Vector Fields
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Vector Fields
1/ 20
Vector Fields
Denition
A vector eld on R2 (or R3 ) is a funct
Parametric Surfaces and Surfaces of Revolution
Math 55 - Elementray Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Parametric Surfaces
1/ 15
Recall
A curve in R3 i
Double Integrals over General Regions
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Double Integrals over General Regions
1/ 14
Recall
If f (
Double Integrals in Polar Coordinates
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Double Integrals in Polar Coordinates
1/ 12
Recall
If f i
Applications of Double Integrals
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Applications of Double Integrals
1/ 21
Volume of a Solid
Let f
Triple Integrals
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Triple Integrals
1/ 12
Triple Integral over a Box
Suppose f (x, y, z) is dened
Triple Integrals in Cylindrical
& Spherical Coordinates
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Cylindrical & Spherical Coordinates
1/
Line Integrals of Vector Fields
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Line Integrals
1/ 7
Work Along a Curve Problem
We know that the
Line Integrals of Scalar Fields
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Line Integrals
1/ 16
Curtain Area Problem
Problem
Let C be a sm
The Fundamental Theorem of Line Integrals
and Greens Theorem
Math 55 -Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
FTLI and Greens Theorem
1/ 19
Recal
Sequences
Math 55 - Elementray Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Sequences
1/ 17
Sequences
A sequence can be thought of as an ordered list of numbers:
Series
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Series
1/ 20
Recall
A sequence cfw_an = cfw_a1 , a2 , . . . is convergent if
lim an = L
Chain Rule and the
Gradient
November 19, 2007
The gradient
Denition: Let f be a function of n variables: x1, x2, . . . , xn, then the gradient is
f (x1, . . . , xn) =
f f
f
,
.,
x1 x2
xn
1
Linear appr
Convergence Tests
(for Series with Positive and Negative Terms)
Math 55 - Elementary Anlaysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Convergence Tests II
Recall
R
D
Surface Integrals
Math 55 - Elementray Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Surface Integrals
1/ 16
Surface Integrals
In the same way that a line integra
Taylor and Maclaurin Series
Math 55 - Elementary Anlaysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Taylor and Maclaurin Series
Recall
A power series centered at a is
Power Series
Math 55 - Elementary Anlaysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Power Series
Recall
R
Alternating Series Test
If a given alternating series
n
(1)
Convergence Tests
(for Series with Non-negative Terms)
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Convergence Tests I
1/ 21
Recall
R
The s
Directional Derivatives and the Gradient Vector
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Directional Derivatives & Gradient
1/ 21
Recall
Relative and Absolute Extrema
Math 55 - Elementray Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Relative and Absolute Extrema
1/ 15
Recall
R
R
R
For a function f
Double Integrals over Rectangles
Math 55 - Elementary Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Double Integrals over Rectangles
1/ 15
Recall
If f (x) is dene
MATHEMATICS 55
Sample First Long Exam
Do as indicated. Show complete and clear solution to get full points.
1. Let the temperature at any point (x, y) be given by T (x, y) = 3x2 + 2xy + 2y 3 .
(a) Fin
Chapter 8
Transformers
Artemio P. Magabo
Professor of Electrical Engineering
Electrical and Electronics Engineering Institute
University of the Philippines - Diliman
Direct Current Power System
The fi
Sinusoidal Steady-State
Analysis
Artemio P. Magabo
Professor of Electrical Engineering
Edited by Michael Pedrasa, June 2013
Electrical and Electronics Engineering Institute
University of the Philippin
Chapter 4
Introduction to Differential
Equations
Transient Analysis of
First-Order Networks
Electrical and Electronics Engineering Institute
University of the Philippines - Diliman
Revised by Michael
University of the Philippines
College of Science
Physics 72
Set A
First Long Exam (Makeup)
First Semester, AY 20152016
Name:
Instructor:
Section/Class Schedule:
Student Number:
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College:
First
Chapter 3
Inductors, Capacitors and
RLC Circuits
Electrical and Electronics Engineering Institute
University of the Philippines - Diliman
Revised by Michael Pedrasa, May 2012
Outline
Fundamental Capac
University of the Philippines
College of Science
Physics 72
Set A
First Long Exam
Second Semester, AY 20142015
Name:
Instructor:
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Course:
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First Long Exa
Chapter 9
Resonance and Electric
Filters
Artemio P. Magabo
Professor of Electrical Engineering
Electrical and Electronics Engineering Institute
University of the Philippines - Diliman
Revised by Luis
University of the Philippines
College of Science
Physics 72
Set A
First Long Exam
First Semester, AY 20152016
Name:
Instructor:
Section/Class Schedule:
Student Number:
Course:
College:
First Long Exam