LECTURE: M-W-F 10:00 - 10:50 ET 202 (Zeman)
DISCUSSION: Tu-Th
10:00 - 10:50 ET202 (Hill)
QUIZZES: January 14 and 21, February 4, 11
and 25 and March 4 SOLUTIONS:
Q1 Q2-1 Q2-2 Q3 Q4
Q5 Q6
MIDTERM 1 Solutions MIDTERM 2 Solutions
FINAL EXAM INFORMATION:
Please bring your student ID -- if you are unable
to produce the student ID you will be denied to take the exam.
There will be six computational problems. The topics covered
are listed below. The difficulty will be about the same as Midterm 2.
Similarly as in the case of Midterm 2, the questions will be posed in
terms of applications, i.e. you will be asked to compute mass, center
of mass, work done by a vector field, rate of flow, etc. ... You will be
expected to decide which kind of integral to use for which application.
I will also include applications of the divergence and Stokes theorem,
but only very basic and straightforward ones because there is not much
time to practice on them.
As in the midterms, all integrals will be relatively
easily evaluable and you will be expected to evaluate them, i.e. get a
number in some form. You will be also expected to use standard integration
techniques: Integration in parts, substitution, change of coordinates
in multiple integrals: polar, cylindrical and spherical coordinates and
also linear transformation of coordinates (i.e. transforming parallelograms
into rectangular areas). Don't forget about Jacobians.
Here is a brief guide on surface integrals.
If you know the material in this guide plus all requisite integration and
parametrization techniques you should not have problems with the final exam.
Topic 1: Evaluating line integrals for computing
work (i.e. integrals of type 2). You will be expected to figure out if
it is more suitable to evaluate directly or using a potential function.
Sample problems: Exercies for Section
15.3 (p.1145): 9, 11, 15, 17, 19, 21, 23
Topic 2: Evaluating line integrals along a closed curve
using Green's theorem. The focus is on evaluating integrals of vector fields
that have discontinuity points enclosed by the curve.
Sample problems: Exercises
for Section 15.4 (p. 1156): 33, 34, 35, 36 Study Example 4.5 in
Section 15.4 (p.1154)
Topic 3: Surface integral of a function over a surface.
Computing mass, center of mass given the density function.
Sample problems: Exercises
for section 15.6 (p.1179): 29, 31, 33, 35 and 49, 50, 51 (compute 51 just as far as you can
without a calculator) 52
Topic 4: Flux integrals. Computing rate of flow
of a fluid, heat.
Sample problems: Exercises
for section 15.6 (p.1179): 37, 39, 43, 44,
47 Study Examples 6.6 and 6.7 (Section 15.6)
Topic 5: Using divergence theorem to evaluate
surface integrals.
Sample problems: Exercises
for Section 15.7 (p.1189): 17,
19, 21, 23
Topic 6: Using Stokes theorem to evaluate line
integrals.
Sample problems: Exercises
for Section 15.8. (p.1199): 1, 3, 15, 17, 19
HOMEWORK PROBLEMS
Note on integrals
W10-Fri: No new problems -- practice the sample problems above.
W10-Wed: Exercises for Section
15.8. (p.1199): 1, 3, 15, 17, 19
W10-Mon: Exercises for Section 15.7 (p.1189):
5, 7, 17, 19, 21, 23
W9-Fri: No new problems. Continue working on the problems from
W9-Wed. Try to use both methods to evaluate the integrals.
W9-Wed: Exercises for section 15.6 (p.1179): 37, 39, 41, 47,
and 63, 65, 67, 69
W9-Mon: Book, Exercises for Section 15.6 (p. 1179): 27, 29,
31, 35, 49, 53, 57, 59 Note: For 57 and 59, you don't have
to necessarily use formulas from 53 and 54.
W8-Fri: Book, Exercises for Section 15.6 (p. 1179): 1 -
8, 9 - 14, 17, 19, 20, 21, 23, 25 You have already seen
the surfaces from 1-8 and 9-14 before and should be able to recognize
them without too much work.
W8-Wed: Book, Exercises for Section 15.4 (p. 1156): 33,
34 and Exercises for Section 15.5 (p. 1156): 7, 10,
13, 16, 26, 37, 38, 39, 47, 48, 51, 52. Do some recap of surface area
and parametrizations of surfaces in 3D.
W8-Mon: Do recap of surface area and parametrizations
of surfaces in 3D. Book, Exercises for Section 15.4 (p.1156):
32
W7-Fri: Try to understand how Green's theorem work --
study the lecture.
W7-Wed: Relax after the midterm.
W7-Mon: Prepare for Midterm 2
W6-Fri: Prepare for Midterm 2
W6-Wed: Section 15.4. Examples 4.1, 4.2 and 4.4.
Exercises for Section 15.4 (p. 1156): 1, 5, 7, 9, 15, 21, 23
W6-Mon: Exercises for Section 15.3 (p. 1145):
7, 11, 15, 17, 21, 23, 27
Bonus: 49
W5-Fri: Study Example 3.1 and Example 3.3 in
Section 15.3. Exercises for Section 15.2 (p. 1134): 37-42
Bonus: 69
W5-Wed: Exercises
for Section 15.2 (p. 1134): 25, 27, 33, 35
W5-Mon: Section 15.2 Example
2.1. Exercises for Section 15.2 (p. 1134): 3, 9, 23, 43, 45
W4-Fri: Section 15.1 Example 1.10. Exercises
for Section 15.1 (p. 1119): 23, 29, 45-47, 55
W4-Wed: No new assignment -- work on Problems
assigned on W3-Fri.
W4-Mon: No new assignment -- prepare for
Midterm 1.
W3-Fri: Exercises for Section 15.1
(p. 1119): 1, 2, 3, 5, 9 Exercise 11 and Exercise 35, 37,
39
W3-Wed: Exercises for Section
14.8 (p. 1101): 1, 3, 5, 13, 15, 17, 19 and 23
Bonus: 33
W3-Mon: Holiday
W2-Fri: Examples 4.1 and 42. for
Section 14.4. and Exercises for Section 14.4 (p732): 1, 5,
7. Bonus: 31
W2-Wed: Example 3.6 on p. 732, Exercises
for Section 10.3 (p. 732): 3, 7 and Exercises
for Section 10.5 (p. 754): 35, 37
W2-Mon: Exercises for
Section 12.6 (p. 905): 13, 15, 19, 23b, 29,
W1-Fri:
Exercises for Section 12.6 (p. 905): 1, 2, 3, 4
W1-Wed: Exercises for Section 11.4 (p. 817) : 11, 15,
17, 21, 49, 51and Exercises for Section 11.5 (p. 826): 1,
3, 11
W1-Mon:
p. 803
Exercise 9,12,13, p. 804 Exercise 17,21, p. 817 Exercise 5 and
p.818 Exercise 10
LECTURES
W10-Fri: Review
W10-Wed: Divergence theorem and Stokes theorem. Applications:
evaluating flux integrals using triple integrals and line integrals using
surface integrals.
W10-Mon: Orientation
of surfaces. Open regions and their boundaries in 3-dimensional space.
Divergence theorem.
W9-Fri: Flux integrals -- recap and practical
computing. Example on heat flow from the book -- evaluatio using
the evaluation
formula (i.e. the method not used in the book to compute the heat
flow).
W9-Wed: Flux integral. Formula for evaluation of flux integral.
Normal vectors and orientation of a surface.
W9-Mon: Integral of a scalar function over a surface. Evaluation
formula of the integral for parametric surfaces. Evaluation formula for
surfaces given by a function. Special case: Integral of the function 1 over
S = Area of S.
W8-Fri: Divergence. The operator "nabla". Laplacian. Computation
with the operator "nabla". Criterion on conservativity. A non-conservative
irrotational field. Idea of surface integral.
W8-Wed: An example of use of Green's theorem for non-simply
connected regions. Curl and divergence of vector
field. Curl and Green's theorem. The "nabla" operator.
W8-Mon: Green's theorem for non-simply connected regions.
Criterion of conservativeness.
W7-Fri: Explanation of Green's theorem.
W7-Wed: Midterm 2
W7-Mon: Holiday
W6-Fri: Review for Midterm 2
W6-Wed: Orientation of closed curves. Simple curves.
Green's theorem and two of its applications: Evaluating challenging
integrals and computing the area of the region enclosed by a curve.
W6-Mon: Fundamental theorem on line integrals,
evaluation of line integrals for conservative fields, line integrals
along closed curves.
W5-Fri: Curve orientation. Sign of line integrals
and curve orientation. Line integrals along piecewise smooth
curves.
Open and connected regions. Path independence for line
integrals. Fundamental theorem on line integrals.
W5-Wed: Line integrals for computing work (i.e.
of type II). Line integrals with respect to x,y,z.
W5-Mon: Smooth curves, line integrals of Type
1, review of work done by a constant force field along a line.
W4-Fri: Directional derivatives, gradient,
potential, conservative vector fields. Methods for verification
if the field is conservative or not.
W4-Wed: Midterm 1
W4-Mon: Review
W3-Fri: Vector fields. Examples of vectore
fields. Flow lines. Computation of flow lines.
W3-Wed: Change of variables in double
integrals. Transformations using polar coordinates.
W3-Mon: Holiday
W2-Fri: Arc length in physics and
geometry. Tangent vector, tangent parallelpiped. Surface area
-- for mula for parametrized surfacts. Formula for surface
area given by a function in rectangular coordinates as a consequence.
W2-Wed: Arc length. The idea. Formulas
for arc length (a) in rectangular coordinates, (b) for parametric
representation and (c) for parametrization with polar coordinates.
W2-Mon: Parametrizations of surfaces
in 3D space. Use of polar, cylindrical and spherical coordinates.
W1-Fri:
Parametrizations of 1-dimensional (lines and curves) and
2-dimensional (surfaces). Polar coordinates. Parametrization
of a circle. Parametrization r = sin ß. Cylindrical and
spherical coordinates. Cylinder in cylindrical coordinate
W1-Wed:
Cross products: basic algebra of cross products.
Geometric meaning of the cross product: area of the induced
parallelogram and the volume of the induces paralell piped. Parametrizations
of lines and curves in plane and 3-D space.
W1-Mon: Review
of vectors. Dot product (11.3) and cross product (11.4),
haven't yet done geometric interpretation of cross product.
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