Section 16.1 - Vector Fields
Denition. Let D be a set in Rn . A vector eld is a function F that assigns a vector F (x1 , . . . , xn )
in Vn to each point (x1 , . . . , xn ) in D.
F (x, y ) = P (x, y )
Section 14.2 - Limits and Continuity
Recall: Let f (x) be a function of one variable dened on some open interval that contains the
number a, except possibly a itself. Then we say the limit of f (x) as
Section 14.3 - Partial Derivatives
Example 1. Consider the function of two variables
f (x, y ) = sin(x y ) + cos(x).
Fixing one of the variables and letting the other vary, we can think of f as a func
Section 14.5 - The Chain Rule
Functions of One Variable
Theorem. If y = f (x) and x = g (t) are dierentiable then y is indirectly a function of t and
dy
dy dx
=
dt
dx dt
Example 1. If y = sin(x) and x
Section 13.4 - Motion in Space: Velocity and Acceleration
Suppose that a particle moves along the curve C traced by the vector function r(t).
TEC V13.4: Velocity and Acceleration Vectors: http:/www.st
Section 13.3 - Arc Length
Recall: (Section 10.2) If C is a plane curve described by the parametric equations x = f (t),
y = g (t) for a t b with f and g continuous on [a, b], and C traversed exactly o
Section 13.1 - Vector Functions and Space Curves
A function f : X Y is a rule that maps each element of a domain X to an element of the range Y .
Denition. A vector-valued function (or vector function
Chapter 12 Review
Three-Dimensional Coordinate Systems
Right hand rule: to choose an orientation on R3 = R R R
Curl ngers around the z -axis in the direction o a counterclockwise rotation from the pos
Section 13.2 - Derivatives and Integrals of Vector Functions
Denition. The derivative r of a vector function r is dened to be
dr
r(t + h) r(t)
= r (t) = lim
.
h0
dt
h
If the derivative r (t) exists, t
Chapter 16: Solutions
(1) (a) Find parametric equations for the line segment C from the point (1, 5, 0) to the point
(1, 6, 4).
x(t) = 1 + 2t
y (t) = 5 + t
z (t) = 4t
0t1
(b) Evaluate the line integra
Section 14.6 - Directional Derivatives and the Gradient Vector
Partial Derivatives
If f is a function of two variables, its partial derivatives at a point (x0 , y0 ) are
f (x0 + h, y ) f (x0 , y0 )
f
Section 14.7 - Maximum and Minimum Values
Denition. A function f (x, y ) has a local maximum at (a, b) if f (x, y ) f (a, b) when (x, y ) is
near (a, b). The number f (a, b) is called a local maximum
Section 14.4 - Tangent Planes and Linear Approximations
Let z = f (x, y ) be a function with continuous partial derivatives and S be the graph of f .
P = (x0 , y0 , z0 )
C1 is the trace of f in the pl
Section 16.4 - Greens Theorem
Let C be a simple closed curve traversed once by a position vector r(t) = x(t), y (t) for a t b.
Let D be the region in the xy -plane enclosed by C .
Denition. We sometim
Section 16.3 - The Fundamental Theorem for Line Integrals
The Fundamental Theorem of Calculus. If F is continuous on [a, b] then
b
F (x) dx = F (b) F (a)
a
Theorem. Let C be a smooth (or piecewise smo
Section 15.4 - Double Integrals in Polar Coordinates
Polar Coordinates (r, )
r is the signed distance from the origin
is the angle measured counter-clockwise
from the positive x-axis
x = r cos()
y =
Section 16.2 - Line Integrals
Arc Length
Let C be a smooth curve traced out by the
position vector r(t) for a t b.
Plane curve:
r(t) = x(t), y (t) , a t b
Space curve:
r(t) = x(t), y (t), z (t) , a t
Section 14.8 - Lagrange Multipliers
Example 1. Find the absolute maximum and minimum values of f (x, y ) = x2 + 2y 2 on D =
cfw_(x, y ) : x2 + y 2 1.
In the interior of the disk: x2 + y 2 < 1
fx = 2x
Section 12.6 - Cylinders and Quadric Surfaces
Denition. The traces (or cross-sections) of a surface are the curves of intersection of the surface
with planes parallel to the coordinate planes.
Plane S
Section 15.1 - Double Integrals Over Rectangles
Denite Integrals of Functions of One Variable
If f (x) 0 then
a and b.
b
a f (x)dx
represents the area under the curve y = f (x) above the x-axis betwee
Section 15.3 - Double Integrals over General Regions
Denition
Let f (x, y ) be a function of two variables and suppose we want to nd the double integral
f (x, y ) dA
D
over a general bounded region D,
Section 15.2 - Iterated Integrals
Let z = f (x, y ) be a function of two variables dened on a rectangle R = [a, b] [c, d].
The double integral
f (x, y ) dA is the
R
signed volume of the region between
Section 12.5: Equations of Lines and Planes
Equations of Lines
To nd the equation of a line, we need to know:
1. a point on the line
2. the direction of the line
In R2 , the slope m =
rise
run
=
y2 y1
Chapter 15 Review
Double Integrals
Let z = f (x, y ) be a function of two variables dened on a rectangle R = [a, b] [c, d].
The double integral
f (x, y ) dA is the
R
signed volume of the region betwee
Chapter 16 Review
Vector Fields
A vector eld is a function F that assigns a vector
F to each point.
F (x, y ) = P (x, y ), Q(x, y )
or
F (x, y, z ) = P (x, y, z ), Q(x, y, z ), R(x, y, z )
A vector el
Chapter 13 Review
Vector Functions: r(t) = f1 (t), f2 (t), . . . , fn (t)
where fi : X R are real-valued functions called the component functions of r.
Limit of a vector function
lim r(t) = lim f1 (t)
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Practice problems from old exams for
math 233
William H. Meeks III
October 26, 2012
Disclaimer: Your instructor covers far more materials that we can possibly t into a four/ve questions exams. These p
Practice problems from old exams for
math 233
William H. Meeks III
October 26, 2012
Disclaimer: Your instructor covers far more materials that we can possibly t into a four/ve questions exams. These p