Name
Multivariate Calculus
MATH 280
Spring 2011
Exam #2
Instructions: Do your work on separate paper. You can work on the problems in any order.
Clearly label your work on each problem with the problem number. You do not need to write
answers on the quest
MATH 280
Name
Multivariate Calculus
Fall 2010
Exam #3
Instructions: Do your work on separate paper. You can work on the problems in any order.
Clearly label your work on each problem with the problem number. You do not need to write
answers on the questio
Name
Multivariate Calculus
MATH 280
Fall 2010
Exam #2
Instructions: Do your work on separate paper. You can work on the problems in any order.
Clearly label your work on each problem with the problem number. You do not need to write
answers on the questio
MATH 280
Name
Multivariate Calculus
Fall 2010
Exam #1
Instructions: Do your work on separate paper. You can work on the problems in any order.
Clearly label your work on each problem with the problem number. You do not need to write
answers on the questio
MATH 280
Multivariate Calculus
Fall 2011
Integration over a surface
Given a surface S in space, we can (conceptually) break it into small pieces each
of which has area dA. In some cases, we will add up these small contributions to
get the total area of th
Name
Calculus and Analytic Geometry I
MATH 180
Spring 2009
Final Exam
Part A Instructions: Do the work for these problems on separate paper. Show enough detail
for me to assess how you arrive at your conclusions. Clearly box or circle the nal answer
for e
MATH 280
Multivariate Calculus
Fall 2011
Gradient vector elds
1. Consider the function z = f ( x, y) = xy. Below is a plot showing level sets for z
from 15 to 15 in steps of 1 in the window 4 x 4, 4 y 4.
(a) On the level curve plot, draw estimates of grad
MATH 280
Multivariate Calculus
Fall 2011
Describing and integrating nonuniform density on a line segment
An object with uniform composition throughout has the same mass density at
each point. Likewise, if charge is spread uniformly throughout a region, th
MATH 280
Multivariate Calculus
Fall 2011
Problems: Total from volume density
1. Relative to a chosen cartesian coordinate system, a solid object sits in the rst
octant bounded by z = 4 x2 y and the coordinate planes. The object
has a non-uniform compositi
Name
Calculus and Analytic Geometry II
MATH 181
Fall 2009
Final Exam
Instructions: Do your work on separate paper. You can work on the problems in any order.
Clearly label your work on each problem with the problem number. You do not need to write
answers
MATH 280B
Multivariate Calculus
Fall 2011
More on equations of planes
So far, we have seen several forms for the equation of a plane:
Ax + By + Cz + D = 0
z = m x x + my y + b
z z0 = m x ( x x0 ) + m y ( y y0 )
standard form
slopes-intercept form
point-sl
MATH 280
Name
Multivariate Calculus
Fall 2010
Exam #4
Instructions: Do your work on separate paper. You can work on the problems in any order.
Clearly label your work on each problem with the problem number. You do not need to write
answers on the questio
MATH 280
Name
Multivariate Calculus
Fall 2010
Exam #5
Instructions: Do your work on separate paper. You can work on the problems in any order.
Clearly label your work on each problem with the problem number. You do not need to write
answers on the questio
MATH 280
Multivariate Calculus
Fall 2011
Divergence of a vector eld
Flux
Given a vector eld F and an oriented surface S in space, we can think of the
surface integral S F d A as a ux. In this interpretation, we think of F as the velocity eld of a uid ow a
MATH 280
Name
Multivariate Calculus
Spring 2011
Exam #4
Instructions: Do your work on separate paper. You can work on the problems in any order.
Clearly label your work on each problem with the problem number. You do not need to write
answers on the quest
MATH 280
Name
Multivariate Calculus
Spring 2011
Exam #3
Instructions: Do your work on separate paper. You can work on the problems in any order.
Clearly label your work on each problem with the problem number. You do not need to write
answers on the quest
MATH 280
Multivariate Calculus
Fall 2011
Curl of a vector eld
Circulation
Given a vector eld F and an oriented closed loop C in space, we can think of
the line integral C F dr as a circulation. In this interpretation, we think of F as the
velocity eld of
MATH 280
Multivariate Calculus
Fall 2011
Integrating a vector eld over a curve
Denition
We are given a vector eld F and an oriented curve C in the domain of F as
shown in the gure on the left below. The general idea of integrating the vector
eld F along t
Name
Multivariate Calculus
MATH 280
Spring 2011
Exam #5
Instructions: Do your work on separate paper. You can work on the problems in any order.
Clearly label your work on each problem with the problem number. You do not need to write
answers on the quest
MATH 280
Multivariate Calculus
Fall 2011
Vector eld plots
For each of the following, use the given grid to sketch the given vector eld F for the
region with 2 x 2 and 2 y 2. Plot an output for each of the points provided
on the grid.
1.
F = x+0
2.
F = y+0
MATH 280
Multivariate Calculus
Fall 2011
Fundamental theorems of calculus
Note: In each of the following theorems, hypotheses on continuity of the integrand
and niceness of the relevant region are omitted in order to focus on other details.
Fundamental Th
MATH 280
Name
Multivariate Calculus
Spring 2011
Exam #1
Instructions: Do your work on separate paper. You can work on the problems in any order.
Clearly label your work on each problem with the problem number. You do not need to write
answers on the quest
MATH 280
Multivariate Calculus
Fall 2011
Problems: Total from area density
1. Charge is distributed on a at rectangular region of dimensions L by W so
that the area charge density is proportional to the distance from one corner,
reaching a maximum of 0 at
MATH 280
Multivariate Calculus
Fall 2011
Equations of planes
You should be familiar with equations of lines in the plane. From this experience, you know that the equation of a line in the plane is a linear equation in two
variables. Well use x and y as th
Name
Multivariate Calculus
MATH 280
Spring 2007
Exam #4
Instructions: You can work on the problems in any order. Please use just one side of each
page and clearly number the problems. You do not need to write answers on the question sheet.
This exam is a
MATH 280B
Multivariate Calculus
Spring 2011
More on equations of planes
So far, we have seen several forms for the equation of a plane:
Ax + By + Cz + D = 0
z = mx x + my y + b
z z0 = mx (x x0 ) + my (y y0 )
standard form
slopes-intercept form
point-slope
MATH 280
Name
Multivariate Calculus
Spring 2007
Exam #2
Instructions: You can work on the problems in any order. Please use just one side of each
page and clearly number the problems. You do not need to write answers on the question sheet.
This exam is a
MATH 280
Multivariate Calculus
Spring 2011
Components of the gradient vector
start with function f : R2 R and point P in the domain where, in a zoomed-in
view, the level curve through P and nearby level curves are parallel lines
dene gradient vector
f a
MATH 280
Multivariate Calculus
Spring 2011
The Greek alphabet
Greek letters are commonly used as variable names in mathematics and statistics. The
purpose of this assignment is to give you some familiarity with the Greek alphabet so that
you become more c