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 function F that assigns to
each (x, y) (or (x, y, z) a two (
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 is continuous on a Type I region D such that
D = cfw_(x,
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 (x, y) 0 for all (x, y) in a closed and bounded region
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 on a rectangular box
B = cfw_(x, y, z) : a x b, c y d,
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/ 23
Cylindrical Coordinates
Consider a point P in space.
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 work W done by a force F in moving an
object is given
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 smooth curve dened by a vector function
R(t) = x(t) + y(t
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
Recall: FTOC
Recall from Math 53:
Theorem
Let f (x) be a fun
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:
a1 , a2 , a3 , a4 , . . . , an , . . .
Every member of
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
n
exists for some real number L. Otherwise, it is dive
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 approximations for functions of n variables
For a function
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
Divergence Test
If lim an = 0, then the series
n
R
an is
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 integral is related to the arclength,
surface integrals are re
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 of the form
cn (x a)n .
n=0
We are able to obtain power
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) bn or
n=1
(1)n1 bn
n=1
satises
(i) lim bn = 0;
n
(ii) b
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 series
an converges if its sequence of partial sums
n=1
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 (x, y) is integrable over the rectangular region
R = [a,
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 is given by a vector function
R(t) = f (t) + g(t) + h(t)
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) Find the directional derivative of T at (1, 0) on the dire
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 first DC Power System was the Pearl Street
Station, built
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 Philippines - Diliman
The Sinusoidal Function
The sinusoid is de
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 Pedrasa, May 2012
Elec Ckts 9ed, Chapter 7
Differential
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:
Course:
College:
First Long Exam (Makeup)
First Semester, AY 20152016
Physics
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 Capacitor Characteristics
Fundamental Inductor Characteristi
University of the Philippines
College of Science
Physics 72
Set A
First Long Exam
Second Semester, AY 20142015
Name:
Instructor:
Section/Class Schedule:
Student Number:
Course:
College:
First Long Exam
Second Semester, AY 20142015
Physics 72
INSTRUCTIONS:
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 G. Sison, Mar 7, 2004
Revised Jhoanna Pedrasa
Sept 2005
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
First Semester, AY 20152016
Physics 72
INSTRUCTIONS: C
Nomenclature of
Hydrocarbons
Alkanes, Alkenes, and Alkynes
Alkanes (CnH2n+2)
Saturated hydrocarbons
Consist of (head-on) overlap of carbon sp3 hybrid orbitals
Occasionally called aliphatic i.e. fat
McMurry (2010). Organic Chemistry. 8th Ed
Straight lin
Lagrange Multiplier
Math 55 - Elementray Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Lagrange Multiplier
1/ 7
Recall
R
R
R
(a, b) is a critical point of f (x, y) if f (a, b) = 0
(fx (a, b) = fy (a, b) = 0) or either
Tangent Planes
Math 55 - Elementray Analysis III
Institute of Mathematics
University of the Philippines
Diliman
Math 55
Tangent Planes
1/ 12
Recall
For f (x, y), the gradient of f is
f (x, y) = fx (x, y), fy (x, y)
and the directional derivative of f at (