13
13
Boundary-Value Problems
in Rectangular Coordinates
EXERCISES 13.1
Separable Partial Differential Equations
1. Substituting u(x, y ) = X (x)Y (y ) into the partial dierential equation yields X Y = XY . Separating variables
and using the separation co
9
Vector Calculus
EXERCISES 9.1
Vector Functions
1.
2.
3.
4.
5.
6.
7.
8.
9.
Note: the scale is distorted in this graph. For t = 0, the graph starts at (1, 0, 1). The upper loop shown
intersects the xz -plane at about (286751, 0, 286751).
438
9.1
10.
11. x
Chapter Eighteen
Stokes
18.1 Stokes's Theorem
Let F : D R 3 be a nice vector function. If
F ( x , y , z ) = p( x , y , z ) i + q ( x , y , z ) j + r ( x , y , z ) k ,
the curl of F is defined by
q p
r q p r
curlF =
j +
k .
i +
x y
y z z x
Here also
Chapter Sixteen
Integrating Vector Functions
16.1 Introduction
Suppose water (or some other incompressible fluid ) flows at a constant velocity v
in space (through a pipe, for instance), and we wish to know the rate at which the water
flows across a recta
Chapter Fifiteen
Surfaces Revisited
15.1 Vector Description of Surfaces
We look now at the very special case of functions r: D R 3 , where D R 2 is a
nice subset of the plane. We suppose r is a nice function. As the point (s, t ) D moves
around in D, if w
Solutions to some of the problems in
Chapters 15 and 16 of Cain & Herod.
1
Section 15.1
15.1, Problem 1
We use x = u and y = v as parameters. The surface can then be parametrized by
r(u, v ) = u, v, u + 2 v 2 , u, v 0.
15.1, Problem 2
If we let x = 2 x, y
MATH 2ZZ3 Winter 2010 (version # 2)
1
TEST #2 (VERSION 2)
19:00 20:15, March 16, 2010
Dr. Protas (C01), Dr. Kovarik (C02), Dr. Atena (C03)
This text paper consists of 8 pages (including this one). You are responsible for ensuring that
your copy of the te
Engineering Mathematics IV (MATH 2ZZ3) Winter 2010
1
HOMEWORK #4: VECTOR CALCULUS:
DOUBLE AND TRIPLE INTEGRALS, INTEGRAL
THEOREMS
Due: one minute after 11:59pm on March 15
Instructions:
The assignment consists of four questions worth, respectively, 3, 2,