Mathematics 264 Advanced Calculus Assignment 4
Due in class Thursday October 14, 2010
1. (a) Calculate the curl of F = rn (x i + y j), where r2 = x2 + y 2 , and n
is an integer, i.e. n = . . . , 2, 1, 0, 1, 2, . . .
(b) For each n for which curl F = 0, nd
Applications to PDE
Math 264 - Advanced Calculus for Engineers Lecture 10 - 5/23/17
In this lecture we apply the ideas from Fourier Series to study boundary value problems for the Laplace, Wave,
and Heat Equation. As before the main idea is to separate va
McGill University
Midterm examination
March 14, 2013
Advanced Calculus for Engineers
Math 264
March 14, 2013
Time: 8:35 - 9:55 AM
Prof. N. Sancho
Student name (last, first)
Student number (McGill ID)
INSTRUCTIONS
1. If you are not registered in this secti
McGill University
Midterm examination
March 15, 2013
Advanced Calculus for Engineers
Math 264
March 15, 2013
Time: 10:05 - 11:25 AM
Prof. R. Choksi
Student name (last, first)
Student number (McGill ID)
INSTRUCTIONS
1. If you are not registered in this sec
Intro to PDE
Math 264 - Advanced Calculus for Engineers
Lecture 8 - 5/17/17
In this lecture we begin discussing how to solve basic boundary value problems for the linear partial differential equations introduced in the last lecture. We start by first focu
Fourier Series
Math 264 - Advanced Calculus for Engineers
Lecture 9 - 5/18/17
In this lecture we introduce Fourier Series and conclude the solutions for the boundary value problems
considered in the previous lecture. Lets start by first defining a Trigono
Conservation Laws
Math 264 - Advanced Calculus for Engineers
Lecture 7 - 5/16/17
Introduction:
In this lecture, building upon our understanding of the various fundamental theorems of calculus in 2 and 3
dimensions, we introduce basic partial differential
FTC I
Math 264 - Advanced Calculus for Engineers
Lecture 5b - 5/9/17
Today we present the fundamental theorems of calculus for planar vector fields: Greens Theorem and the
planar Divergence Theorem. The proofs of the theorems are rather straightforward ap
Line Integrals
Math 264 - Advanced Calculus for Engineers
Lecture 3 - 5/3/17
Today we seek to understand how to compute integrals on curves. Next time we will study integration on
surfaces. We first consider line integrals with scalar functions f (x, y, z
Div, Grad, Curl
Math 264 - Advanced Calculus for Engineers
Lecture 5a - 5/9/17
Today we study differential operators for vector fields. Recall, for scalar functions of several variables we
defined the gradient operator, , as a natural generalization of th
FTC II
Math 264 - Advanced Calculus for Engineers
Lecture 6 - 5/11/17
Today we present the Stokes Theorem and some basic applications. The Stokes Theorem is a natural generalization of Greens theorem but to surfaces with boundary in R3 . In many ways this
Div, Grad, Curl
Math 264 - Advanced Calculus for Engineers
Lecture 5a - 5/9/17
Today we study differential operators for vector fields. Recall, for scalar functions of several variables we
defined the gradient operator, , as a natural generalization of th
Sturm-Liouville
Math 264 - Advanced Calculus for Engineers Lecture 12 - 5/25/17
In this lecture we generalize Fourier Series to more general orthogonal expansions and consider the main
results of Regular Sturm-Liouville Theory. In one dimensions we see th
Name and ID:
Math 264 Practice Midterm
1. Compute the line integral
R
C
F~ d~r for the following vector fields F~ and curves C:
a) F~ = (y, x) and C is the line segment starting at (2, 0) and ending at (0, 2)
b) F~ = (x, y, 2z) and C is the twisted cubic
Name and ID:
Math 264 Practice Midterm
1. Compute the line integral
R
C
F~ d~r for the following vector fields F~ and curves C:
a) F~ = (y, x) and C is the line segment starting at (2, 0) and ending at (0, 2)
b) F~ = (x, y, 2z) and C is the twisted cubic
Surface Integrals
Math 264 - Advanced Calculus for Engineers
Lecture 4 - 5/4/17
Remark 0.1. Today we turn our attention to studying surfaces and defining integrals on them. Before we
start we make some motivating remarks about what we have been doing this
Review
Math 264 - Advanced Calculus for Engineers
Lecture 1 - 5/1/17
In todays lecture we discussed some topics from multivariable calculus that will be useful for the rest of the
course. One topic that we did not discuss in lecture was Mutlivariable Inte
Math 264 Fall 2014 ~ Written Assignment 3
Due date: Monday Oct. 20 by 4:25pm. AH students must, hand in this assignment at the
room of Tutorial Section 3.
Instructions: All students in the course should hand in this assignment as in~
structed above, regar
Math 264 Fall 2014 Written Assignment 2
Due date: Monday Sept. 29 by 5:25pm. All students must hand in this assignment at the
room of Tutorial Section 6.
Instructions: All students in the course should hand in this assignment as instructed above, regardle
616
Chapter I 0. Partial Differential Equations and Fourier Series
EXAMPLE
1
that the innite series of Eq. (19) converges and also satises Eqs. (1) and (4). To
satisfy the initial condition (3), we must have
u(x,0) = E 2c" sin :x m f(x). (20)
12:1
In
Math 264 - Advanced Calculus for Engineers
Euler Equation
5/23/17
Today in lecture we saw how the Euler Equation naturally appears when solving the Dirichlet Problem for
the Laplace Equation on the unit disc by separation of variables. Below we present th
FTC I
Math 264 - Advanced Calculus for Engineers Lecture 5b - 5/10/17
Today we present the fundamental theorems of calculus for planar vector fields: Greens Theorem and the
planar Divergence Theorem. The proofs of the theorems are rather straightforward a
Math 264 Formula Sheet
Paramaterization:
~r(, ) = (a sin cos , a sin sin , a cos ) (Spherical Coordinates for Sphere of Radius a)
Divergence Theorem
ZZZ
~ F~ dV =
I
~
F~ dS
S
V
Stokes Theorem
ZZ
F~ d~r =
C
ZZ
~ F~ dS
~
S
For F~ = (F1 , F2 , F3 )
~ F~ = F1
Fourier Series II
Math 264 - Advanced Calculus for Engineers Lecture 11 - 5/24/17
In this lecture we study generalized Fourier Series. We aim to give a unified explanation for the various
Fourier Series we have seen previously as well as motivate the cons