Math 417-517
Prof. Dimock
Fall 2016
name.SOLUTIONS.
class (417 or 517?) .
Exam 1
Directions: Show all work. Cross out wrong work. Simplify answers as much as possible.
1. (10 points) Let z = f (u, v)
Math 417-517
Prof. Dimock
Fall 2016
name.SOLUTIONS.
class (417 or 517?) .
Exam 3
Directions: Show all work. Cross out wrong work. Simplify answers as much as possible.
1. (10 points) Let u(R) = |R|2 .
Math 417-517
Prof. Dimock
Fall 2016
name.SOLUTIONS.
class (417 or 517?) .
Exam 2
Directions: Show all work. Cross out wrong work. Simplify answers as much as possible.
1. (10 points) In polar coordina
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
h
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 degre
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 f
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.
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
s
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
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
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 t
INVERSE FUNCTION THEOREM EXAMPLES
w2
1.
Example 1. Consider
x = f1 (u, v) = u2 v 2 ,
y = f2 (u, v) = 2uv.
Then
y
2u
v=
and
x = u2
y2
.
4u2
Then
4u4 4xu2 y 2 = 0
so
u2 =
=
16x2 + 16y 2
8
4x
x 1
2 2
x
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 R
THE LAPLACIAN
1.
The Meaning of the Laplacian
The Laplacian of a function f is
f =
f = div gradf.
The divergence theorem gives us that for a region R with boundary
surface S ,
f n d.
f dV =
R
S
If the
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
=
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 regio
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 denomina
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 p
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
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
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 dierentia
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
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 +
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
FUBINI'S THEOREM
Denition. Let R = cfw_(x, y) : a x b, c y d be a rectangle and
let
B R
be a subset of
R.
For an integer
n > 0,
we consider the
ba
and the subdivision
subdivision of [a, b] into n subi
Math 417-517
Prof. Dimock
Fall 2017
name.SOLUTIONS.
class (417 or 517?) .
Exam 2
Directions: Show all work. Cross out wrong work. Simplify answers as much as possible.
1. (10 points) Let a = a1 i + a2
Syllabus for Math 417-517
Fall 2017
Prof. Dimock
1. This is a course in advanced calculus, with emphasis on applications.
The first 2/3 of the course covers calculus in several variables. It is
partly
Math 417-517
Prof. Dimock
Fall 2017
name.SOLUTIONS.
class (417 or 517?) .
Exam 1
Directions: Show all work. Cross out wrong work. Simplify answers as much as possible.
1. (10 points) Consider the equa