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The final exam for math 550: vector calculus, taught by prof. Girardi during spring 2000. The exam covers topics from vector calculus by marsden & tromba, including div and curl of vector fields, linear transformations, and double integrals. Students are required to work on the exam in a logical fashion, show all their work, and indicate their reasoning.
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Prof. Girardi Math 550 29 April Spring 2000 Final Exam
MARK BOX problem points 1 10 2 10 3 10 4 10 5 10 6 10 total 60 % 100
1
1a. Figure 4.4.9 from the text, which is shown on the first page of this exam,
shows some flow lines and moving regions for a fluid moving in the plane field velocity field V~.
Where is div V~ > 0?.
Where is div V~ < 0?.
Intuitively explain your answers in (a few) complete sentences.
1b. Figure 4.4.7 from the text, which is shown on the first page of this exam,
shows the movement of a small rigid paddle wheel that is placed in moving fluid. The fluid has velocity field V~.
What can you say about the curl~ V~?.
Intuitively explain your answer in (a few) complete sentences.
D =
(x, y) :
( (^) x
a
( (^) y
b
(u, v) : u^2 + v^2 ≤ 1
So D is an ellipse (inside and boundary included) and D∗^ is the unit circle (inside and boundary included). Let A be the area of D.
3a. A 1-to-1 transformation T so that T (D∗) = D is:
T (u, v) = 〈 , 〉.
3b. Expressed as a double integral over D∗,
3c. By integrating the integral in 3b (and perhaps using the fact that the area
of D∗^ is π), we can compute that A =.
x + 2y + z = 6.
Make a (rough) sketch of V. The volume V of V can be expressed as the following triple integrals:
dz dx dy
dx dz dy.
You do NOT need to actually perform the integration.
More space for the last problem: