Calculus of One and Several Variables and Linear Algebra, Analytical Track
C 1

Spring 2013
MA 1C (SECTION 11) RECITATION 1
TONY WING HONG WONG
1. Introduction
Here are some basic information about the TA:
Oce: 304 Kellogg (connected to Sloan);
Phone: 3954027;
Email: tonywong@caltech.edu;
Webpage: http:/www.math.caltech.edu/tonywong/ma1c.ht
Calculus of One and Several Variables and Linear Algebra, Analytical Track
C 1

Spring 2013
MA 1C (SECTION 11) RECITATION 5
TONY WING HONG WONG
DISCLAIMER: This note may not cover all materials in the midterm. It should only be
used as an auxiliary tool for revision. Please refer to my other recitation notes, lecture notes
and the textbook for
Calculus of One and Several Variables and Linear Algebra, Analytical Track
C 1

Spring 2013
MA 1C (SECTION 11) RECITATION 4
TONY WING HONG WONG
1. Hessian Test
Let f : D Rn R be a C 2 function, i.e. the second derivative exists and is continuous.
The Hessian matrix at a D is
2 f (a)
2 f (a)
x1 xn
x1 x1
.
. .
.
.
H (a) = .
.
.
2 f (a)
xn x1
Calculus of One and Several Variables and Linear Algebra, Analytical Track
C 1

Spring 2013
MA 1C (SECTION 11) RECITATION 2
TONY WING HONG WONG
1. Total Differentiation
f : D Rn Rm is dierentiable at a int(D) if and only if there exists linear L : Rn Rm
such that
f (a + h) f (a) L(h) m
= 0.
lim
h0
hn
The total derivative of f at a, denoted as Ta
Calculus of One and Several Variables and Linear Algebra, Analytical Track
C 1

Spring 2013
MA 1C (SECTION 11) RECITATION 6
TONY WING HONG WONG
1. Midterm statistics
Whole class
HW
Average
79.31
75percentile 87.24
Median
80.61
25percentile 73.98
Midterm Average
58.11
75percentile 68
Median
57
25percentile 45.5
Overall
Average
65.65
75perc
Calculus of One and Several Variables and Linear Algebra, Analytical Track
C 1

Spring 2013
MA 1C (SECTION 11) RECITATION 7
TONY WING HONG WONG
1. Conservative Vector Fields
Let f : D Rn Rn , D open, f continuous. f is conservative if there is a C 1 : Rn R
such that f = .
f is conservative if and only if C f d = 0 for all piecewise C 1 closed cu
Calculus of One and Several Variables and Linear Algebra, Analytical Track
C 1

Spring 2013
MA 1C CALCULUS OF ONE AND SEVERAL VARIABLES AND
LINEAR ALGEBRA, ANALYTICAL TRACK
LECTURE 15
1. Examples and Applications
1.1. Volume below the graph of a function.
Let B = [a1 , b1 ] [a2 , b2 ], and let f : B [0, ) be a continuous function. Let C =
cfw_(x
Calculus of One and Several Variables and Linear Algebra, Analytical Track
C 1

Spring 2013
MA 1C (SECTION 11) RECITATION 10
TONY WING HONG WONG
1. Trigonometric Identities
sin x + sin y = 2 sin x+y cos xy
2
2
sin x sin y = 2 cos x+y sin xy
2
2
cos x + cos y = 2 cos x+y cos xy
2
2
cos x cos y = 2 sin x+y sin xy
2
2
2
2
cos 2x = 1 2 sin x = 2 co
Calculus of One and Several Variables and Linear Algebra, Analytical Track
C 1

Spring 2013
MA 1C (SECTION 11) RECITATION 9
TONY WING HONG WONG
1. Surface Integral
This is an analogue of line integral.
Let U be simply connected and U be piecewise C 1 Jordan curve. Let : U R2 R3
be C 1 . is simple if is oneone (injective), and is regular at (u0
Calculus of One and Several Variables and Linear Algebra, Analytical Track
C 1

Spring 2013
MA 1C (SECTION 11) RECITATION 8
TONY WING HONG WONG
1. Change of Variables
Let : U Rn V Rn be a dieomorphism, i.e. is bijective and both and 1 are
in C 1 . Let f : V R be integrable. Then f : U R is integrable and
f (x1 , . . . , xn ) dx1 . . . dxn = U f