Section 11.3
The Dot Product of Two Vectors
Definition of Dot Product
Theorem 11.4 Properties of the
Dot Product
Theorem 11.5
Angle Between
Two Vectors and
Figure 11.24
Alternative form of dot product
and Figure 11.25
Definition of Orthogonal Vectors
Figu
Section 11.2
Space Coordinates and
Vectors in Space
Next, we want to extend the
ideas of vectors into three
Before we can dimensions we need to
do that, though,
be able to identify points in three-space.
In this text we use the Righthanded system.
common
Section 11.4
Definition of Cross Product of
Two Vectors in Space
Or you can think of it as the determinant of
the 3x3 matrix
ijl
u1 u2 u3
v1 v2 v3
One way to quickly find this
determinant
i
j
u1 u2
v1 v2
k
u3
v3
i
j
u1 u2
v1 v2
Form products along the dia
Calculus III Formulas
u v
cos =
uv
(u v )
proj v u = 2 v
v
W= F PQ
u v = u
V=
A=
T (t ) =
= r2 + z2
z
= arccos
2
2
r +z
a = Dt
T
b
s=
[]
v
[ x' (t )]
2
va
= aT =
v
2
a
K=
r ' (t )
b
r ' (t ) dt
a
y' '
[1 + ( y' ) ]
2
3
2
a (t ) N (t )
=2
v (t )
(
Section 12.5
Arc Length and Curvature
2
Example:
Sketch the space curve and find its length
over the indicated interval.
2
r(t)=(cost+tsint)i+(sint-tcost)j+t k
Definition of Arc
Length Function and
Figure 11.29
Curvature is the measure of how sharply a
cu
12.3
Velocity and Acceleration
Projectile Motion
Examples
A baseball player at second base throws a ball 90
feet to the player at first base. The ball is thrown at
50 miles per hour at an angle of 15 above the
horizontal. At what height does the player at
Section 11.1
Vector Valued Functions
Definition of Vector-Valued Function
How are vector valued functions
traced out?
In practice it is often easier to
rewrite the function.
Sketch the curve represented by the vectorvalued function and give the orientatio
11.7
Cylindrical and Spherical
Coordinates
The Cylindrical Coordinate System
1.
In a cylindrical coordinate system, a point P in
space is represented by an ordered triple( r , , z )
( r , ) is a polar representation of the projection
P in the xy-plane.
2.
11.6
Surfaces in Space
Definition of a Cylinder
Let C be a curve in a plane and let L be a line
not in a parallel plane. The set of all lines
parallel to L and intersecting C is called a
cylinder. C is called the generating curve (or
directrix), and the
Section 13.4
Differentials
Total Differentials
Uses for Differentials
1) To approximate the change in a function
value.
Evaluate f(1,2) and f(1.05, 2.1) and calculate the
change in z, z and then use the differential
dz to approximate z
Use in approximatin
Section13.5
ChainRulesforFunctionsofSeveral
Variables
Theorem13.6ChainRule:
OneIndependentVariable
andFigure13.39
Examples
dw
FindusingtheappropriateChainRule.
dt
y
1) w = ln
x
w = xyz
x =t
y = 2t
2
2)
z =e
t
x = cos t
y = sin t
Example
Differentiateimpli
Section 14.7
Triple Integrals in Cylindrical and
Spherical Coordinates
In general, to convert from
rectangular triple integrals to
cylindrical
2 g 2 ( ) h2 ( r cos , r sin )
f( x, y, z)dV =
Q
f (r cos , r sin , z)rdzdrd
1 g1 ( ) h1 ( r cos , r sin )
T
Section 14.5
Surface Area
Figure 14.42 and Figure 14.43
Definition of Surface Area
Example:
Find the area of the surface given by
z=f(x,y) over the region R.
2
1) f(x,y)= y
R: square with vertices (0,0), (0,4), (3,4) (3,0)
9 + x2 y2
2) f(x,y)=
x 2 + y 2 4
Section 14.3
Changes of Variable:
Polar Coordinates
Some double integrals are MUCH
easier to evaluate using polar
coordinates
Figure 14.28
Section 14.4
Center of Mass and Moments of
Inertia
Definition of Mass
of a Planar Lamina
of Variable Density
and Fig
Section 13.10
Lagrange Multipliers
Theorem 13.19 Lagrange's
Theorem
Method of Lagrange
Multipliers
Example
Use Lagrange Multipliers to find the indicated
extrema, assuming that x and y and z are
positive.
1) Minimize f ( x, y ) = 3 x + y + 10
Subject to t
Section 13.8
Extrema of Functions of Two
Variables
Peaks and Valleys
Definition of Relative
Extrema and Figure
13.64
Theorem 13.16 Relative Extrema
Occur Only at Critical Points
As in single
variable
calculus, not
all critical
values yield
extrema
To help
Section 13.7
Tangent Planes and Normal Lines
Definition of Tangent Plane and
Normal Line
Find an equation of the tangent plane to the
surface at the indicated point.
x + y + z = 36
2
2
2
(2, -2, 4)
If the surface is given by z=f(x,y)
you can define the f
Section 13.6
Directional Derivatives and
Gradients
Figure 13.42, Figure 13.43, and
Figure 13.44
Find the directional derivative of the
function at P in the direction of v
1)
h( x, y ) = e sin y
2)
g ( x, y, z ) = xye
P (1, )
2
x
z
v=-I
P(2,4,0) Q(4,3,1)
D
Section 13.3
Partial Derivatives
To find f x you consider y constant and
differentiate with respect to x.
Similarly, to find f y you hold x
constant and differentiate with respect
to y.
Examples:
f ( x, y ) = x 3 y + 7
xy
z= 2
2
x +y
2
2
Geometrically spe
Section 13.2
Limits and Continuity
Limits of functions of several
variables have the same
properties as do limits of
functions of single variables.
Examples:
( 5 x + y + 1)
( x , y ) ( 0 , 0 )
lim
lim
( x , y ) ( 0 , 0 )
1
2
2
x +y
Examples
Section 13.1
Introduction to Functions of
Several Variables
Examples:
f ( x, y ) = 4 x 4 y
2
V ( r , h) = r 2 h
f ( x, y ) = ln( xy 6)
2
The graph of a function of two
variables is the set of all points
(x,y,z) for which z=f(x,y) and (x,y)
is in the domai
Calculus III - 11.3 Velocity and Acceleration
Definition of Velocity and Acceleration
If x and y are twice-differentiable functions of t, and r is a vector-valued function given by r(t) = x(t)I + y(t)j,
then the velocity vector, acceleration vector and sp
Calculus III Section 11.1 Vector-Valued Functions
A function of the form r(t) = f(t)i + g(t)j + h(t)k is a vector-valued function where the component functions f, g,
and h are real-valued functions of the parameter t.
The domain is the intersection of the
Slide 1
_
_
_
Vectors in the Plane
Section 10.1
_
_
_
_
Slide 2
_
Why do we need vectors?
_
_
Some quantities can be characterized by a
single real number (these are called scalar
quantities) but others that involve both
magnitude and direction cannot. A
12.6Directional Derivatives and Gradients
Definition of Directional Derivative
Let f be a function of two variables x and y and let u= cos i + sin j be a unit vector.
Then the directional derivative of f in the direction of u, denoted by D u f ( x, y ) is
Slide 1
Stokess Theorem
CalculusI I I
Sect ion14.8
Slide 2
Orientation of a Surface
Consider the diagram on p. 1081
St okess Theorem givest herelat ionshipbet ween
asurfaceint egraloveranorient edsurfaceSand
alineint egralalongaclosedspacecurveC
form ingt
Slide 1
The Dot Product of Two Vectors
Section 10.3
Slide 2
Definition of Dot Product
The dot product of u = u1 , u2
is
u v = u1v1 + u2 v2
and v = v1 , v2
The dot product of u = u1 , u2 , u3 and v = v1 , v2 , v3
is
u v = u1v1 + u2 v2 + u3v3
Slide 3
Proper