Algebra I
Lectured in Spring 2015 at Imperial College by Dr. J. R. Britnell.
Humbly typed by Karim Bacchus.
Caveat Lector: unofficial notes. Comments and corrections should be sent to
kb514@ic.ac.uk. Other notes available at wwwf.imperial.ac.uk/kb514.
Syl
14.6: DIRECTIONAL DERIVATIVES AND THE
GRADIENT VECTOR
KIAM HEONG KWA
1. Directional Derivatives
The directional derivative of a function f (x, y ) at (x0 , y0 ) in the direction of a unit vector u = a, b is dened by
f (x0 + ha, y0 + hb) f (x0 , y0 )
h0
h
14.5: THE CHAIN RULE
KIAM HEONG KWA
We shall assume all functions are dierentiable in this section.
1. The Chain Rule
Suppose that f is a function of the n variables x1 , x2 , , xn and
each xi is a function of the m variables t1 , t2 , c , tm , then
f
=
t
14.4: TANGENT PLANES AND LINEAR
APPROXIMATIONS
KIAM HEONG KWA
1. Tangent Planes
Suppose f (x, y ) is continuously dierentiable in the sense that it
has continuous partial derivatives. Then an equation of the tangent
plane to the surface S dened by the equ
14.3: PARTIAL DERIVATIVES
KIAM HEONG KWA
Let f be a function of two variables whose domain D includes points
arbitrarily close to (a, b). The partial derivative of f with respect to x
is given by
f (a + h, b) f (a, b)
fx (a, b) = lim
h0
h
provided the lim
14.2: LIMITS AND CONTINUITY
KIAM HEONG KWA
1. Limits of Functions
Let f be a function of two variables whose domain D includes points
arbitrarily close to (a, b). We say that the limit of f (x, y ) as (x, y )
approaches (a, b) is the number L and we write
14.1: FUNCTIONS OF SEVERAL VARIABLES
KIAM HEONG KWA
1. Functions of Two Variables
A function f of two variables is a rule that assigns to each ordered
pair of real numbers (x, y ) in a set D R2 a unique real number
denoted by f (x, y ).
The domain of f i
Autumn Quarter 2010
Mon
September 20
27
The Ohio State University
Tue
21
28
14.3
Wed
22
14.1, 14.2
29
Thu
23
4
5
6
Quiz 1
24
14.2, 14.3
30
14.4
14.6
Fri
October 1
14.5
14.7
7
8
14.8
Last day to drop
without a W
11
12
15.1
13
14
15.2
18
19
Midterm 1
25
20
14.7: MAXIMUM AND MINIMUM VALUES
KIAM HEONG KWA
1. Local Extreme Values and Critical Points
A function f of two variables is said to have a local maximum at a
point (xM , yM ) if
f (x, y ) f (xM , yM ) for all (x, y ) suciently close to (xM , yM ).
The nu
Analysis II
Lectured in Autumn 2015 at Imperial College by Prof. M. V. Ruzhansky.
Typed by Karim Bacchus.
Caveat Lector: unofficial notes. Comments and corrections should be sent to
kb514@ic.ac.uk. Other notes available at wwwf.imperial.ac.uk/kb514.
Note:
Chapter 1
Representing Uncertainty
Probability theory provides a framework for handling uncertainty.
UNCERTAINTY the absence of perfect knowledge of the
state of nature.
STATE OF NATURE some aspect of the real world, either current
or yet to be observed.
From the integration theory in single variable calculus, were familiar with the
following so-called Cavalieris Principle . Let E be a solid lying between the two
parallel planes x = a and x = b. Then the volume of E is given by
b
A(x) dx
a
where A(x) repr
Suppose f is a two variable real valued function, and wed like to know the rate
of change of f at (x0 , y0 ) in some arbitrary direction, say v = a, b 1. This is
accomplished by computing
d
f(x(t), y(t)
dt
.
t=0
In order to eectively compute the derivativ
Wed like to know the rate of change of f at (x0 , y0 ) in the direction v = a, b .
The following limit computes exactly this
lim
h0+
f(x0 + ah, y0 + bh) f(x0 , y0 )
.1
(x0 + ah x0 )2 + (y0 + bh y0 )2
We call the above, the directional derivative of f at (
Now that we know about directional derivatives, we want to use them to answer the following question: where are the relative maximums and minimums of a
function f(x, y)? First, we recall the denition of a relative maximum/minimum.
Denition 1. We call (x0
Wed like to extend the notion of integral from single variable calculus to multivariable calculus. To do this, we recall the single variable denition.
Let f be a bounded function on [a, b]. We partition [a, b]
a = x0 < x1 < x2 < . . . < xn1 < xn = b,
so t
Now that we know how to identify local extrema, wed like to know the following:
given a region D and a function f dened on D,
when does f attain its absolute maximum/minimum on D
how do we locate the points (x, y) at which f attains these extrema.
Lucki
Let f be a single variable real valued function. Then f is dierentiable at a means
lim
x a
f(x) f(a)
=: f (a)
xa
exists. Alternatively, we could rewrite the above and say that f is dierentiable at
a means there exists a real number, say A, such that
f(x)
Consider the graph of a function f(x, y). Let (x, y, f(x, y) be a point on the
graph. The graph of f denes a surface. Its a natural to ask the following: if
youre standing at the point (x, y, f(x, y), whats the slope of the surface in the
direction parall
Denition 1. A function f : D R2 R means a rule that assigns to each
(x, y) D a unique real number z := f(x, y)1. We call D the domain of f.
2
In the denition of function we have that its domain, D, is specied. But often
times, no mention of the domain of
The fundamental philosophy of calculus is to a) approximate, b) rene the approximation, and c) apply a limit process. In order to deal with c), well need the
concept of a limit, specically, for functions of several variables.
Denition 1. The limit as (x,