MTH 437/537 Project
Fall 2015
Introduction
Diary
Team rules
Option selection
Option 1
Option 2
Introduction
According to our course policies document, the project is due at noon on Monday, Nov 23. We
can discuss this.
The project may be done individually
MTH 417/517 Sample Exam Two Dr. Faran
November 2015
1. Find the area of the surface dened by x + y + z = 1, x2 + 2y 2 1.
Parametrizing the
surface by (x, y, 1 x y), the normal is (1, 1, 1)
and so the area 3 times the area of the ellipse x2 + 2y 2 1, whic
MTH 417/517 Sample Exam Two Dr. Faran
November 2015
1. Find the area of the surface dened by x + y + z = 1, x2 + 2y 2 1.
2. Let F = x3 + y 3 + z 3 k. Find the surface integral of F over the unit
i
j
sphere (with outward pointing normal) using the divergen
DISTRIBUTIONS, THE DIRAC DELTA FUNCTION
AND HELMHOLTZ DECOMPOSITION
1.
Distributions and the Dirac Delta Function
If two functions
all functions
,
f
and
g
on
R3
f dV =
satisfy
then we must have
f = g.
g dV
for
This is the idea behind ex-
tending the notio
THE HYPERBOLIC PLANE:
AN APPLICATION OF METRIC COEFFICIENTS
Let
H = cfw_(x, y) R2 | y > 0 be the upper
1
ds2 = 2 dx2 + dy 2
y
half-plane, and let
be the metric quadratic form.
Consider curves of the form
y = y (x), a x b.
Such a curve will
have length
b
1
A CLASS OF REAL INTEGRALS
We examine integrals of the form
+
+
R (x) cos x dx and
R (x) sin x dx,
where R is a rational function of x whose denominator has no real zeroes
and whose numerator has degree at most one less than the degree of
the denominator.
SURFACES
A surface is a particular kind of subset of space. Not every subset
is a surface, and so we need to have some idea what is a surface and
what isn't.
We restrict our discussion to subsets of R3 , though it is possible to
have surfaces in higher di
EQUALITY OF MIXED PARTIALS
FROM MARSDEN AND TROMBA,
CALCULUS
VECTOR
, 1ST EDN, P. 120
If f is C 2 (twice continuously dierentiable) then the mixed
partials are equal; that is,
Theorem.
2f
2f
=
.
xy
yx
Proof. Consider the expression
(1.1) f (x0 + x, y0 +
DIFFERENTIATION UNDER THE INTEGRAL SIGN
f (t, x)
and
d
dt
Theorem 1. If
f
t
b
(t, x)
exist and are continuous, then
b
f (t, x) dx =
a
a
f
(t, x) dx.
t
We want to show that
Proof.
lim
b
a f
h0
or
(t + h, x) dx
h
b
a f
(t, x) dx
b
=
a
f
(t, x) dx,
t
b
1
h0
CALCULUS OF V
ARIATIONS
Theorem 1.
If F (x, y, y ) is independent of y , then
Fy (x, y )
is constant along a solution of the Euler-Lagrange equation.
Proof. It follows immediately from
0 = Fy
d
Fy
dx
that if Fy = 0, then Fy is constant along a solution o
MTH 417/517 Sample Exam One Dr. Faran
September 30, 2015
1. Let
if (x, y) = (0, 0) ,
if (x, y) = (0, 0) .
x3 x2 y
x2 +y 2
f (x, y) =
0
(a) Is f continuous at (0, 0)? Why or why not?
(b) Is f dierentiable at (0, 0)? Why or why not?
2. Show that there is a
MTH 417/517 Sample Exam One Dr. Faran
September 30, 2015
1. Let
f (x, y) =
x3 x2 y
x2 +y 2
0
if (x, y) = (0, 0) ,
if (x, y) = (0, 0) .
(a) Is f continuous at (0, 0)? Why or why not?
|x3 x2 y|
x2 + y 2
|x|3 + |x|2 |y|
x2 + y 2
|f (x, y) f (0, 0)| =
3/2
(x2
DIVERGENCE AND CURL IN ORTHOGONAL COORDINATES
1.
Divergence
We use the divergence theorem on the image R of a box in (u1 , u2 , u3 )-space. We
take one corner of the box to be (u1 , u2 , u3 ) and the opposite corner (u1 + u1 , u2 + u2 , u3 + u3 ).
Then
1
CR EQUATIONS IMPLY DIFFERENTIABILITY
If u (x, y) , v (x, y) have continuous partial derivatives which satisfy
the CR equations near a point, then f (z) = u (x, y) + iv (x, y) is dierentiable at
that point and
Theorem 26.
f (z) =
v
v
u
u
+i
=
i .
x
x
y
y
P
ez
dz
z 5 5z 4 + 8z 3 8z 2 + 7z 3
|z|=2
The integrand here is analytic except where the denominator vanishes.
Thus the rst trick is to nd out where the roots of the denominator
are. Here, the denominator factors.
ez
dz =
z 5 5z 4 + 8z 3 8z 2 + 7z 3
|z|=2
PATH INDEPENDENCE OF INTEGRALS OF CLOSED
FORMS
Suppose we have a form
M dx + N dy
on a region
D
of
R2
which is
closed, that is
M
N
=
.
y
x
Suppose also that we have functions x = x (s, t)
2
some region D R to the region D . Then
and
y = y (s, t)
from
dx =
ERROR CORRECTION: AN INTERESTING EXAMPLE
1.
Error Correction
It is not true that any function satisfying the Cauchy-Riemann equations is dierentiable. The condition that the real and imaginary parts
have continuous partial derivatives is necessary.
Exampl
SPECIAL FUNCTIONS
1.
Trigonometric Functions
We have dened
cos z =
1 iz
e + eiz
2
sin z =
1 iz
e eiz .
2i
We rst note that
1 iz
1 iz
2
2
e + eiz
e eiz
4
4
1 2iz
1 2iz
=
e + 2 + e2iz
e 2 + e2iz
4
4
= 1,
cos2 z + sin2 z =
just as in the real case.
The sum
CHANGE OF V
ARIABLES IN TRIPLE INTEGRALS
EXAMPLE
We will consider a solid V R which lies in the rst octant (x 0,
y , z 0) above the cone z = x + y and below the plane
0
z = 2/2. Suppose this solid has density = 1. We want to nd the
total mass and the vari
ADDITIONAL VECTOR IDENTITY F
ACTS
(1) Double Takes
(a) u = 0
(b) ( v) = 0
This is also the equality of mixed partials. If v = v1 +
i
v2 + v3 k,
j
(
(c)
(d)
/x /y /z
v) = det /x /y /z .
v1
v2
v3
The terms here involving any one coordinate of v is simply
t
Nigel asks:
One thing I don't get is how to locate the roots of an equation if I do not have a guess as to where they
might be located, or if they are very close to one another. The methods we've covered seem to require
intuition or extra information abou
Quiz #6 Questions answered
Dante asks:
Instead of using symbolics to define a general function, could one make a general
contour plot formula more efficient with broadcasting and automatic differentiation?
Making a good contour plot is a difficult task, e
Introduction to Numerical Analysis I, Fall 2015
Project
The Project is currently due Nov 23. Instructions are here.
Exam 2
Exam 2 is Wednesday, Nov 4. It will cover what we've talked about from Sep 18 (start of linear systems)
through Oct 26 (linear least
EXISTENCE OF EIGENVECTORS OF A SYMMETRIC
MATRIX
Let A be a symmetric real n n matrix.
We start by considering the real valued function on Rn
f (x) = xT Ax.
We can calculate its derivative as follows.
f (x + hv) f (x)
h0
h
(x + hv)T A (x + hv) xT Ax
= lim
CHAIN RULE
1.
Preliminaries
In order to get the chain rule, we need to have some tools at hand.
Theorem
Rn ,
(Cauchy-Schwarz Inequality). For any two vectors x, y
|x y| |x| |y| .
Proof. See Greenberg, p. 424. One way to look at it is that the component o
Figure 1.1.
1.
Let R be the region bounded by x = y , x + y = 2 and x = 2y. (See Fig. 1.1.)
Suppose we want
A Different Change of Variables
2
x dxdy.
R
Setting this up in Cartesian coordinates gives us an iterated integral that must be
split up into two i
MTH 417/517
Sample Final Exam
Dr. Faran
December 9, 2015
1. (10 points) Let
f (x, y) =
x3 x2 yxy 2 +y 3
x2 +y 2
0
if (x, y) = (0, 0) ,
if (x, y) = (0, 0) .
(a) Is f continuous at (0, 0)? Why or why not?
(b) Is f dierentiable at (0, 0)? Why or why not?
2.