17.1 Parameterized Curves
Curves in a plane may be represented by two parametric equations:
x = f ( t ) and y = g ( t ) . For a given value of t, the point ( x, y ) = ( f ( t ) , g ( t ) ) locates a
specific point in the plane. As t varies, the points ( x
16.1 The Definite Integral of a Function of Two Variables
For functions of one variable, definite integrals are constructed as the limit of Riemann
n
b
i =1
a
Sums: lim f ( x i )x = f ( x)dx where x 0 refers to decreasing the width of the
x 0
rectangles i
16.5 Integrals in Cylindrical and Spherical Coordinates
Just as some double integrals are easier to compute as iterated integrals in polar
coordinates, for some triple integrals it is advantageous to work in either cylindrical or
spherical coordinates. In
17.3 Vector Fields
A vector field is a function whose input is a point in some vector space and whose output
is a vector quantity. For example, given the function:
f ( x, y ) or g ( x, y, z ) , f and g are vector fields because it assigns a vector quantit
17.2 Motion, Velocity and Acceleration
Velocity is a vector quantity v and so it has magnitude and direction.
r (t )
Average velocity is defined as before as
where r ( t ) is the vector function defining
t
the position of an object in space. Instantaneous
16.2 Iterated Integrals
In the previous section we extended the notion of Riemann Sums to double integrals.
Iterated integrals enable us to find exact values for some integrals involving several
variables through iteration.
Reconsider the fox population d
16.3 Triple Integrals
Triple Integrals are an extension of the notion of Riemann Sums to a domain that is a
volume and a range that is a function of location within the volume:
If w = f ( x, y, z ) is such a function, then we can construct a Riemann Sum b
16.4 Double Integrals in Polar Coordinates
f ( x, y )dA by overlaying a grid on the
coordinate system and forming the limit of a Riemann Sum lim f ( x , y )xy by
Recall that double integrals were defined as
R
x 0
y 0 i , j
i
j
shrinking the rectangular g
Homework #8 Solutions
Homework 8: Page 813-815: 4,10,23,33,41
4.) For the function f ( x, y ) = e 2 xy , calculate all four second order partial derivatives and
check that f xy = f yx
Solution:
f ( x, y ) = e 2 xy f x = 2 ye 2 xy f y = 2 xe 2 xy so:
f xx
Homework #6 Solutions
Page 785-789 #10,15,23,34,38,66,76,C1,C2
10.) Find the gradient of the function: f ( x, y ) = ln ( x 2 + y 2 )
f f
2x
2y
Solution: grad ( f ) = f = i +
j= 2
i+ 2
j
2
x
y
x +y
x + y2
15.) Find the gradient of f ( x, y ) = x 2 y + 7
Homework #7 Solutions
Homework 7: Page 803-806 #2,11,23,29
2.) Given z = x sin y + y sin x, x = t 2 and y = ln t
dz z dx z dy
1
Find:
= + = ( sin y + y cos x )( 2t ) + ( x cos y + sin x ) =
dt x dt y dt
t
x cos y sin x
2t sin y + 2ty cos x +
+
t
t
sin (
Review #1 With Some Examples and Solutions
Calculus III Multivariable
1.) Representing, manipulating, and evaluating functions of several variables
a.) Given f x, y or f x, y, z described algebraically, be able to represent the function
with a table of va
Homework #3 Solutions
Page 740-743 #3,6,7,9,18,53,C1,C2
For Exercises 19, perform the following operations on the given 3-dimensional vectors.
a =2j+k
b = 3i + 5 j + 4k
c =i+6j
y = 4i 7 j
z = i 3j k
3.) ab = 10 + 0 + 4 = 14
6.) a c + y =
14.3 Local Linearity and the Differential
Linear approximation for functions of one variable amounts to finding a linear function
that approximates the function near a point a in the domain of f ( x) . We accomplished
this by using the geometry to find a
12.2 Graphs of Functions of Two Variables
A function of two variables is an ordered triple ( x, y, z ) where each input is
an ordered pair and the output z is a unique output related to each input.
so we represent the function f ( x, y ) = z by either not
12.1 Functions of two variables
A function of two variables is an ordered triple ( x, y, z ) where each input is
an ordered pair and the output z is a unique output related to each input.
Just as in a function of two variables where we commonly use the or
14.2 Computing Partial Derivatives
z
z
and f y ( x, y ) =
are calculated the same way as
x
y
derivatives of a single variable because one of the two variables is held constant.
Partial derivatives f x ( x, y ) =
The derivative function may be thought of a
13.4 Cross Product
We introduce a second type of vector product, the cross product.
The cross product is denoted by v w . Unlike the dot product, the cross product is a
vector quantity. The cross product has a geometric and an algebraic definition which
14.4 Gradients and Directional Derivatives in the Plane
Recall that the plane tangent to f ( x) at ( a, b ) is given by the linear function
L( x) = f x ( a, b )( x a ) + f y ( a, b )( y b ) + f ( a, b )
where f x ( a, b ) is the slope in the x-direction a
14.1 The Partial Derivative
Consider the table of values of the function of two variables that describes the
temperature T ( x, y ) of a unevenly heated metal plate in two coordinates.
3 85
90
110 135 155 180
2 100 110 120 145 190 170
1 125 128 135 160 17
13.3 The Dot Product
Recall the following mathematical properties of vectors:
Commutative Law: v w w v
uv wu vw
Vector addition Associative Law:
Scalar multiplication Associative Law: v v
Scalar addition Distributive Law: v v v
Vector addition Distr
Review #2 Part 2
Calculus III Multivariable
Section 14.4 Gradients and the directional derivative in the plane
x y
1.) Given: f x, y
Find the directional derivative at P = (1,-2) in the direction:
2
x
1
i.) v 3i 2 j
ii.) v i 4 j
iii.) Find the direct
12.3 Contour Diagrams
The graph of a function f ( x, y ) is a surface with the input being an ordered pair
( x, y ) and the output being the value of the function at the point ( x, y ) . The graph is a
surface in three space. To create an image of the sur
12.5 Functions of Three Variables
Functions of one variable are mappings of an input x to a unique output y f x . These
are lines in two space with zero thickness such as a straight line, parabola, hyperbola,
semicircle.
Functions of two variables are map
12.4 Linear Functions
Linear functions in one variable y = f ( x) are defined by the notion that the rate of change,
y
is constant. For linear functions of two variables z = f ( x, y ) , the plot is a surface
x
which is a flat plane. The rate of change wo
13.3 The Dot Product
Recall the following mathematical properties of vectors:
Commutative Law: v + w = w + v
u+v +w=u+ v+w
Vector addition Associative Law:
Scalar multiplication Associative Law: v = ( ) v
Scalar addition Distributive Law: ( + ) v = v
13.2 Vectors in General
Besides displacement vectors there are many quantities that have both magnitude and
direction and hence are vector quantities. For example, velocity is a vector quantity. The
magnitude of velocity is commonly called speed. Speed is
13.1 Displacement Vectors
The displacement vector from one point to another is an arrow with its tail at the first
point and its tip at the second point. The magnitude of a displacement vector is the
distance between the points, represented by the length