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 vario
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 curv
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
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 of
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
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
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 ge
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
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
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+h
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 secti
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
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
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 pag
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
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 e
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
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 eigenvalu
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