Math 425 Assignment 4 Due October 7 , 2013
Part I: There is no webwork this week. Webwork. Links for the Webwork part of the
homework are on the course web page. Your user name is the part of your student email
address before the @ symbol, so if your em
Math 425 Notes 10
Line Integrals
Our goal is to nd a methods to compute the quantities such as:
Given a curved wire in space with specied density at each point, nd the total mass
of the wire.
Given a portion of surface is space with specied charge densi
Math 425 Notes 11 Surface Integrals
Our goal is to nd a method of calculating
surface area,
total mass of a surface given density,
ux through a surface.
Parametrizing Surfaces
Let S be a surface in R3 . A parametrization of S is a nice map
x(u, v )
u
Figure 1: Functions Viewed Optically
Math 425 Lecture 4
The Chain Rule
The Chain Rule in the One Variable Case
We explain the chain rule in the one variable case. To do this we view a function f as a
(impossible) lens. A light ray leaves the domain copy o
Math 425 Notes 12: Greens, Divergence, Stokes
We state these basis results.
Theorem 1 (Divergence (planar versiion). Let F be a vector eld in the plane. Let R be
a nice region of the plane and let R be its boundary. Let n be the outward normal to R,
let d
Math 425 Lecture 3
The Derivative
Our goal is to to give a denition of the derivative that is equivalent to the old denition,
but will provide better guidance when using the derivative. We rst examine the derivative
in the one dimensional case.
The Deriva
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Math 425 Notes 9
Multiple Integrals: Dention
We state a rough version of the fundamental theorem of calculus. Almost all calculations involving integrals lead back to this.
Denition 1. Let f : R R and let [a, b] be a closed interval. Let
a = x0 < x1 < < x
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Math 425 Lecture 5
Approximations
Cauchys Mean Value Theorem
Theorem 1 (Mean Value Theorem). Let f be a continuous function on an interval [a, b]
with a derivative for all points in the open interval (a, b). Then there is a point c (a, b)
so that
f (b) f
Notes 8: Div and Curl
F1
Let F = F2 be a vector eld in R3 , and set
F3
=
,
x y z
.
The divergence of F is
F1 F2 F3
+
+
=
x
y
z
F.
.
Let T : R3 R be temperature, so T is heat ow. We have seen that the
divergence of a vector eld measures the rate at which
Math 425
Notes 6: Lagrange Multipliers
Example 1. We are to bring water to a cow on the other side of a eld. To do this we
have to take the bucket to the river and ll it with water. What is the shortest path?
We could solve the problem on a map as follows
Curvature of Plane Curves
What is arc length parametrization?
Let
: [a, b] R R2 ,
t (s) = (x(s), y (s)
be a nice curve. We dene its arc length from t = a to t = b to be
b
 (t)dt.
a
We say that (s) is an arc length parametrization provided  (s) 1. We
Math 425 Assignment 3 Due September 27, 2013
Part I: Webwork. Links for the Webwork part of the homework are on the course web
page. Your user name is the part of your student email address before the @ symbol, so if
your email address is [email protected]
Homework 1 Math 425 Fall 2013 Due Wednesday September 11
1: Find two nonparallel vectors both orthogonal to (1, 2, 3).
2: Assume the earth is at and that the yaxis points north. If a whale is a position (1, 3)
and a rock is at position (4, 7) what is th
Math 425 Assignment 5 Due October 14 , 2013
Part I: There is no webwork this week. Webwork. Links for the Webwork part of the
homework are on the course web page. Your user name is the part of your student email
address before the @ symbol, so if your e
Math 425 Assignment 8 Due November 15 , 2013
Part I: There is webwork this week. Links for the Webwork part of the homework are on
the course web page. Your user name is the part of your student email address before the
@ symbol, so if your email addres
Math 425 Assignment 7 Due November 4 , 2013
Part I: There is webwork this week. Links for the Webwork part of the homework are on
the course web page. Your user name is the part of your student email address before the
@ symbol, so if your email address
Figure 1: Tetrahedron
Math 425 Assignment 2 Due September 18, 2013
1: This is not your usual homework problem. It is an experiment. I am not sure it will
be a good problem. Possibly it will be a great one and possibly it may not work at all as
a homework
Math 425 Practice Problems for Exam October 16, 2013
1. Let F : Rn R. What is
F ? ( Just give a formula.)
2. Let : R R2 . What is the curvature of ? Give the denition. The formula is
not necessary.
3. Show that the divergence of the curl of a vector eld i
Math 425 Practice Problems for Final Exam
1. State Greens theorem. Use Greens Theorem to compute the area of the ellipse
2
x2
+ y2 = 1 with a line integral.
a2
b
2. Let F (x, y ) = (x, 3y ) be a uid ow. Compute the net ow out of the region below
y = 1 x2
Math 425 Practice Problems for Exam October 16, 2013
1. Find the critical points of z = 2(x2 + y 2 )e(x
2 +y 2 )
.
Hint: There is at least a curve of critical points.
2. Use Lagrange multipliers to nd the max of z = x + y restricted to the unit sphere.
3.