MA330: Assignment 5
Required Reading.
• Read Chapter 5, § 47-51
To be turned in May 2nd at the start of class.
1. Textbook, page 128, #3
2. Textbook, page 128, #9
3. Textbook, page 128, #13
4. In class we integrated over a certain region to show
as a line integral.
∂Q
dA
r ∂x
RR
=
H
Γ
Qdy. Repeat the procedure to rewrite
∂P
R ∂P
RR
dA
5. Let (r, θ) be polar coordinates. Define the surface r = hr cos θ, r sin θ, ri for r ∈ [0, 2] and θ ∈ [0, 2π).
(a) Compute
dr
dr
×
dr
dθ .
(b) Compute the area of the surface.
6. A pipe with a circular cross-section has its central axis aligned with the z-axis. The velocity of water in the pipe
is given by f (r) = Vmax (1 − (r/R)2 )k where R is the radius of the pipe, r is the radial coordinate from the pipe’s
central axis, and Vmax is a positive constant with units of distance/time. Imagine a square of side length s < R,
with normal j + k centered on the central axis of the pipe.
(a) Sketch the pipe and the disk. Clearly label R, s, and r.
(b) Compute the rate of flow through the square (in units of volume/time). You may leave your answer in the form
of an integral, provided the integrand has been simplified.
7. Suppose f is twice differentiable
in the region R and has the property that its normal derivative along the boundary
RR
of R is 0. Prove that R ∆f dA = 0.
1