LECTURE 1F
SHENGYANG ZHANG
Chain Rule. Given f : E1 Rn Rm , g : E2 Rm Rk .
Consider x E1 s.t. f (x) E2 . Suppose f diff at x, g diff at f (x) then
(g f ) diff at x and
D(g f )x = Dgf (x) Dfx
Recall. Dfx : Rn Rm , Dgf (x) : Rm Rk .
D(g f )x : Rn Rk linear
LECTURE 3F
SHENGYANG ZHANG
Recap: Directional Derivatives. u unit vector in Rn , f : E Rn Rm
(x)
(Du f )x = limt0 f (x+tu)f
t
Theorem. (Du f )x = Dfx u.
Theorem. If kDfp k M, p in some set containing segment xy
kf (y) f (x)k M kx yk.
Proof. Think: start a
LECTURE 1M
SHENGYANG ZHANG
Preview of differentiation.
Recall. f : E R
(x)
()f (x) = limh0 f (x+h)f
if the limit exists else say f not diff at x.
h
Some equivalents to ().
f (x + h) f (x)
(1)
lim
f (x) = 0
h0
h
there is some constant A s.t.
f (x + h) f (
LECTURE 1W
SHENGYANG ZHANG
Last time. f : E Rn Rm , E Rn open
(x)Ahk
=0
We say Dfx equals A L(Rn , Rm ) if limh0 kf (x+h)f
khk
equivalent: f (x + h) = f (x) + Ah + r(h) with limh0
to zero faster then linear.
kr(h)k
khk
= 0, r(h) goes
Normed vector space:
LECTURE 3W
SHENGYANG ZHANG
Theorem. f : E Rn Rm . If
then f is differentiable on E.
Heading toward:
fi
xj
: E R all exist and are continuous
Inverse Function theorem. casually stated: If Dfx is invertible (as a
matrix) then f has an inverse function on sa
LECTURE 2W
SHENGYANG ZHANG
Recap. Partial derivatives
f : E Rn Rm , E open write f = (f1 , , fn )
Note fi (x) = i f (x)
i : Rm R, x xi
f (x+tej )fi (x)
fi
(x) = limt0 i
(Dj fi )(x) = x
t
j
f
xj (x)
Theorem. If f is diff at x (f as above). Then (Df )x ej
LECTURE 2F
SHENGYANG ZHANG
Theorem. Continuous partial Differentiable
Given E Rn open, f : E Rm . Suppose for x E partial
some open neighborhood of x and are continuous at x.
Then f is differentiable at x.
fi
So Df = ( x
)i,j
j
fi
xj
exists on
f
Idea. f x
LECTURE 2M
SHENGYANG ZHANG
Recap: f : E Rn Rm , E open.
(x)Ahk
=0
Dfx = A meanings limh0 kf (x+h)f
khk
n
m
A L(R , R ), kAk = supkvk=1 kAvk
Chain rule (casual written): D(g f )x = Dg(f (x) Dfx .
Inverse. L(Rn , Rm ) all linear maps Rn Rm
n m matrix as vec
MATH118C LECTURE 4M
SHENGYANG ZHANG
1a. f : R Rn vector valued functions
f (x) make sense (my notation f (x) vector in Rm )
Dfx matrix: R Rm
f (x) :vector in Rm
(x)
make sense
f (x) = limh0 f (x+h)f
h
m
makes sense vector R scalar
Claim: f (x) = Dfx (1)
P
Self-Assessment Questions Math 118B,
Winter 2010
1. Let D Rn be a bounded set and let f be uniformly continuous on D.
Prove that f is bounded on D.
2. Let f be a function dened on a set E which is such that it cen be uniformly approximated within on E by
Self-Assessment Questions Math 118A, Fall
2009
1. Every rational number can be written in the form x = m/n, where
n > 0, and m and n are integers without any comon divisors. When
x = 0, we take n = 1. Consider the functions f dened on R by
f (x) =
0 if x
Homework 1 Math 118C, Spring 2010
Due on Tuesday, April 6th, 2010
1. Prove that there exist constants > 0 and c > 0 such that
N
n=1
1
c
log N .
n
N
The number is called Eulers constant, and 0.57721 .
2. If an xn has radius of convergence R, prove that it
Homework 2 Math 118C, Spring 2010
Due on Tuesday, April 13th, 2010
1. Suppose f R on [0, A] for all A < , and f (x) 1 as x +.
Prove that
etx f (x) dx = 1.
lim t
+
t0
0
2. Suppose 0 < < , f (x) = 1 if |x| , f (x) = 0 is < |x| , and
f (x + 2 ) = f (x) for a
Homework 3 Math 118C, Spring 2010
Due on Tuesday, April 20th, 2010
1. Let KN be the Fjer kernel, dened in the previous assignment. If
e
f R and f (x+), f (x) exist for some x, prove that
lim N (f ; x) =
N
f (x+) + f (x)
.
2
2. Dene
f (x) = x3 sin2 x tan
Homework 4 Math 118C, Spring 2010
Due on Tuesday, April 27th, 2010
1. Give a matrix A Rnm , prove that
n
i=1
|aij |.
(a) A
1
= max1j m
(b) A
2
= max | is an eigenvalue of AT A .
= max1in m |aij |.
j =1
A 2 A F n A 2 , where A 2 =
F
maxi,j |ai j | A 2 mn m
Homework 5 Math 118C, Spring 2010
Due on Tuesday, May April 4th, 2010
1. In this problem you are asked to prove the following existence and
uniqueness theorem for Ordinary Dierential Equations, known as Picards theorem:
Theorem 0.1 (Picards Existence Theo
Homework 6 Math 118C, Spring 2010
Due on Tuesday, May 18th, 2010
1. Dene f in R2 by
f (x, y ) = 2x3 3x2 + 2y 3 + 3y 2 .
(a) Find the four points in R2 at which the gradient of f is zero. Show
that f has exactly one local maximum and one local minimum in
R
Homework 7 Math 118C, Spring 2010
Due on Tuesday, June 1st, 2010
1. Verify Stokes theorem for = x dx + xy dy with D as the unit square
with opposite vertices at (0, 0) and (1, 1).
2. Evaluate the integral of = (x y 3 ) dx + x3 dy around the circle
x2 + y
Self-Assessment Questions Math 118A, Fall
2009
1. For any sequence cfw_cn of positive numbers,
lim inf
n
cn+1
cn+1
lim inf n cn lim sup n cn lim sup
.
n
cn
cn
n
n
2. For any two real sequences cfw_an , cfw_bn , prove that
lim sup(an + bn ) lim sup an +
Self-Assessment Questions Math 118A, Fall
2009
1. Evaluate the limit
n
lim n
n
e
1
1+
n
n
.
without using LHospitals rule.
2. Prove that
n
1
1+
n
lim
n
exists using the following steps:
(a) Prove that
n
an =
1
1+
n
1
1
n
n
bn =
and
are both increasing se
LECTURE 3M
SHENGYANG ZHANG
Theorem (Continuous Partials Differentiable. We had E Rn
open, f : E R (for now m = 1 case)
Want to show if partial exists near x are continuous at x then Dfx =
f
f
( x
(x)
(x), . x
n
1
We had: limh0
kf (x+h)f (x)
f (x + h) f (x