Second Order Partial Derivatives; the Hessian
Matrix; Minima and Maxima
Second Order Partial Derivatives We have seen that the partial derivatives of a dierentiable function (X) = (x1, x2, ., xn) are
again functions of n variables in their own right, deno
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Math 3214 Project; Due: 04/30/12.
Basic Setting:
We consider the elliptical region
R R2 ; R = cfw_x, y
4x2 + y 2 4 .
The surface is the portion of the graph of the function
z = (x, y ) = 1
y2
x2
4
4
lying above R in R3 . The curve C is the boundary
Math 3214: Practice Exam 2 and Take Home
Problem to be Turned in with 4-4-12 Exam
1.
Let (x, y ) = y sin x + x cos y and let C be the curve described parametrically by x = t, y = t2 , 0 t 1. With F (X ) = F (x, y ) = (x, y ), compute
the line integral C F
Practice Exam 1: Math 3214; Spring, 2012
1.
Let P = ( 1 1 1 ) , Q = ( 2 3 2 ) , R = ( 1 1 1 ).
a)
Find an equation for the plane passing through Q and orthogonal to Q P .
b)
Let L be the line through P in the direction of R and let ( x y z ) be the
point
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Quiz 4, Math 3214, Spring 2012
Due in 418 McBryde, Friday, April 27, 2012. Use this page as
cover sheet and staple, please. This quiz is worth 32 points.
Use spherical coordinates to compute the integral
1.
A
x 2 + y 2 + z 2 dx dy dz
where A is the h
Vector Functions of a Vector Variable
General Definition of a Vector Function A vector function F
is a mathematical rule of correspondence, expressed via Y = F (X),
which assigns to each vector X in a region D(F ) Rn , called the
domain of the function F
Arc Length; Integration with Respect to
Arc Length
Arc Length
Let us suppose we have a curve in Rn :
x1 (t)
x2 (t)
C : X(t) =
, a t b,
.
.
xn (t)
X(t) being continuously dierentiable with respect to t. If we fix an
initial value of t, say t = a, and
1
Notes for Numerical Analysis
Math 4445
by
S. Adjerid
Virginia Polytechnic Institute
and State University
(A Rough Draft)
2
Chapter 1 4
Nonlinear Algebraic Equations
In this chapter we will discuss numerical methods for nding zero of algebraic
problems f
Chapter 3
Iterative Methods for Algebraic
Systems
3.1
Introduction
For large systems Ax = b having for instance n = 106 equations Gaussian
elimination for a full matrix A requires O(2n3 /3) oating operations. On a
teraop computer, 1012 oating-point operat
Chapter 2
Solving Linear Systems
2.1
Review
We start with a review of vectors and matrices and matrix-matrix and matrixvector multiplication and other matrix and vector operations.
Vectors operations and norms:
Let x = [x1 , x2 , , xn ]t and y = [y1 , y2
1
Notes for Numerical Analysis
Math 4445
by
S. Adjerid
Virginia Polytechnic Institute
and State University
(A Rough Draft)
2
Chapter 1
Error Analysis
1.1
Introduction
In Science and Engineering there are several sources of errors that aect
scientic result
Notes for Numerical Analysis
Math 4445
by
S. Adjerid
Virginia Polytechnic Institute
and State University
(A Rough Draft)
1
2
Chapter 1
Solving Eigenvalue Problems
1.1
Basic facts about eigenvalues
Matrix eigenvalue problems have many important application
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Classroom ID
Quiz 2: Math 3214; Due: Monday, Feb. 13, 2012
Note: The symbol
is attached.
indicates the transpose of the vector to which it
1. Let C be the curve in R 2 given by y = x3 , 0 x 1/2.
Compute the arc length integral
C
2.
f (x, y ) ds,
f (x
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Exam 2: Math 3214; April 4, 2012
1. Let (x, y ) = y cos x + x sin 2y and let C be the curve described parametrically by x = t2 , y = t3 , 1 t 2. With
F (X ) = F (x, y ) = (x, y ), compute the line integral C F (X )dX .
2. Dene R = cfw_(x, y ) (x 1)2
Scalar Valued Functions of Several Variables;
the Gradient Vector
Scalar Valued Functions
Let us consider a scalar (i.e., numerical, rather than
vector) valued function of n variables:
y = (X ) = (x1 , x2, ., xn), X D,
where the domain D is a region in Rn