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 region R has volume V ,
1
V
f dV =
R
1
V
f n d.
S
If w
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
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.
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
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
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
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
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
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
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
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
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 =
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
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
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
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
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
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
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
x2 + y 2 .
Now this is usually negative if we take the m
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 subintervals of length
n
dc
of [c, d] into n subintervals o
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) be a differentiable function. Define z as a function of
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 coordinates the position, velocity, and acceleration are given
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 . Find the Laplacian u three ways
(a) In Cartesian coord
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
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