1. (20 points) Find the total ux upward throught the upper hemisphere
(z 0) of the sphere x2 + y 2 + z 2 = a2 of the vector eld T(x, y, z ) =
x3
3
i + y z 2 + e zx j + (zy 2 + y + 2 + sin(x3 )k.
Hint Use the divergence theorem around some simple region. y
Surprise Quiz!
Name:
August 1, 2007
1. Find the ux, S F dS, of the vector eld F(x, y, z ) = z 3 k through S, the upper
hemisphere of x2 + y 2 + z 2 = 1 (z 0), with orientation away from the origin.
1
Surprise Quiz!
Name:
August 1, 2007
1. Find the ux, S F dS, of the vector eld F(x, y, z ) = xi z j + y k through S, the
part of the sphere x2 + y 2 + z 2 = 4 in the rst octant, with orientation toward the origin.
1
(Not so) Surprise Quiz!
Name:
July 26, 2007
1.(a) Compute C y dx x dy where C is the circle with center the origin, radius 2, and clockwise orientation using two methods: (a) by calculating directly, and (b) by using Greens
Theorem (recall Greens theorem:
Surprise Quiz!
Name:
July 12, 2007
1.(a) Compute
z dV where E is the solid region bounded by the xz plane and the
E
hemispheres y = 9 x2 z 2 and y = 16 x2 z 2 .
1
Surprise Quiz!
Name:
July 6, 2007
3
x2
1.(a) Compute 2 1/x x2 y dydx.
(b) Write this iterated integral in dxdy order. You are not asked to evaluate the integral
this wayjust write it down.
1
1. (20 points) Find the total ux upward throught the upper hemisphere
(z 0) of the sphere x2 + y 2 + z 2 = a2 of the vector eld T(x, y, z ) =
zx
x3
3
i+
yz2 + e
j + (zy 2 + y + 2 + sin(x3 )k.
Hint Use the divergence theorem around some simple region. youl
Formula sheet for nal exam in calc IV, summer 2007
Polar coordinates
x = r cos y = r sin
r2 = x2 + y 2 dA = rdrd
Cylindrical coordinates
x = r cos y = r sin r2 = x2 + y 2
dV = rdzdrd
Spherical coordinates
x = sin cos y = sin sin z = cos
2 = x2 + y 2 + z
1. (10 points) Find the ux of
xi + y j + z k
3
(x2 + y 2 + z 2 ) 2
radius 1 with outward orientation. Explain.
through the sphere centered at (2, 0, 0) of
2. (20 points) Suppose a vector eld is dened by F(x, y, z ) = 2xy i + (x2 + 2yz )j + y 2 k.
a) Deter
Practice exam solutions for calc IV, Summer 2007 Exam 1
96
1
1
1. (15 points) a) (1 2 1 ) (3)3 1 (3)4 5 (3)5 ) = .
5
2
5
1
(y +1) 2
2
2 y 2
1
2
c) 7 2y x dxdy + 1 2y x dxdy
2. b) u = x y 2 , v = y , for instance. So x = u + v 2 and y = v
(x, y )
1 2v
= 1
Practice exam for calc IV, Summer 2007 Exam 1
Exam 1 will consist of roughly 100 points of similar problems to those below. No notes,
texts or calculators will be allowed on the exam. You will be provided with a formula sheet.
Answers to these problems wi
Formula sheet for exam 1 in calc IV, summer 2007
Polar coordinates
x = r cos y = r sin
r2 = x2 + y 2 dA = rdrd
Cylindrical coordinates
x = r cos y = r sin r2 = x2 + y 2
dV = rdzdrd
Spherical coordinates
x = sin cos y = sin sin z = cos
2 = x2 + y 2 + z 2
Calculus IV S1202Q Section 002, Summer
2007 Exam 1
Thomas D. Peters
Name:
July 19, 2007
Do all problems, in any order.
Show your work. An answer alone may not receive full credit.
No notes, texts, or calculators may be used on this exam.
Problem Possible
Calculus IVA
Complex variables. Lecture II
November 1, 2003
1
Complex line integrals
Consider an oriented curve C in the plane. We have dened the line integral
P dx + Qdy .
C
If
x = x(t), y = y (t) , a t b
is a parametrization of the curve, then
P dx + Qd
Calculus IVA
Complex variables. Lecture I
October 28, 2003
Contents
1 Introduction
1
2 Relation with the partial derivatives
4
3 Rules of dierentiation
7
4 The logarithm
9
5 Harmonic functions
10
6 Complex power series
12
7 Exercises
14
1
Introduction
So
3
Contour integrals and Cauchys Theorem
3.1
Line integrals of complex functions
Our goal here will be to discuss integration of complex functions f (z ) =
u + iv , with particular regard to analytic functions. Of course, one way to
think of integration is
2
2.1
Complex Functions and the Cauchy-Riemann Equations
Complex functions
In one-variable calculus, we study functions f (x) of a real variable x. Likewise, in complex analysis, we study functions f (z) of a complex variable z C (or in some region of C).
1
1.1
Review of complex numbers
Complex numbers: algebra
The set C of complex numbers is formed by adding a square root i of 1
to the set of real numbers: i2 = 1. Every complex number can be written
uniquely as a + bi, where a and b are real numbers. We u