Chapter 3. Determinants.
Let F = R or C and Mn the set of all n by n matrices over F . The text denes
the determinant as a function from Mn into F satisfying certain axioms. It goes on to
derive various properties of such a function and to show that there
Chapter 9
The Theorems of Stokes and
Gauss
1
Stokes Theorem
This is a natural generalization of Greens theorem in the plane to parametrized
surfaces in 3-space with boundary the image of a Jordan curve. We say
that is smooth if every point on it admits a
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b, Analytical; Homework Set 1
Due: 10am, Monday, January 14, 2013
Read the rst 9 sections of Chapter 1 of Apostol. That is read pages 3 through 13.
You can collaborate on the problems as l
Chapter 7
Div, grad, and curl
7.1
The operator
and the gradient:
Recall that the gradient of a dierentiable scalar eld on an open set D in
Rn is given by the formula:
,
,.,
.
(7.1)
x1 x2
xn
It is often convenient to dene formally the dierential operator
Chapter 5
Line Integrals
A basic problem in higher dimensions is the following. Suppose we are given a function
(scalar eld) g on Rn and a bounded curve C in Rn , can one dene a suitable integral of g
over C ? We can try the following at rst. Since C is b
Chapter 6
Greens Theorem in the Plane
Recall the following special case of a general fact proved in the previous
chapter. Let C be a piecewise C 1 plane curve, i.e., a curve in R2 dened
by a piecewise C 1 -function
: [a, b] R2
with end points (a) and (b)
Chapter 4
Multiple Integrals
4.1
Basic notions
We will rst discuss the question of integrability of bounded functions on closed rectangular
boxes, and then move on to integration over slightly more general regions.
Recall that in one variable calculus, th
Chapter 3
Tangent spaces, normals and extrema
If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any
fold or cusp or self-crossing, we can intuitively dene the tangent plane to S at a as follows.
Consider a plane which lies
VECTOR CALCULUS
Notes for Math1c, D. Ramakrishnan
Chapter 1
Subsets of Euclidean space, vector
elds, and continuity
1.1
Introduction
The subject matter of this course is the study of functions
f : Rn Rm
for general n and m (one may summarize the previous
Chapter 2
Dierentiation in higher dimensions
2.1
The Total Derivative
Recall that if f : R R is a 1-variable function, and a R, we say that f is dierentiable
)
at x = a if and only if the ratio f (a+hhf (a) tends to a nite limit, denoted f (a), as h tends
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b, Analytical; Homework Set 2
Due: Noon, Tuesday January 22, 2012
Read sections 1 through 5 and 9 through 16 of Chapter 2. Skip sections 6 and 7
of Chapter 2 as the discussion of inverses
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b, Analytical; Homework Set 8
Due: 10am, Monday, March 11, 2013
1. Problem 14 on page 126 of the text.
2. Prove that for each 2 by 2 matrix A over C, exactly one of the following holds:
(1
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b, Analytical; Homework Set 7
Due: 10am, Monday, March 4, 2013
Read sections 11-16 in Chapter 1, and sections 1-10 and 19 in Chapter 5 in the text.
Sections 12-18 in Chapter 5 are optional
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b, Analytical; Homework Set 4
Due: 10am, Monday, February 4, 2013
Read sections 17 and 18 from Chapter 2 of Apostol. That is read pages 58 through
65.
1. Let V = V2 be the space of ordered
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b, Analytical; Homework Set 5
Due: 10am, Tuesday, February 19, 2013
Read Chapter 3 in the text and read the notes on determinants on the course web
page.
1. Problem 3 on page 80 of the tex
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b, Analytical; Homework Set 6
Due: 10am, Monday, February 25, 2013
Read Chapter 4 in the text and read the notes on eigenvalues and eigenvectors on the
course web page.
1. Problem 14 on pa
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
Math 1b, Analytical; Homework Set 3
Due: 10am, Monday, January 28, 2013
1. Problem 26 on page 43 of Apostol. In part (b) prove that T and S are 1-1
correspondences (rather than proving S and T a
Chapter 4. Eigenvalues and Eigenvectors.
In this chapter V is an n-dimensional vector space over F = R or C. Recall L(V ) is
the vector space of all linear maps from V to V .
Let f L(V ). An eigenvalue for f is an element a F such that there exists a
nonz
Chapter 8
Change of Variables,
Parametrizations, Surface Integrals
8.1
The transformation formula
In evaluating any integral, if the integral depends on an auxiliary function of the
variables involved, it is often a good idea to change variables and try t