Vector Functions
We first saw vector functions back when we were looking at the Equation of Lines. In that
section we talked about them because we wrote down the equation of a line in in terms of a
vector function (sometimes called a vector-valued functio
Relative Minimums and Maximums
In this section we are going to extend one of the more important ideas from Calculus I into
functions of two variables. We are going to start looking at trying to find minimums and
maximums of functions. This in fact will be
Limits
In this section we will take a look at limits involving functions of more than one variable. In
fact, we will concentrate mostly on limits of functions of two variables, but the ideas can be
extended out to functions with more than two variables.
B
Partial Derivatives
Now that we have the brief discussion on limits out of the way we can proceed into taking
derivatives of functions of more than one variable. Before we actually start taking derivatives of
functions of more than one variable lets recal
Spherical Coordinates
In this section we will introduce spherical coordinates. Spherical coordinates can take a little
getting used to. Its probably easiest to start things off with a sketch.
Spherical_G1
Spherical coordinates consist of the following thr
Interpretations of Partial Derivatives
This is a fairly short section and is here so we can acknowledge that the two main interpretations
of derivatives of functions of a single variable still hold for partial derivatives, with small
modifications of cour
Equations of Lines
In this section we need to take a look at the equation of a line in . As we saw in the previous
section the equation does not describe a line in , instead it describes a plane. This doesnt mean
however that we cant write down an equatio
Tangent, Normal and Binormal Vectors
In this section we want to look at an application of derivatives for vector functions. Actually,
there are a couple of applications, but they all come back to needing the first one.
In the past weve used the fact that
1. State Stokestheorem.
Let S be an oriented piecewise-smooth surface that is bounded by a
simple, closed, piecewise-smooth boundary curve C with positive orientation. Let F be a vector eld whose components have continuous
partial derivatives on an open r
1. Evaluate
ZZ
the origin.
ex
2 +y 2
dA where D is the disk of radius 2 with center at
D
R2 R2
R 2 h1
r2
r2
ir=2
In polar coordinates this is 0 0 e r dr d = 0 2 e
d =
r=0
R
2
1
(e4 1) d = (e4 1).
2 0
ZZ
2. Evaluate
sin y 2 dA where D is the triangle with
1. Let F (x; y) be a function whose partial derivatives take on the following
values:
(0; 3) ( 1; 2) (0; 1) (3; 0)
Fx
1
5
3
2
Fy
2
3
4
1
Let g (t) = F (t2
t). Compute g 0 (0) and g 0 (1).
1; 2
We have dF = Fx dx + Fy dy = Fx 2t dt
g 0 (t) =
so g 0 (0) =
2
1. State Greens theorem.
Let D be the region bounded by a positively oriented, picewise-smooth,
simple closed curve C in the plane. If P and Q have continuous partial
derivatives on an open region containing D, then
ZZ
Z
P dx + Q dy =
(Qx Py ) dA
C
D
2. L
1a. Convert (4; 7 =4; =6) from spherical to rectangular coordinates.
7
7
4 sin cos ; 4 sin sin ; 4 cos
6
4
6
4
6
=
=
p
p ! p !
4 2 4
2
4 3
;
;
2 2 2
2
2
p
p p
2;
2; 2 3
1b. Convert ( 1; 2; 3) from rectangular to cylindrical coordinates.
q
p
r = ( 1)2 + 22
1. Find the directional derivative of the function
(a) ln (x2 + y 2 ) at the point (1; 2) in the direction of the vector h 1; 3i.
2x
2y
which is equal to h2=5; 4=5i
r ln (x2 + y 2 ) =
; 2
2
2
x + y x + y2
at the point (1; 2).
p
The unit vector in the dire
1. Find the unit tangent vector to the curve
The derivative is
1
p ;
2 t
p
1 3t2
;
t2 1 + t3
t; 1=t; ln (1 + t3 ) at t = 1.
.
At t = 1, this is h1=2; 1; 3=2i. The length of this vector is
p
1p
1=4 + 1 + 9=4 =
14
2
so the unit tangent vector is
h1=2; 1; 3=
1. What is a conservative vector eld? Give an example of a vector eld
that is not conservative.
A vector eld F is conservative if F = rf for some scalar function f .
See page 1061 of the text. Dont write down everything youve ever
heard about conservative
1. Find the local maximum and minimum values, and the saddle points,
of the function f (x; y) = xy 2 x2 2y 2 .
rf = hy 2 2x; 2xy 4yi so the critical points are obtained by solving
the equations y 2 2x = 0 and 2xy 4y = 0. From the second equation
we get 2y
1. Dene the dot product of two vectors.
The dot product of two vectors ha1 ; a2 ; a3 i and hb1 ; b2 ; b3 i is the scalar
a1 b 1 + a2 b 2 + a3 b 3 .
2. Dene what the cross product of two vectors is.
The cross product of two vectors ha1 ; a2 ; a3 i and hb1
1. Let F (x; y) =
Z
y2
arctan t dt. Find Fx (2; 3) and Fy (2; 3).
x
Use the fundamental theorem of calculus to compute Fx (x; y) = arctan x
and Fy (x; y) = 2y arctan y 2 . So Fx (2; 3) = arctan 2 and Fy (2; 3) =
6 arctan 9.
2. Suppose x2 2y 3 + 3z 4 = 6 d