Math 53 Syllabus
Course outline.
In single variable calculus, you studied functions of a single
variable, limits, and continuity. You then learned about derivatives,
which describe how functions change, and which can be used to help
find maxima and minima of functions. You then learned about integrals
which describe the aggregate behavior of a function over an interval,
such as the area under a curve or the average of a varying quantity.
The derivative and the integral are then tied together in the
fundamental theorem of calculus, one version of which relates the
integral of the derivative of a function over an interval to the
values of the function at the endpoints of the interval.
While single variable calculus is extremely useful, in many
applications one needs to consider functions of more than one
variable. For example, in understanding the weather, the temperature
is a function of three variables: latitude, longitude, and altitude.
In this course we will generalize the ideas from single variable
calculus mentioned above to functions of two or three variables.
(Most of what we will do can also be extended to functions of any
number of variables, although this requires a bit more abstraction,
and the thorough understanding of two and three variables that we will
develop in this course will provide the intuition for understanding
functions of more than three variables, should you encounter them
later.)
-
We will begin with some preliminaries on functions and on two- and
three-dimensional geometry, from chapters 10, 12, and 13 of the book.
-
We will then study derivatives of functions of multiple variables
(partial derivatives, directional derivative, gradient), and we will
use them to find
maxima and minima of functions of more than one variable. This
material is in chapter 14 of the book.
-
We will then consider
integration of functions of two or three variables (chapter 15). The
symbol manipulation involved is not much different than integration of
functions of one variable, but more geometric reasoning is required to
set up the limits of integration, because one is integrating over two-
and three-dimensional regions which are more complicated than
intervals.
- Finally, the climax of the course is vector calculus, covered
in Chapter 16. Here we will connect partial differentiation and
multiple integration
in four big theorems: the Fundamental Theorem for Line Integrals,
Green's Theorem, Stokes' Theorem, and the Divergence Theorem. These
can all be regarded as generalizations of the fundamental theorem of
calculus, and are very important for example in physics. (These four
theorems are actually all special cases of a more general theorem,
called "Stokes' theorem with differential forms", which holds in any
number of dimensions and which you can learn about in a more advanced
course.)
Since the heart of the course comes at the end, we will move somewhat
rapidly through the early, warm-up material, in order to devote
sufficient time to the Big Theorems that come later.
Lecture schedule and homework assignments.
Below each lecture is the corresponding homework assignment; homework
from a Tuesday lecture is due on Friday, and homework from a Thursday
lecture is due the following Monday.
- (Tues 8/26) Introduction. Parametric curves. Section 10.1.
- 10.1: 1, 5, 9, 11, 13, 14, 19, 28, 29.
- (Thurs 8/28) More about parametric curves: tangents, area, arc
length. Sections 10.2 and 10.3.
- 10.2: 1, 3, 5, 9, 17, 21,
31
- 10.3: 7, 17, 23.
- (Tues 9/2)
Polar coordinates, sections 10.4 and 10.5. Brief review
of conic sections, which are discussed in section 10.6. (We will be
skipping section 10.7, and all of Chapter 11.)
- 10.4: 17, 21, 25, 29, 35, 56, 59.
- 10.5: 3, 5, 29, 45, 49.
- (Thurs 9/4)
Vectors, lines, and planes. Sections 12.1 through 12.5.
- 12.1: 15, 21.
- 12.2: 4, 25, 27.
- 12.3: 25, 27, 32, 63.
- 12.4: 3, 7, 29, 41.
- 12.5: 3, 5, 15, 19, 49.
- (Tues 9/9)
Quadric surfaces; cylindrical and spherical
coordinates. Sections 12.6 and 12.7.
- 12.6: 3, 5, 17, 21-28, 47.
- 12.7: 37, 39, 45, 51, 53, 57, 59.
- (Thurs 9/11)
Vector-valued functions and space curves. Sections 13.1
and 13.2. (We will skip sections 13.3 and 13.4, but you might enjoy
reading about Kepler's Laws in section 13.4.)
- 13.1: 3, 5, 7-12, 21, 31.
- 13.2: 3, 5, 9, 17, 23, 33, 39.
- (Tues 9/16)
Functions of several variables, limits and continuity.
Sections 14.1 and 14.2.
- 14.1: 21, 23, 25, 30, 35, 37, 43, 51-56.
- 14.2: 1, 3, 7, 9, 11.
- (Thurs 9/18)
Partial derivatives, tangent planes, differentials.
Sections 14.3 and 14.4.
- 14.3: 11, 13, 15, 21, 33, 51, 55.
- 14.4: 1, 3, 5, 11, 13, 31, 33.
- (Tues 9/23)
The chain rule for functions of several variables.
Section 14.5.
- 14.5: 1, 3, 5, 7, 9, 11, 19, 21, 25, 27, 29, 45, 47.
- (Thurs 9/25)
The gradient. Section 14.6.
- 14.6: 3, 5, 7, 9, 11, 13, 21, 23, 37, 39, 45, 47.
- (Tues 9/30)
Optimization: maxima and minima. Section 14.7.
- The first midterm will cover the
lectures through 9/25. In preparing for this and the subsequent
exams, it might be helpful to look at the review problems at the ends
of the chapters of the book.
- (Thurs 10/2) MIDTERM #1.
- 14.7: 1, 3, 5, 7, 11, 27, 29, 31, 41, 43.
- (Tues 10/7)
Optimization problems with constraints, using Lagrange
multipliers. Section 14.8.
- 14.8: 1, 3, 5, 7, 15, 17, 19, 21, 23, 39.
- (Thurs 10/9)
Double and interated integrals. Sections 15.1 and 15.2.
- 15.1: 1, 5, 11, 13.
- 15.2: 3, 5, 7, 15, 19, 21, 27, 29.
- (Tues 10/14)
Double integration over more general regions. Sections
15.3 and 15.4.
- 15.3: 9, 11, 15, 17, 25, 33, 39.
- 15.4: 1, 3, 5, 7, 9, 11, 13, 21.
- (Thurs 10/16)
Applications of double integrals; Triple integrals.
Sections 15.5 and 15.7. (We will skip section 15.6, but we will
essentially cover this material later in section 16.6.)
- 15.5: 1, 3, 7, 13.
- 15.7: 2, 3, 7, 9, 11, 13, 15, 35.
- (Tues 10/21)
Triple integrals over more general regions, and in
cylindrical and spherical coordinates. Sections 15.7 and 15.8.
- 15.7: 17, 19, 31.
- 15.8: 1, 3, 5, 7, 13, 17, 19, 21, 29, 33, 35.
- (Thurs 10/23)
Change of variables in multiple integrals; Jacobians.
Section 15.9.
- 15.9: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21.
- (Tues 10/28)
Vector fields and line inegrals. Sections 16.1 and 16.2.
- 16.1: 5, 11-14, 15-18, 21, 23.
- 16.2: 1, 3, 5, 7, 11, 15, 17, 19, 21.
- (Thurs 10/30)
The Fundamental Theorem for Line Integrals. Section 16.3.
- 16.3: 3, 5, 7, 9, 11, 13, 15, 19, 21, 23.
- (Tues 11/4)
Green's Theorem. Section 16.4.
- 16.4: 1, 3, 7, 9, 15, 17, 19, 21, 27.
- (Thurs 11/6)
Curl and divergence. Section 16.5.
- 16.5: 1, 3, 5, 7, 9, 11, 15, 17, 21, 25, 27, 31.
- (Tues 11/11) Holiday.
- The second midterm will cover the lectures through
11/4 (and mainly after 9/25).
- (Thurs 11/13) MIDTERM #2.
- (Tues 11/18) Parametric surfaces. Section 16.6.
- 16.6: 1, 3, 11-16, 17, 19, 33, 35, 37, 39.
- (Thurs 11/20) Surface integrals. Section 16.7.
- 16.7: 5, 7, 9, 11, 13, 17, 21, 23, 27, 41, 43.
- (Tues 11/25) Stokes' Theorem. Section 16.8.
- 16.8: 1, 3, 5, 7, 9, 13, 15, 17, 19.
- (Thurs 11/27) Holiday.
- (Tues 12/2) The Divergence Theorem (aka. Gauss's theorem). Section 16.9.
- 16.9: 1, 3, 5, 7, 9, 14, 15, 21, 23, 25, 29, 30.
- (Thurs 12/4) Review.
- Memorize the chart on page 1118.
- (Thurs 12/18, 8-11 am) FINAL EXAM.
This will cover the entire course, with emphasis on the
later lectures.
- Vacation!
- To complete your study of basic calculus, you might want to
take Math 54, if you have not already done so.
- To prepare for the study of more advanced calculus (analysis),
and to learn how to prove calculus theorems rigorously, you can
take Math 104.
Up to Math 53 home page.