zdz =
P1 (x, y)dx + Q1 (x, y)dy + i P2 (x, y)dx + Q2 (x, y)dy
Γ
Γ
Γ
for some real functions Pi and Qi . Note that the real integrals on the right side of the equation are evaluated
over the same geometric path in the (x, y) plane as the complex integral in the complex plane on the left side
of the equation.
(b) Use the methods of vector calculus to evaluate these integrals, if possible. If evaluating them without knowing
the path is impossible, explain why.
6. Consider the field f (r) = r ln (krk).
(a) Show that the field is irrotational. Explain why this is relevant to the existence of a function F (r) such that
f = ∇F away from r = 0.
(b) Compute such an F . Check that it is correct by computing ∇F .
Rr
(c) Evaluate r12 f (r) · dr along any path avoiding the origin.
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