In exercises 1-8, nd a parametric representation of the
surface.
1. z=3x+4y 2. x2+y2+z2=4
3. x2+y2-zz=l 4. xZy2+z2=4
5. Theportionofx2+y2=4fromz=0toz=2
6. Theportionofy2+zz=9fromx=1tox=1
7. The portion of z=4x2 y2 above thexy-plane
8. The portion of z=x2+
Vector-Valued Functions DEFINITION I.I
A vector-valued function [(1) is a mapping from its domain D C IR to its range
R C V3, so that for each I in D, r0) = v for exactly one vector v 6 V3. We can
always write a vector-valued function as
1'0) = f (0i + 30
THEOREM I.I
For any vectors a, b and c in V2, and any scalars d and a in R, the following hold:
(1) a + b: b + a (commutativity)
(ii) a + (b + c)_ (a + b) + c (associativity)
(zero vector)
(additive inverse)
(v) d (a + b) = da + db (distributive law)
(vi)
Lets look for the line that passes through the point P1(x1, y1,z1) and that is parallel
to the position vector a = (a1, a2, (13) (see Figure 10.39). For any other point P(x, y, z) on
the line, observe that the vector m will be parallel to :1. Further, two
Multiple Integrals f (0:)
FIGURE I3.Ia FIGURE I3.Ib
Approximating the area on the Area under the curve
subinterval [n-1, xi] FIGURE I3.4b FIGURE I 3.44: FIGURE I3.4d
Approximating the volume above Approximate volume Approximate volume DEFINITION I.2
For
CONSTRAINED OPTIMIZATION
AND LAGRANGE MULTIPLIERS THEOREM 8.I
Suppose that f (x, y, z) and g(x, y, z) are functions with continuous rst partial
derivatives and Vg(x, y, z) 75 0 on the surface g(x, y, z) = 0. Suppose that either
(i) the minimum value of f
EXAMPLE 2.6
Evaluate
lim
(x,y)(0,0)
Proving That a Limit Exists
x2y
.
x 2 + y2
Solution As we did in earlier examples, we start by looking at the limit along several
paths through (0, 0). Along the path x = 0, we have
0
= 0.
(0,y)(0,0) 0 + y 2
lim
Similar
r(t) = h f (t), g(t), h(t)i,
where t represents time and where t [a, b]. We observed in section 11.2 that the value of
r (t) for any given value of t is a tangent vector pointing in the direction of the orientation
of the curve. We can now give another in
Functions of Several Variables
and Partial Differentiation
that depend on more than one variable, that is, functions whose domain is multidimensional.
A function of two variables is a rule that assigns a real number f (x, y) to each ordered
pair of real n
MATH 11102 Dr. Feng Quiz #2 9 Feb 2016
Name: V ID:
ESE DS I I1
0 Due 25 minutes after the beginning of class
a Only basic scientic (non-graphing) calculators allowed
go Whomfewerit
Box or circle your nal answer
0 Be sure to check the backs of pages for al
Chapter 10
Laplace Transform
Methods
p. 576 to p. 600
Chapter 10
Section 10.1.
Laplace Transforms
p. 576 to p. 586
Laplace Transforms
Solving linear differential equations with constant
coefficients.
Systematic way of solving initial value problems.
So
Chapter 7
Linear Systems
of
Differential Equations
p. 396 to p. 463
Chapter 7
Section 7.1.
First-Order Systems
of
Differential Equations
p. 396 to p. 404
Motivation
Many natural phenomena are complex.
To describe them accurately, we often need more
than o
Differential Equations
&
Linear Algebra
MATH - 211
Book for the course
Differential Equations & Linear Algebra
Third Edition
by
C. Henry Edwards & David E. Penney
Pearson International Edition
Chapter 1.
First-Order Ordinary Differential
Equations
1.1. Di
Chapter 6
Eigenvalues and Eigenvectors
p. 366 to p. 395
Context
Chapters 3 & 4:
Solving
Ax = b
Ax = 0
Constructing a basis for Row(A), Col(A), Nul(A)
Chapter 6:
For a matrix A , need to find l and x such that
Ax =lx i.e. (A lI)x = 0
A is a square n x
Chapter 7 Short Revision
Linear Systems
of
Differential Equations
Section 7.1
Section 7.2
Section 7.3
Section 7.5
p. 396
p. 407
p. 418
p. 446
to
to
to
to
p. 404
p. 417
p. 431
p. 451
Examples of Coupled Systems
x ' 3 x y
y ' 2 x 2 y 40sin(3t )
x' ' 6tx 2 y
Chapter 7
Linear Systems
of
Differential Equations
p. 396 to p. 463
Chapter 7
Section 7.3.
The Eigenvalue Method
for
Linear Systems
p. 418 to p. 431
Homogeneous First-Order Linear System with
Constant Coefficients
x1' a11 x1 a12 x2
a1n xn
x2' a21 x1 a22
Chapter 5
Higher-Order Linear Differential Equations
Second-Order Linear Differential Equations
p. 286 to p. 351
Recall: First-Order Linear Differential Equation
dy
P x y Q x
dx
P x ,Q x : known continuous functions x I
Integrating factor:
General soluti
LINEAR SYSTEMS
AND
MATRICES
Systems of Linear Equations : Recall
Chapter 3
3.1 Introduction to Linear System : p.147 to p.154
p.155 Problems: 1 to 22
3.2 Matrices and Gaussian Elimination : p.156 to p.165
p.165 Problems: 1 to 27
3.3 Reduced Row-Echelon Ma
Linear Algebra & Diff. Equations
Summer 2014
Student ID:
Quiz 2
Student Name:
1) Let
,
Find the following:
(a) The row reduced echelon matrix that is row equivalent to A.
[
]
(b) A basis for the column space of A.
[ ] and
[ ]
(c) A basis for the row space
Linear Algebra & Diff. Equations
Summer 2014
Student ID:
Quiz 4
Student Name:
Question 1 Find the general solution for the following differential equation:
The characteristic equation:
(
)
Then the general solution is:
( )
(
)
Question 2 Solve the followi
Linear Algebra & Diff. Equations
Summer 2014
Student ID:
Quiz 1
Student Name:
1) Consider the following linear system:
a) write the system in a matrix form.
[
][ ]
[ ]
b) use matrix multiplication to check whether the vector (6,-4,1) satisfies the
homogen