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.)
-
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.
Lecture schedule, reading, and homework assignments.
Note that topics listed for a given lecture may begin in the previous
lecture or end in the following lecture. Readings are given in
parentheses (S = Stewart, W = Worksheets). Below each lecture is the
corresponding homework assignment. All problems are from Stewart, 5th
edition. Homework from a Tuesday lecture is due in section on Friday, and
homework from a Thursday lecture is due the following Monday.
- (Tues 1/16) Introduction. Parametrized curves. (S10.1;
W1)
- 10.1: 5, 9, 13, 15, 17, 21, 28, 31, 33.
- (Thurs 1/18) More about parametrized curves: tangents, area, arc
length. (S10.2; W2)
- 10.2: 1, 3, 7, 11, 17, 25, 43, 53, 59.
- (Tues 1/23)
Polar coordinates (S10.3,4; W3). Brief review
of conic sections (S10.5; you should be able to sketch a conic
section from the equation, but you don't need to know about the focus and
directrix).
- 10.3: 17, 21, 25, 35, 39, 54, 57.
- 10.4: 3, 5, 29, 45, 47.
- (Thurs 1/25)
Vectors, dot product, cross product, determinant. (S12.1-4; W4).
- 12.1: 15, 21.
- 12.2: 4, 25, 35.
- 12.3: 23, 28, 51, 59.
- 12.4: 3, 7, 9, 15, 29, 41.
- (Tues 1/20)
Lines, planes, and quadric surfaces. (S12.5,6; W4,5)
- 12.5: 3, 5, 19, 21, 23, 53, 55.
- 12.6: 3, 5, 17, 21-28, 47.
- (Thurs 2/1)
Cylindrical and spherical
coordinates. (S12.7)
- 12.7: 37, 39, 45, 51, 53, 57, 59, 65.
- (Tues 2/6)
Vector-valued functions and space curves. (S13.1,2; W6)
- 13.1: 3, 5, 19-24, 25, 35.
- 13.2: 3, 5, 9, 17, 23, 33, 39.
- (Thurs 2/8)
Functions of several variables, limits and continuity.
(S14.1,2; W8,9)
- 14.1: 21, 23, 25, 30, 37, 45, 53-58.
- 14.2: 1, 3, 7, 9, 11.
- (Tues 2/13)
Partial derivatives, tangent planes, differentials.
(S14.3,4; W10,11)
- 14.3: 13, 15, 17, 27, 35, 53, 57.
- 14.4: 1, 3, 5, 11, 13, 31, 33.
- (Thurs 2/15)
The chain rule for functions of several variables.
(S14.5; W12)
- 14.5: 1, 3, 5, 7, 9, 11, 21, 23, 27, 29, 31, 47, 49.
- (Tues 2/20)
Directional derivatives and gradient. (S14.6; W13)
- 14.6: 5, 7, 9, 11, 13, 21, 23, 39, 41, 47, 49.
- (Thurs 2/22)
Optimization: maxima and minima. (S14.7; W14).
- Study for the first midterm. To study for this and the
subsequent exams, the chapter review problems in Stewart
are highly recommended.
- (Tues 2/27) MIDTERM #1. Will cover the material through 2/20.
- 14.7: 1, 3, 5, 7, 11, 27, 29, 33, 41, 43.
- (Thurs 3/1)
Optimization with constraints: Lagrange
multipliers. (S14.8; W15)
- 14.8: 1, 3, 5, 7, 15, 17, 19, 21, 23, 39.
- (Tues 3/6)
Double and interated integrals. (S15.1,2; W16)
- 15.1: 1, 5, 11, 13.
- 15.2: 3, 5, 7, 15, 19, 21, 27, 29.
- (Thurs 3/8)
Double integration over more general regions. (S15.3; W17)
- 15.3: 9, 11, 15, 17, 23, 37, 39, 41, 43, 45, 57.
- (Tues 3/13)
Double integration in polar coordinates.
(S15.4; W18)
- 15.4: 1, 3, 5, 9, 11, 13, 15, 23, 25, 29.
- (Thurs 3/15)
Applications of double integrals; Surface area. (S15,5,6; W19,20)
- 15.5: 1, 3, 5, 9, 13.
- 15.6: 3, 5, 9, 21, 23.
- (Tues 3/20)
Triple integrals.
(S15.7; W21)
- 15.7: 2, 3, 7, 9, 11, 13, 15, 17, 19, 31, 37.
- (Thurs 3/22) Triple integrals in cylindrical and spherical
coordinates. (S15.8; W21,23)
- 15.8: 1, 3, 5, 7, 13, 17, 19, 21, 29, 31, 33.
- Spring break!
- (Tues 4/3)
Change of variables in multiple integrals; Jacobians.
(S15.9, W22)
- 15.9: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21.
- (Thurs 4/5)
Vector fields and line inegrals. (S16.1,2; W24,25)
- Study for the second midterm.
- (Tues 4/10) MIDTERM #2. Will cover the material up through 4/3,
and mostly after 2/20.
- 16.1: 5, 11-14, 15-18, 21, 23.
- 16.2: 1, 3, 5, 7, 11, 15, 17, 19, 21.
- (Thurs 4/12)
The Fundamental Theorem for Line Integrals. (S16.3; W26)
- 16.3: 3, 5, 7, 9, 11, 13, 15, 19, 21, 23.
- (Tues 4/17)
Green's Theorem. (S16.4; W27)
- 16.4: 1, 3, 7, 9, 15, 17, 19, 21, 27.
- (Thurs 4/19)
Curl and divergence. (S16.5; W28)
- 16.5: 1, 3, 5, 7, 9, 11, 15, 17, 21, 25, 27, 31.
- (Tues 4/24)
Parametrized surfaces and their areas. (S16.6; W29)
- 16.6: 1, 3, 11-16, 17, 19, 35, 39, 41, 43.
- (Thurs 4/26) Surface integrals. (S16.7; W30)
- 16.7: 5, 7, 9, 11, 13, 17, 21, 23, 27, 41, 43.
- (Tues 5/1) Stokes' Theorem. (S16.8; W31)
- 16.8: 1, 3, 5, 7, 9, 13, 15, 17, 19.
- (Thurs 5/3) The Divergence Theorem. (S16.9; W32)
- 16.9: 1, 3, 5, 7, 9, 13, 15, 21, 23, 25, 29, 30.
- (Tues 5/8) Review.
- Memorize the chart on page 1134.
- (Fri 5/18, 12:30-3:30) FINAL EXAM. Will cover the entire course,
with a little over half of the exam devoted to the last third of the
course.
- 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.