1. Exercises from Sections 3.1
Review of the implicit function theorem
Let F (x, y) : Rn+k Rk be a differentiable function (here x = (x1 , . . . , xn ) and y = (y1 , . . . , yk ).
The implicit function theorem gives sufficient conditions for which we can
1. Exercises from 3.3
Today we are going to work with surface paramaterizations as preparation for doing integration over
surfaces.
Problem 1. Folland 3.3.1(b)
Let f : R2 R3 be given by f (u, v) = (au cos v, bu sin v, u) for a, b > 0.
What happens if we
1. Exercises from 5.2
Remind them of the definition of line integrals Show them Greens theorem in plane
Problem 1. (Folland 5.2.1(a,c)
Part a): Let C be the unit circle, traversed clockwise. Last time, we computed the following line integral
by hand
Z
(
MAT 237, PS3

Due, Friday Nov 7, 2:00 pm.
in SS Math Aid Center
FAMILY NAME:
FIRST NAME:
STUDENT ID:
Please note:
1. Your problem set must be submitted on this form. Please provide your nal, polished solutions in the
spaces provided. Remember, it is a
1. Exercises from 3.2
In this tutorial well study different ways of representing smooth curves. There are three principle
ways to do this. For curves in R2 , they are as follows:
(1) Explicitly, as the graph of a function y = f (x)
(2) Explicitly by a par
Problem Set 16
1. Easy (Details) (/problemSet/question/113/)
Determine the integral of each function on the specified rectangle:
f (x, y) = ex cos(y) for 0 x 1 and 2 y ,
2. f (x, y) = exy for 0 x 1 and 2 y 1,
3. f (x, y) = x2 6 3xy 2 for 1 x 2 and 1 y 1,
Problem Set 10
1. Easy (Details) (/problemSet/question/75/)
Find all the critical points of the following functions. Say whether the critical points are local maxima, local
minima, saddle points, or otherwise.
= x4 2x2 + y 3 6y
2. f (x, y) = (x 1)(x2 y 2
Problem Set 11
1. Easy (Details) (/problemSet/question/83/)
Determine whether the equation sin(xyz)
neighbourhood of the point (x, y, z)
(Text:
Solution:
= z may be solved for z as a function of x and y in a
= (/2, 1, 1).
)
2. Easy (Details) (/problemSet
1. Exercises from 4.1
This tutorial will focus on some aspects of the theory of Riemann integrals. Lets first remind
ourselves of the relevant definitions.
A partition of an interval [a, b] is a collection of numbers P = cfw_a = x0 < < xj < < xn = b.
If
1. Exercises from 5.1
Today were going to practice computing some line integrals.
Problem 1. (Folland 5.1.1(a,b,c,d) Find the arclengths of the following paths
In all of these examples, we use Theorem 5.11 on p. 220.
Part a) (t) = (a cos t, a sin t, bt),
MAT 237, PS4

Due, Friday Nov 28, 2:00 pm.
in SS Math Aid Center
FAMILY NAME:
FIRST NAME:
STUDENT ID:
Please note:
1. Your problem set must be submitted on this form. Please provide your nal, polished solutions in the
spaces provided. Remember, it is
2
Figure 1: Graph of the function f (x, y) = (xy)2 dened on R2 cfw_0. As
x2 +y
(x, y) approaches (0, 0) along the line x = y, f (x, y) approaches 0. However,
as (x, y) approaches 0 along the line x = y, f (x, y) approaches 2.
1
Figure 1: Let f (x) = 1+1/x, and let an = f n (1). In the problem session, well
study the convergence of the sequence cfw_a1 , a2 , . . . = cfw_1, f (1), f (f (1), . . ..
This gure shows a graphical way to think about iterated applications of f .
On the l
1. Exercises from 4.3/4.4
Problem 1. (Folland 4.3.13) An example of a nonintegrable function of two variables
Theorem 4.21 says that if S is measurable, and f : R2 R is bounded, and discontinuous on
a set with zero content, then f is integrable on S.
T
1. Exercises from 5.7/5.8
Theorem 1. Let S be an oriented surface in R3 which is bounded by a piecewise smooth curve S,
where the boundary is given an orientation consistent with S, then if F(x, y, z) is a C 1 vector field defined
on a neighbourhood of S
1. Exercises from 5.3
Today were going to practice doing different types of surface integrals. Briefly explain the concept
of a surface integral.
Problem 1. (Folland 5.3.1) Find the area of the surface z = xy inside the cylinder x2 + y 2 = a2
We pick a p
1. Exercises from 4.3
Today were going to practice some techniques for doing multiple integrals.
Problem 1. Folland 4.3.1(a)
Draw a picture of the region of integration, upper half of the unit disk
We do the yintegral first, then the xintegral
x
Z
3
1
Problem Set 14
1. Medium (Details) (/problemSet/question/102/)
Using any equivalent definition of integration (see Question 4), show that integration is a linear operator;
that, show that if f1 and f1 are integrable on [a, b], then c1 f1
+ c2 f2 is integr
Problem Set 17
1. Easy (Details) (/problemSet/question/125/)
Determine the Jacobian of the following transformations. Whenever possible, write the infinitesimal
area/volume element in terms of one another:
(x, y) = (e , 3 ),
2. (x, y) = (5u 2v, u + v),
3.
Problem Set 12
1. Easy (Details) (/problemSet/question/93/)
For each of the following manifolds, determine which can be written as the graph of a function, the zerolocus of a function, and parametrically. Give the appropriate functions in each case.
1. Th
Surface Normals and Tangent Planes
Normal and Tangent Planes to Level Surfaces
Because the equation of a plane requires a point and a normal vector to the
plane, nding the equation of a tangent plane to a surface at a given point
requires the calculation
Taylors Theorem in 2 Variables
Taylors Theorem tells us how to approximate a function near a given point by a
polynomial, provided we know the derivatives of that function at this point.
Let us recall what this means for functions of one variable: Let a b
1
Implicit Functions
Reading [Simon], Chapter 15, p. 334360.
1.1
Examples
So far we were dealing with explicitly given functions
y = f (x1 , ., xn ),
like y = x2 or y = x2 x3 .
1 2
But frequently the dependence of endogenous variable y on exogenous
varia
DIFFERENTIABILITY IN SEVERAL VARIABLES: SUMMARY OF BASIC
CONCEPTS
1. Partial derivatives If f : R3 R is an arbitrary function and a = (x0 , y0 , z0 ) R3 , then
f
f (a + tj) f (a)
f (x0 , y0 + t, z0 ) f (x0 , y0 , z0 )
(a) := lim
= lim
t0
t0
y
t
t
etc. pro
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