Math 550 Spring 2000 Final Exam: Vector Calculus, Exams of Vector Analysis

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.

Typology: Exams

Pre 2010

Uploaded on 10/01/2009

koofers-user-nu2
koofers-user-nu2 🇺🇸

9 documents

1 / 7

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
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
NAME:
SSN:
INSTRUCTIONS:
1. To receive credit you must:
(a) work in a logical fashion, show all your work,
indicate your reasoning
(b) when applicable put your answer on/in the line/box provided
(c) if no such line/box is provided, then box your answer
(d) if you use your calculator on a particular problem, then indicate so.
2. The mark box indicates the problems along with their points.
Check that your copy of the exam has all of the problems.
3. As indicated on the syballus:
(a) allowed is a calculator (but not a computer)
(b) allowed are the class handouts: table of integrals, calculus for-
mula sheet, and Spring 2000 informal summary (along with your
personal scribbles on them)
(c) not allowed are books and other notes.
4. During this exam, do not leave your seat. If you have a question, raise
your hand. When you finish: turn your exam over, put your pencil
down, and raise your hand.
5. This exam covers (from Vector Calculus by Marsden&Tromba, 4th ed.):
Ch. 1, Ch. 2, Section 3.1, Chs. 4 7 .
1
pf3
pf4
pf5

Partial preview of the text

Download Math 550 Spring 2000 Final Exam: Vector Calculus and more Exams Vector Analysis in PDF only on Docsity!

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

NAME:

SSN:

INSTRUCTIONS:

  1. To receive credit you must: (a) work in a logical fashion, show all your work, indicate your reasoning (b) when applicable put your answer on/in the line/box provided (c) if no such line/box is provided, then box your answer (d) if you use your calculator on a particular problem, then indicate so.
  2. The mark box indicates the problems along with their points. Check that your copy of the exam has all of the problems.
  3. As indicated on the syballus: (a) allowed is a calculator (but not a computer) (b) allowed are the class handouts: table of integrals, calculus for- mula sheet, and Spring 2000 informal summary (along with your personal scribbles on them) (c) not allowed are books and other notes.
  4. During this exam, do not leave your seat. If you have a question, raise your hand. When you finish: turn your exam over, put your pencil down, and raise your hand.
  5. This exam covers (from Vector Calculus by Marsden&Tromba, 4th^ ed.): Ch. 1, Ch. 2, Section 3.1, Chs. 4 – 7.

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.

  1. Let a, b > 0 and

D =

(x, y) :

( (^) x

a

( (^) y

b

D∗^ =

(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∗,

A =.

3c. By integrating the integral in 3b (and perhaps using the fact that the area

of D∗^ is π), we can compute that A =.

  1. Let V be the solid in the first octant bounded by the plane

x + 2y + z = 6.

Make a (rough) sketch of V. The volume V of V can be expressed as the following triple integrals:

V =

dz dx dy

V =

dx dz dy.

You do NOT need to actually perform the integration.

More space for the last problem: