MATH 2934 - Differential and Integral Calculus III (Honors), Sec. 002 - Spring 2016
Tue and Thu 12:00-1:15 in 356 PHSC,
Wed 12:30-1:20 in 222 PHSC
Instructor: Prof. Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu.
Office Hours:
Mon 2:30-3:30 p.m., Tue 9:30-10:30 a.m., or by appointment, in 802 PHSC.
First day handout
OU Math Center (PHSC 209) - open MT 9:30-7, WRF 9:30-5:30, Sun 3-7
Course catalog description:
Prerequisite: 2924 with grade of C or better. Duplicates one hour of 2433 and three hours of 2443.
Vectors and vector functions, functions of several variables, partial differentiation and gradients,
multiple integration, line and surface integrals, Green-Stokes-Gauss theorems. (F,Sp,Su)
Text:
J. Stewart, Calculus (7th ed), Brooks/Cole, 2012, ISBN: 978-0-8400-5818-8
MyMedia repository of worked examples:
A link to MyMedia,
instructions how to use it,
and a list
of the examples by section.
Check out the
OU Math Blog!
It is REALLY interesting!
Course content:
-
Vectors and the geometry of space:
brief review of vectors and vector algebra,
equations of lines and planes, cylinders and quadric surfaces.
-
Vector functions:
vector functions and space curves, derivatives and integrals of vector functions,
arc length for space curves (curvature if time permits),
motion in space (velocity and acceleration).
-
Partial derivatives:
functions of several variables, limits and continuity of functions of several variables,
partial derivatives, tangent planes and linear approximations,
chain rule for partial derivatives,
directional derivatives and gradients, maximum and minimum values for functions of several variables,
Lagrange multipliers.
-
Multiple integrals:
double integrals over rectangles, iterated integrals, double integrals over general regions,
double integrals in polar coordinates, surface area, triple integrals,
triple integrals in cylindrical coordinates, triple integrals in spherical coordinates.
-
Vector calculus:
vector fields, line integrals, the fundamental theorem for line integrals,
Green's theorem, curl and divergence,
parametric surfaces and their areas surface integrals,
Stokes' theorem, divergence theorem.
Homework:
-
Homework 1 (problems given on January 19, 20, 21), due January 25 (Tuesday)
-
Homework 2 (problems given on January 26, 27, 28), due February 2 (Tuesday)
-
Homework 3 (problems given on February 2, 3, 4), due February 9 (Tuesday)
-
Homework 4 (problems given on February 9, 10, 11), due February 16 (Tuesday)
-
Homework 5 (problems given on February 16, 18), due February 25 (Thursday)
-
Homework 6 (problems given on February 23, 24, 25), due March 1 (Tuesday)
-
Homework 7 (problems given on March 1, 2, 3), due March 8 (Tuesday)
-
Homework 8 (problems given on March 8, 9, 10), due March 22 (Tuesday)
-
Homework 9 (problems given on March 22, 24, 29, 31), due April 5 (Tuesday)
-
Homework 10 (problems given on April 5, 6, 7), due April 12 (Tuesday)
-
Homework 11 (problems given on April 12, 13, 14), due April 19 (Tuesday)
-
Homework 12 (problems given on April 19, 20, 21), due April 26 (Tuesday)
Content of the lectures:
-
Lecture 1 (Tue, Jan 19):
Three dimensional coordinates systems:
what does "dimension" mean?;
coordinates; the space Rn;
Cartesian (adjective from the family name of
René Descartes)
coordinates in three dimensions, right-hand rule;
two halfs of R,
four quadrants of R2,
eight octants of R3;
projection of a point onto a coordinate plane and onto a coordinate axis;
Cartesian product of sets,
R3 as R×R×R;
equations of objects in R3;
distance formula in R3
(and a derivation based on the Pythagorean Theorem);
equation of a straight line in R2,
equation of a circle in R2,
equation of a sphere in R3
[Sec. 12.1]
Vectors:
displacement vector, initial and terminal points;
what does u=v mean?;
addition of vectors ("the triangle rule"
and "the parallelogram rule"),
"scalar"="number",
multiplication of a vector by a scalar,
difference of vectors (defined through the other two operations);
components of a vector, position vector;
vector addition and multiplication of a vector
by a scalar in components
[pages 815-819 of Sec. 12.2]
Homework:
Exercises
12.1 / 7, 9, 12, 16, 22, 28, 30, 38;
12.2 / 4, 8, 16.
FFT: Exercises
12.1 /
5,
13,
21,
27,
33,
37,
41;
12.2 /
3,
7,
13.
Remark:
The FFT ("Food For Thought")
exercises are to be solved like regular homework problems,
but do not have to be turned in.
-
Lecture 2 (Wed, Jan 20):
Vectors (cont.):
magnitude (length) of a vector,
fundamental properties of vectors;
standard basis vectors
i=⟨1,0,0⟩,
j=⟨0,1,0⟩,
k=⟨0,0,1⟩;
a=⟨a1,a2,a3⟩=a1i+a2j+a3k;
the unit vector in the direction of a≠0
is a/|a|
[pages 819-821 of Sec. 12.2]
The dot product:
definition of dot product (scalar product, inner product);
elementary properties of dot product;
angle θ between vectors;
a⋅b=|a||b|cos(θ)
(read the proof from the book),
expressing cos(θ) through a⋅b;
perpendicular (orthogonal) vectors,
condition for orthogonality expressed through the dot product;
direction angles and direction cosines,
vector projection projab
of b onto a,
scalar projection compab
of b onto a,
geometric meanings and formal expressions for
projab
and compab
[pages 824-829 of Sec. 12.3]
The cross product:
matrices; determinants of square matrices
of size 1×1, 2×2, and 3×3;
definition of a cross product (vector product);
examples [pages 832-833 of Sec. 12.4]
Reading assignment (mandatory):
Theorems 8 and 9 on the direction and the length of a×b
[page 834 of Sec. 12.4]
Homework:
Exercises
12.2 / 20, 42(a,b),
45(a,d);
12.3 /
27,
38, 39, 54 (justify!),
61,
62.
FFT: Exercises
12.2 /
25,
28,
51;
12.3 /
11,
19,
26,
45,
47,
55,
58.
-
Lecture 3 (Thu, Jan 21):
The cross product (cont.):
basic facts about the cross product:
(a×b)⊥a,
(a×b)⊥b,
|a×b|=|a||b|sin(θ),
direction of a×b (the right hand rule),
condition for a≠0 to be perpendicular to b≠0
(a×b=0),
geometric interpretation of |a×b|
as the area of a parallelogram;
further properties of the cross product;
triple product a⋅(b×c),
geometric interpretation of the triple product
as (plus or minus) the volume of a parallelopiped,
properties of the triple product, examples
[pages 834-837 of Sec. 12.4]
Equations of lines and planes:
lines: vector equation r=r0+tv,
parameter t∈R; parametric equations;
direction numbers; symmetric equations;
vector equation of a line segment connecting two points; examples
[pages 840-843 of Sec. 12.5]
Homework:
Exercises
12.4 / 12, 20,
31,
33,
45,
48,
49;
12.5 / 10, 20, 22.
FFT:
Exercises
12.4 / 9, 11,
13,
15,
19,
37, 38, 46,
53;
12.5 / 1,
7,
13,
17,
19.
The complete Homework 1 (problems given on January 19, 20, 21) is due on January 25 (Tuesday).
-
Lecture 4 (Tue, Jan 26):
Equations of lines and planes (cont.):
normal vector to a plane; vector, scalar, and linear equations of a plane;
example: equation of a plane through three non-collinear points;
example: intersection between a line and a plane;
angle between planes, parallel planes;
example: equation of a line of intersection of two planes;
example: distance from a point to a plane
[pages 843-847 of Sec. 12.5]
Reading assignment (mandatory):
Example 9: distance between two parallel planes;
Example 10: distance between two skew lines [page 847 of Sec. 12.5]
Homework:
Exercises
12.5 / 26, 28, 30, 34, 37, 57, 64, 73.
FFT:
Exercises
12.5 /
51,
63,
67, 68,
75.
-
Lecture 5 (Wed, Jan 27):
Cylinders and quadric surfaces:
review of curves in R2 defined by quadratic equations
- ellipse (x/a)2+(y/b)2=1,
hyperbola (x/a)2−(y/b)2=1,
parabola ax2+y=b,
cone a2x2=y2;
visualising a surface in R3 - traces (cross-sections of the surface with planes
parallel to the coordinate planes);
cylinders, rulings, examples;
quadric surfaces; standard forms of quadric surface equation:
Ax2+By2+Cz2+J=0
or
Ax2+By2+Iz=0;
examples
[pages 851-855 of Sec. 12.6]
Homework:
Exercises
12.6 / 34, 44, 46, 49.
FFT:
Exercises
12.6 / 21-28;
Chapter 12 Concept Check on page 858 / 1-18;
Chapter 12 True-False Quiz on pages 858-859 / 1-22
(the answers are given on page A115).
-
Lecture 6 (Thu, Jan 28):
Vector functions and space curves:
general idea of a function, sequences as functions a:N→R;
vector functions; component functions;
limits and continuity of vector functions;
space curves, parametric equations of a space curve,
curves obtained as intersections of surfaces, examples
[Sec. 13.1]
Derivatives and integrals of vector functions:
derivative of a vector function;
differentiation rules;
examples (|r(t)|=const
implies that r'(t)⋅r(t)=0,
d|r(t)|/dt=r'(t)⋅r(t)/|r(t)|;
definite integrals of vector functions
[pages 871-874 of Sec. 13.2]
Homework:
Exercises
13.1 / 2, 4, 18, 28, 30, 42, 48 (hint: see Exercise 13.1/47);
13.2 / 8, 16,
34, 36, 51, 55.
FFT:
Exercises
13.1 /
21-26,
27,
41;
13.2 /
15,
53.
The complete Homework 2 (problems given on January 26, 27, 28) is due on February 2 (Tuesday).
-
Lecture 7 (Tue, Feb 2):
Derivatives and integrals of vector functions (cont.):
a vector function determines a single space curve,
a space curve corresponds to infinitely many vector functions
(differing by the choice of parameterization);
higher-order derivatives;
unit tangent vector to a space curve at a point
[page 872 of Sec. 13.2]
Arc length and curvature:
derivation of the expression for arc length by representing it
as a limit of Riemann sums;
arc length function s(t);
the arclength function is strictly increasing
(s'(t)>0), therefore invertible, with inverse
function t=Q(s);
parameterizing a curve with equation r(t)
by using the arc length function:
R(s):=r(Q(s));
velocity, speed, odometer, speedometer
[pages 877-879 of Sec. 13.3]
Homework:
Exercises
13.2 / 18, 23, 27, 42,
53;
13.3 /
3,
13;
additional problem
(additional problems are not FFT problems - they should be turned in with the regular homework).
FFT:
Exercises
13.2 /
3,
19,
25;
13.3 /
5.
-
Lecture 8 (Wed, Feb 3):
Arc length and curvature (cont.):
definition of curvature κ of a curve in R3;
practical formula for computing the curvature in terms of
r'(t) and r''(t)
for an arbitrary parameterization r(t) of the curve
(Theorem 10)
[pages 879-881 of Sec. 13.3]
Motion in space: velocity and acceleration:
velocity vector r(t):=r'(t);
acceleration vector a(t):=v'(t);
speed v(t):=|v(t)|;
computing v(t) and r(t)
by integrating a(t)
and using the initial conditions
r(t0) and v(t0)
[pages 886-889 of Sec. 13.4]
Reading assignment (optional):
Normal N and binormal B vectors to a curve;
normal plane (spanned by N and B);
osculating plane (spanned by T and N);
osculating circle at a point r(t)
(a circle of radius ρ(t)=1/κ(t) in the osculating plane
through the point r(t))
[pages 882, 883 of Sec. 13.3]
Reading assignment (mandatory):
Tangential and normal components of the acceleration
[pages 890, 891 of Sec. 13.4]
Homework:
Exercises
13.3 /
17,
23, 49;
13.4 /
22,
39,
45.
FFT:
Exercises
13.3 /
33;
13.4 /
11;
Chapter 13 Concept Check on page 897 / 1-5, 6(a,b,c), 8;
Chapter 13 True-False Quiz on page 897 / 1-14 (the answers are given on page A118).
-
Lecture 9 (Thu, Feb 4):
Functions of several variables:
functions of two variables, independent variables, dependent variable, domain, range;
geometric way of representing functions of two variables: graph, level curves, color coding;
examples;
functions of three or more variables
[Sec. 14.1]
Limits and continuity:
recall: rigorous definition of the limit of a function of one variable;
definition of the limit of a function of one and several variables;
proving discontinuity of a function at a point
by finding different values when approaching the point along different paths,
Examples 1-3, using polar coordinates to understand the nature of the discontinuity
of the function from Example 1;
extending or modifying a function to make it continuous (not always possible!);
limits and continuity of functions of three and more variables
[read pages 72-74 and Examples 2 and 3 on pages 75-77 of Sec. 1.7;
Sec. 14.2]
Reading assignment (mandatory):
Continuity of polynomials,
continuity of rational functions where the denominator is non-zero,
continuity of compositions of continuous functions
[pages 921-922 of Sec. 14.2]
Reading assignment (optional):
Example 4 on page 920 of Sec. 14.2.
Homework:
Exercises
14.1 / 18, 20,
47;
14.2 / 5 (see Example 5),
9,
15 (see Example 3 and consider, e.g., the path x=0 and the path y=2x2),
19 (see Example 9).
FFT:
Exercises
14.1 /
15,
19,
21,
25,
32,
36, 39-42,
59-64;
14.2 /
13,
21,
25.
The complete Homework 3 (problems given on February 2, 3, 4) is due on February 9 (Tuesday).
-
Lecture 10 (Tue, Feb 9):
Limits and continuity (cont.):
using polar coordinates to find limits and/or prove continuity.
Partial derivatives:
definition of partial derivatives for functions of two variables;
practical rules for finding partial derivatives, examples;
definition of partial derivatives for functions of n variables;
higher derivatives; Clairaut's Theorem;
ordinary and partial differential equations;
heat equation
ut(x,y,z,t)=α2Δu(x,y,z,t)
and wave equation
utt(x,y,z,t)=c2Δu(x,y,z,t),
where α and c are positive constants, and
Δ:=∂xx+yy+zz is the Laplace operator (Laplacian);
for any pair of smooth functions φ and ψ (each of them is a function of one variable),
the function
u(x,t)=φ(x+ct)+ψ(x−ct)
is a solution of the (1+1)-dimensional wave equation
utt(x,t)=c2uxx(x,t);
physical meaning of the terms
φ(x+ct) and ψ(x−ct)
as waves propagating to the left, resp. to the right, at a speed c
[pages 924-927, 929-933 of Sec. 14.3]
Homework:
Exercises
14.2 / 17 (use polar coordinates),
39,
40;
14.3 / 18, 26, 30, 34, 40, 41,
52,
64, 71, 78(a).
FFT:
Exercises
14.2 /
25,
33 (see Example 9 on page 922);
14.3 /
21,
29,
33,
73.
-
Lecture 11 (Wed, Feb 10):
Partial derivatives (cont.):
geometric meaning of the partial derivatives of a function of two variables:
ƒx(x0,y0) is the slope
of the tangent line if we "go to the East",
ƒy(x0,y0) is the slope
of the tangent line if we "go to the North";
the tangent vector to the intersection line of the vertical plane
{y=y0} with the graph {z=ƒ(x,y)} of ƒ
is ⟨1,0,ƒx(x0,y0)⟩,
the tangent vector to the intersection line of the vertical plane
{x=x0} with the graph {z=ƒ(x,y)} of ƒ
is ⟨0,1,ƒy(x0,y0)⟩
[pages 927, 928 of Sec. 14.3]
Tangent planes and linear approximations:
equation of the tangent plane to a surface defined as a graph of a function:
(r−R(x0,y0))⋅N(x0,y0)=0,
where the equation of the graph of ƒ is written as
R(x,y)=xi+yj+ƒ(x,y)k,
and
N(x0,y0)=Rx(x0,y0)×Ry(x0,y0)
is the normal vector to the tangent plane to the graph of ƒ at the point
(x0,y0,ƒ(x0,y0)),
and Rx(x0,y0)=⟨1,0,ƒx(x0,y0)⟩,
Ry(x0,y0)=⟨0,1,ƒy(x0,y0)⟩
[pages 939, 940 of Sec. 14.4]
Reading assignment (mandatory):
Implicit differentiation
[Example 4 on page 929 of Sec. 14.3; reviewing Sec. 2.6 will be useful]
Homework:
Exercises
14.3 / 50, 53,
92,
95;
14.4 / 42.
FFT:
Exercise
14.3 / 59, 74,
93,
96.
-
Lecture 12 (Thu, Feb 11):
Partial derivatives (cont.):
implicit differentiation
[Exercise 14.3 / 92 on page 938]
Tangent planes and linear approximations (cont.):
linear approximation (tangent plane approximation) of a function at a point;
increments Δx=dx and Δy=dy
of the independent variables, increment
Δz=ƒ(a+Δx,b+Δy)−ƒ(a,b)
of the function value for z=ƒ(x,y),
and differential
dz=ƒx(a,b)dx+ƒy(a,b)dy
of the function z=ƒ(x,y) at the point
(a,b) for given increments dx and
dy of the independent variables;
using differentials (i.e., linear approximation)
to estimate approximate the increment of the value of the function
due to small changes of the values of the arguments;
an example: estimating the metal needed to make a cylindrical can
(i.e., estimating the change of the volume of a cylinder
when its radius and height increase by small amounts);
example: exact change
ΔA=(x0+dx)(y0+dy)−x0y0=y0dx+x0dy+(dx)(dy)
and approximate change
dA=y0dx+x0dy
of the area of a rectangle
whose sides change from (x0,y0)
to (x0+dx,y0+dy),
geometric interpretation of all terms
[pages 183-184 of Sec. 2.9; pages 940, 941, 943-945 of Sec. 14.4;
skip Equations 5, 6, Definition 7, and Theorem 8 on page 942]
Reading assignment (mandatory):
Linear approximation and differentials
of functions of more than two variables
[pages 945, 946 of Sec. 14.4]
Thinking assignment (mandatory):
The centrifugal force acting on a body of mass m rotating
on a circle of radius r
with angular velocity ω and linear velocity v=ωr
can be written as
F=mv2/r
or as
F=mω2r;
does F increase or increase when r increases?
Homework:
Exercises
14.4 / 6,
11
(only find the linearization),
19,
21,
30,
31.
FFT:
Exercises
14.4 /
35.
The complete Homework 4 (problems given on February 9, 10, 11) is due on February 16 (Tuesday).
-
Lecture 13 (Tue, Feb 16):
Tangent planes and linear approximations (cont.):
a function ƒ(x,y) is said to be differentiable
at (a,b) if
ƒ(a+Δx,b+Δy)=ƒ(a,b)+ƒx(a,b)Δx+ƒy(a,b)Δy+φ(a,b,Δx,Δy), where
φ(a,b,Δx,Δy)/Δx→0
and φ(a,b,Δx,Δy)/Δy→0
as (Δx,Δy)→(0,0);
informally, a function ƒ(x,y) is said to be differentiable
if it has a tangent plane;
if ƒ is differentiable at (a,b), then
ƒx(a,b)
and ƒy(a,b) exist;
the continuity of the partial derivatives of a function
implies the differentiability of the function (Theorem 8);
the differentiability of a function at a point implies continuity
of the function at that point;
for an example of a function that has partial derivatives but is not differentiable
see Exercise 14.4/46
[pages 941, 942, 948 of Sec. 14.4]
The chain rule:
computing
dƒ(g(t),h(t))/dt;
computing
∂ƒ(g(s,t),h(s,t))/∂s
and ∂ƒ(g(s,t),h(s,t))/∂t;
computing
∂ƒ(g1(t1,...,tk),...,gn(t1,...,tk))/∂ti
for i=1,...,k
(skip the proofs of the chain rules)
[pages 948-952 of Sec. 14.5;
reviewing pages 148-152 of Sec. 2.5 may be useful]
Homework:
Exercises
14.5 / 2, 8, 14, 16, 23,
35,
39,
49.
FFT:
Exercise
14.4 / 46,
additional FFT problem
(not to be turned in!);
14.5 /
5,
11,
17,
47.
-
Lecture 14 (Wed, Feb 17):
Exam 1
[on Sec. 12.1-12.6, 13.1-13.4, 14.1-14.4 covered in Lectures 1-12]
-
Lecture 15 (Thu, Feb 18):
The chain rule (cont.):
implicit differentiation of a function of one variable; examples
[pages 952-953 of Sec. 14.5; reviewing Sec. 2.6 may be useful]
Directional derivatives and the gradient vector:
directional derivative Duƒ(x0,y0)
of a function of two variables;
particular cases: Diƒ(x0,y0)=ƒx(x0,y0),
Djƒ(x0,y0)=ƒy(x0,y0);
expressing
Duƒ(x0,y0)
in terms of the partial derivatives of ƒ
and the components of the unit vector u;
definition of the gradient vector
∇ƒ(x0,y0);
expressing the directional derivative as a scalar product
of a unit vector and the gradient vector:
Duƒ(x0,y0)=∇ƒ(x0,y0)⋅u;
gradient and directional derivative of a function of n variables
[pages 957-962 of Sec. 14.6]
Reading assignment (mandatory):
Implicit differentiation of a function of more than one variables;
Implicit Function Theorem
[the text after Example 8 on pages 952-953 of Sec. 14.5; reviewing Sec. 2.6 may be useful]
Homework:
Exercises
14.5 / 28, 34, 55, 59
(differentiate Fx(x,y(x))+Fy(x,y(x))y'(x)=0 with respect to x and use Equation 6 on page 953);
14.6 / 5, 9, 13 (v is not a unit vector!), 40;
additional problem.
FFT:
Exercises
14.5 / 58;
14.6 /
19.
The complete Homework 5 (problems given on February 16, 18) is due on February 25 (Thursday).
-
Lecture 16 (Tue, Feb 23):
Directional derivatives and the gradient vector (cont.):
maximizing the directional derivative Duƒ(x0,y0)
of a function ƒ at a point (x0,y0) by choosing n;
significance of the gradient vector
- the tangent line to level curve at (x0,y0)
is perpendicular to ∇ƒ(x0,y0);
path ⟨x(t),y(t)⟩ of steepest descent/ascent
on the graph z=ƒ(x,y) of a function ƒ(x,y)
- the tangent vector ⟨x'(t),y'(t)⟩
at each point of the path
is proportional to the gradient vector ∇ƒ(x(t),y(t));
directional derivatives of functions of three and more variables;
tangent planes to level surfaces of a function F(r):
the normal vector n to the surface at a point r1 on the surface
can be chosen to be equal to the gradient vector ∇F(r1)
[pages 962-966 of Sec. 14.6]
Homework:
Exercises
14.6 /
23,
29,
37(a,b,d), 46, 49, 56.
FFT:
Exercises
14.6 /
27,
33,
38, 51.
-
Lecture 17 (Wed, Feb 24):
Maximum and minimum values:
a reminder from Calculus I - finding extrema of (differentiable) functions of one variable;
local minima and maxima; global (absolute) minima and maxima;
if a differentiable function has an extremum (i.e., a minimum or a maximum) at (a,b),
then ƒx(a,b)=0 and
ƒy(a,b)=0 (Theorem 2);
critical (stationary) points of a function; examples:
(1)
ƒ(x,y)=x2+y2:
(0,0) is a critical point and a local (and global) mimimum;
(2)
ƒ(x,y)=x2−y2:
(0,0) is a critical point but is not a local extremum;
(3)
ƒ(x,y)=|x|:
all points of the form (0,y) are critical points
and they are all local (and global) minima;
(4)
ƒ(x,y)=|x|+y:
all points of the form (0,y) are critical points
but none of them is a local extremum;
determinants of 2×2 matrices;
second derivatives test - determining whether a function ƒ(x,y)
reaches its extremum at a critical point (x0,y0)
based on the signs of
ƒxx(x0,y0)
and the determinant D(x0,y0)
of the matrix of second partial derivatives of ƒ at
(x0,y0);
Examples 1-3
[pages 970-973 of Sec. 14.7]
Homework:
Exercises
14.7 / 2, 19,
41,
additional problem.
FFT:
Exercises
14.7 /
1,
3,
13,
55;
Chapter 14 Concept Check on page 991 (skip questions 9, 18, and 19);
Chapter 14 True-False Quiz on pages 991-992 (the answers are given on page A123).
-
Lecture 18 (Thu, Feb 25):
Maximum and minimum values (cont.):
more examples of finding local extrema:
finding the minimum distance from a point to a plane (Example 5),
minimizing the material used to make a box (Example 6);
open ε-ball centered a point r0 in Rn:
Bε(r0)={r∈Rn:|r−r0|<ε};
definition of a boundary point of a set S⊆Rn
- a point (not necessarily in S) such that each
open ε-ball centered at the point contains points that belong to S
and points that do not belong to S;
boundary ∂S of a set S - the set of all boundary points of S;
definition of an open set - a set that does not contain any points from its boundary;
definition of an closed set - a set that contains all points from its boundary;
examples of open/closed sets;
algorithm for finding the absolute (global) minima and maxima of a function ƒ on a region S
- find the critical points of ƒ in the interior of S (i.e., in the points of S
that do not belong to ∂S) and the value of ƒ at these points,
then parameterize each part of ∂S and look for the maximum/minimum values of ƒ
on each part of ∂S, finally compare the extreme values of ƒ
in the interior of S and on the boundary of S and among these values
choose the maximum one (this is the global maximum) and the minimum one (this is the global minimum);
an example (parameterizing the segment of a straight line connecting (0,2) and (4,0)
and finding the extremal values on this line segment in Exercise 14.7/30)
[pages 974-976 of Sec. 14.7]
Reading assignment (mandatory):
Definition of a bounded set;
extreme value theorem for continuous functions of two variables on a closed bounded set (Theorem 8);
Example 7
[pages 975-976 of Sec. 14.7]
Reading assignment (optional):
Proof of part (a) of Theorem 3
[page 977 of Sec. 14.7]
Homework:
Exercises
14.7 / 30,
43,
51;
Chapter 14 Review / 42, 66 (page 994).
Remark: In Exercise 14.7/30 you have to parameterize the segment of a straight line
connecting (0,2) and (4,0) by using x as a parameter, i.e., by writing the segment as
x(t)=t, y(t)=..., t∈[a,b]; what are a and b?).
FFT:
Chapter 14 Concept Check on page 991, question 18.
The complete Homework 6 (problems given on February 23, 24, 25) is due on March 1 (Tuesday).
-
Lecture 19 (Tue, Mar 1):
Double integral over rectangles:
areas and single integrals in one-dimensional Calculus
to compute the area under the graph of a function ƒ (assumed nonnegative)
over a finite interval [a,b],
choose a natural number n∈N, a partition
a=x0<x1<...<xn=b
of [a,b], points
xi*∈[xi−1,xi],
and construct the corresponding Riemann sum
as sum of the areas
ƒ(xi*)Δxi
of the rectangles (where
Δxi=xi−xi−1);
definition of the area as a limit of Riemann sums
as n→∞ and maxi(Δxi)→0;
volumes and double integrals: given a rectangle
R=[a,b]×[c,d]
and a function ƒ:R→R,
choose natural numbers m∈N and n∈N,
partitions
a=x0<x1<...<xm=b
and
c=y0<y1<...<yn=d
of [a,b] and [c,d], respectively,
points
(xij*,yij*) with
xij*∈[xi−1,xi],
yij*∈[yi−1,yi],
and construct the double Riemann sum
as the double sum over i and j of
ƒ(xij*,yij*)ΔAij,
where ΔAij is the area of the (ij)th small rectangle;
the limit of the double Riemann sum (if it exists) is the double Riemann integral
∫∫Rƒ(x,y)dA;
midpoint rule for double integrals; average value;
linearity of double integrals:
∫∫R(ƒ+γg)dA=∫∫RƒdA+γ∫∫RgdA,
monotonicity of double integrals: if ƒ(x,y)≤g(x,y)
for all (x,y)∈R, then
∫∫RƒdA≤∫∫RgdA
[Sec. 15.1]
Homework:
Exercises
15.1 / 12, 14, 18.
FFT:
15.1 /
7,
13,
17.
-
Lecture 20 (Wed, Mar 2):
Iterated integrals:
back to Calculus I: to compute a definite integral of a function of one variable
over a finite interval, one can do it directly from the definition as a Riemann sum,
or use the connection between integration and differentiation provided by the
Fundamental Theorem of Calculus, examples;
the concept of an iterated integral;
Fubini's Theorem; examples
[Sec. 15.2]
Thinking assignment (mandatory):
In Solution 1 of Example 3 on page 1009, when the integration over x was performed
(one first computes the integral that is inside, and after that the outside integral),
the integration variable x was changed to ξ=xy,
which implies that dξ=ydx (because while integrating over x,
we treat y as a constant), and the limits of integration in the integral over ξ
became ξlower=xlowery=1⋅y=y
and ξupper=xuppery=2⋅y=2y.
Reading assignment (mandatory):
Read the intuitive reasons why Fubini's Theorem should be valid [page 1008]
Homework:
Exercises
15.2 / 2, 14, 22, 38;
4.2 / 72 (page 309);
Chapter 6 Review / 118 (page 484).
FFT:
Exercises
15.2 /
17,
19,
23,
27,
35.
-
Lecture 21 (Thu, Mar 3):
Iterated integrals (cont.):
a detailed example on solving a double integral as an iterated integral,
with a special emphasis on the procedure of changing variables
in the "inside" integral (namely, treat the "outside" variable as constant).
Double integrals over general regions:
definition of a double integral of a function ƒ
over a general region D by:
(1) choosing a rectangle R that contains D,
(2) extending the function ƒ:D→R
to a function F:R→R
(defined as F:=ƒ on D, F:=0
at the points from R that are not in D),
and (3) computing the double integral of F
over the rectangle R by using Fubini's Theorem;
regions of Type I (defined by the inequalities
a≤x≤b and
g1(x)≤y≤g2(x))
and of Type 2 (defined by the inequalities
c≤y≤d and
h1(y)≤x≤h2(y));
deriving an expression for integrals over a region of Type I
(boxed formula 5 on page 1014);
read Example 1
[pages 1012-1015 of Sec. 15.3]
Thinking assignment (mandatory):
Write down an expression for integrals over a region of Type II
(boxed formula 6 on page 1014)
in analogy with the formula for integrals over a region of Type I;
do Example 2 considering the region D in it as
a region of Type I and as a region of Type II
[pages 1014-1015 of Sec. 15.3]
Homework:
Exercises
15.3 / 9, 14, 15,
17;
in Exercise 15 only set up interated integrals for both orders of integration
(i.e., consider the region D first as a region of Type I and then
as a region of Type II), do not solve the iterated integrals!
FFT:
Exercises
15.3 / 11, 12.
The complete Homework 7 (problems given on March 1, 2, 3) is due on March 8 (Tuesday).
Lecture 22 (Tue, Mar 8):
Double integrals over general regions (cont.):
more examples;
computing interated integrals by using Fubini's Theorem to change the order of integration (Example 5);
properties of double integrals: linearity, monotonicity,
additivity in the domain, normalization;
using the properties to derive the double-sided bound
on a double integral (formula (11)) and a double-sided bound
on the average value of a function over a region;
computing integrals over general regions
by subdividing the region D into non-overlapping
regions D1, ..., Dk
of type I or type II, and using the property that
∫∫DƒdA=∫∫D1ƒdA+⋅⋅⋅∫∫DkƒdA
[pages 1016-1019 of Sec. 15.3]
Double integrals in polar coordinates:
polar coordinates; geometric meaning of the curves r=const (circles
centered at the origin) and θ=const (half-lines starting at the origin);
polar rectangles; subdividing a polar rectangle
into smaller polar rectangles;
derivation of the expression for the area
ΔAij≈ri−1(Δri)(Δθj)
of an infinitesimally small polar rectangle;
constructing a Riemann sum
[pages 1021-1022 of Sec. 15.4]
Homework:
Exercises
15.3 /
44, 46, 48, 51, 58,
62,
64.
FFT:
Exercises
15.2 / 40;
15.3 /
25,
37, 43, 45,
47,
53, 57.
-
Lecture 23 (Wed, Mar 9):
Double integrals in polar coordinates (cont.):
area element in Cartesian coordinates,
dA=dx dy,
and in polar coordinates,
dA=r dr dθ;
change to polar coordinates in double integrals
(two things to remember: to include the factor r in the expression
for the area element and to replace x and y
with rcos(θ) and rcos(θ) in the integrand);
example: calculating the area of a circle using different methods:
(1) by a single integral (as in Calculus I),
(2) by a double integral in Cartesian coordinates,
(3) by a double integral in polar coordinates
with a coordinate system centered at the center of the circle,
(4) by a double integral in polar coordinates
with a coordinate system centered at the periphery of the circle
and the x-axis passing through the center of the circle,
with two different orders of integration
[pages 1022-1025 of Sec. 15.4]
Homework:
Exercises
15.4 / 2, 4,
11,
13,
25
(only set up the integral in polar coordinates, do not evaluate it),
31 (only set up the integral in polar coordinates, do not evaluate it).
FFT:
Exercises
15.2 / 40 (page 1012);
15.4 / 6, 29,
39,
40.
-
Lecture 24 (Thu, Mar 10):
Triple integrals:
defining a triple integral over a box as the limit of a triple Riemann sum;
defining a triple integral over an arbitrary bounded region;
Fubini's Theorem for triple integrals;
type-1, type-2, and type-3 regions in R3;
computing triple integrals over a type-1 region,
over a type-2 region, and over a type-3 region;
detailed solutions of Exercise 15.7/7
and Exercise 15.7/33
[pages 1041-1045 of Sec. 15.7]
Thinking assignment (optional):
Think about using single integrals to compute the total mass of a thin wire
of length L of linear density λ(x)
and using double integrals to compute the total mass of a
thin sheet shaped as a 2-dimensional region D
with a given area density σ(x,y).
Homework:
Exercises
15.7 / 9, 34, 36, 55(a);
additional problem.
FFT:
Exercises
15.7 /
23,
27,
29, 31, 33
(a detailed solution),
35
(the answers of Exercises 29, 31, 33, 35 are given on pages A125, A126).
The complete Homework 8 (problems given on March 8, 9, 10) is due on March 22 (Tuesday).
-
Lecture 25 (Tue, Mar 22):
Triple integrals (cont.):
more examples
[Sec. 15.7]
Homework:
No homework problems assigned.
FFT:
Exercises
15.7 /
13,
19
(see Example 5 of Sec. 15.7 on page 1046,
and Example 4 of Sec. 15.3 on pages 1016-1017).
-
Lecture 26 (Wed, Mar 23):
Exam 2
[on Sec. 14.4-14.7, 15.1-15.4, 15.7 covered in Lectures 13, 15-24]
-
Lecture 27 (Thu, Mar 24):
Triple integrals in cylindrical coordinates:
cylindrical coordinates in R3
- definition, geometric meaning, surfaces r=const>0
(an infinitely long cylinder with radius r and axis
coniciding with the z-axis),
θ=const∈[0,2π) (a half-plane starting at the z-axis),
and z=const (a horizontal plane);
volume element in cylindrical coordinates:
dV = r dr dθ dz;
computing triple integrals in cylindrical coordinates
[Sec. 15.8]
Homework:
Exercises
15.8 / 8 (see Example 2), 10, 16,
21,
25(a), 29 (see Example 4).
FFT:
Exercises
15.8 / 3, 5, 6, 7 (see Example 3),
9.
-
Lecture 28 (Tue, Mar 29):
Triple integrals in spherical coordinates:
spherical coordinates in R3
- definition, geometric meaning, surfaces ρ=const>0
(a sphere of radius ρ centered at the origin),
θ=const∈[0,2π) (a half-plane starting at the z-axis),
and φ=const (a cone with axis coinciding with the z-axis);
volume element in spherical coordinates:
dV = ρ2 sin φ dρ dθ dφ;
computing triple integrals in spherical coordinates; examples
[Sec. 15.9]
Homework:
Exercises
15.9 / 4, 8, 10, 20, 28, 40.
FFT:
Exercises
15.9 / 1,
5,
6, 7, 9, 11, 13, 15,
17,
19.
-
Lecture 29 (Wed, Mar 30):
Triple integrals in spherical coordinates (cont.):
more examples on using spherical coordinates;
definition of the ball Bn(R)
and the sphere Sn(R)
of radii R in Rn;
volume Vn(R)
of Bn(R),
area An(R)
of Sn(R);
Vn(R)=Vn(1)Rn,
An(R)=An(1)Rn−1;
An(R)=dVn(R)/dR;
computing the volume of a ball of radius R in Rn
(see handout).
Homework:
No homework problems assigned.
-
Lecture 30 (Thu, Mar 31):
Change of variables in multiple integrals:
changing variables in R2
from (x,y) to (u,v):
x=X(u,v),
y=Y(u,v);
inverse change of variables:
u=U(x,y),
v=V(x,y);
transformed function:
F(u,v):=ƒ(X(u,v),Y(u,v));
Jacobian ∂(x,y)/∂(u,v)
of a transformation (x,y)→(u,v);
formula for change of variables in a double integral under a change of variables;
an example;
changing variables in Rn
from x=(x1,...,xn)
to u=(u1,...,un):
x=X(u);
inverse change of variables in Rn:
u=U(x);
transformed function:
F(u):=ƒ(X(u));
Jacobian ∂(x)/∂(u)
of a transformation (x)→(u);
formula for change of variables in a multiple integral under a change of variables;
comparison with the change of variable in the 1-dimensional case
[Sec. 15.10]
Homework:
Exercises
15.10 / 4, 14, 20, 28 (in Exercise 28 use the change of variables
u=x+y, v=y);
additional problems.
FFT:
Exercises
15.10 /
7,
13;
Chapter 15 Concept Check on page 1073 (skip questions 4(b,d),
8(b,d), and 10); Chapter 15 True-False Quiz on pages 1073-1074 (skip question 8;
the answers are on page A127).
The complete Homework 9 (problems given on March 22, 24, 29, 31) is due on April 5 (Tuesday).
-
Lecture 31 (Tue, Apr 5):
Vector fields:
vector fields on subsets of R2 and R3;
plots of vector fields;
gradient (conservative, potential) vector fields;
potential function of a conservative vector field;
examples (Newton's law of gravity, Coulomb's law of interaction
between stationary electric charges);
an example of a vector field that is not conservative:
F(x,y)=−yi+3xyj
(to show that F is not conservative, we assume that it is
conservative, i.e., that
F=Pi+Qj=∇ƒ=ƒxi+ƒxj,
and use Clairaut's Theorem from Sec. 14.3 to come to a
contradiction because
(ƒx)y=Py=(−y)y=−1
is different from
(ƒy)x=Qx=(3xy)x=3y
[Sec. 16.1]
Line integrals:
motivation: mass of a thin wire with a known shape and linear density;
line integral (with respect to arc length)
of a function along a curve C in R2 and R3;
piecewise-smooth curves;
velocity v(t)=r'(t) and speed
v(t)=|v(t)|;
expressing the arc length as
ds=v(t)dt
[pages 1087-1090, 1092-1094 of Sec. 16.2]
Homework:
Exercises
16.1 / 10,
23,
24;
16.2 /
11,
33,
35.
FFT:
Exercises
16.1 / 1, 3,
5,
11-14,
13,
15-18,
29-32;
16.2 /
3.
-
Lecture 32 (Wed, Apr 6):
Line integrals (cont.):
expressing work as an integral of
F(r)⋅dr=P(x,y,z)dx+Q(x,y,z)dy+R(x,y,z)dz
over a curve C;
expressing the integral of F(r)⋅dr over a curve C
as an integral of of F(r(t))⋅r'(t)
integrated over the parameter t from its initial value a to its final value b;
an example;
behavior of line integrals when the direction of traversing the curve C is reversed
(i.e., when the integration is over −C instead of over C):
the integrals with respect to arc length do not change,
while the integrals of type F(r)⋅dr
change sign
[pages 1090-1096 of Sec. 16.2]
Reading assignment:
Expressing the integral of F(r)⋅dr over a curve C
as an integral of F⋅Tds,
where T=r'(t)/|r'(t)|
is the unit tangent vector
[pages 1094-1095 of Sec. 16.2]
Thinking assignment:
Parameterizing a segment of a straight line
from the point r0 to the point r1 as
r(t)=(1−t)r0+tr1,
t∈[0,1]
(so that r(0)=r0,
and r(1)=r1).
Homework:
Exercises
16.2 /
7,
14,
21,
39.
FFT:
Exercises
16.2 /
17,
18,
52;
additional FFT problems
(not to be turned in!).
-
Lecture 33 (Thu, Apr 7):
The Fundamental Theorem for line integrals:
the Fundamental Theorem for line integrals
(with proof using the Chain Rule and the Fundamental Theorem of Calculus);
independence of path - definition;
vanishing of the line integral
over any closed path is a necessary and sufficient
condition for path independence (Theorem 3);
definitions of open region and connected region,
path independence of a vector field over an open
connected region implies that the vector field
is conservative (statement of Theorem 4, will be proved in next lecture);
for a conservative vector field in R2,
F(x,y)=P(x,y)i+Q(x,y)j,
the equality
∂P/∂y=∂Q/∂x holds
(Theorem 5);
generalization: for a conservative vector field in R3,
F(x,y,z)=P(x,y,z)i+Q(x,y,z)j+R(x,y,z)k,
the equalities
∂P/∂y=∂Q/∂x,
∂Q/∂z=∂R/∂y,
∂R/∂x=∂P/∂z hold
(an extension of Theorem 5 for R3);
practical recipe for finding a function ƒ(x,y)
such that F(x,y)=∇ƒ(x,y):
integrate the condition P=ƒx with respect to x
to find ƒ(x,y) up to an arbitrary function
φ(y), and then impose the condition Q=ƒy
to determine φ(y)
[alternatively, integrate the condition Q=ƒy with respect to y
to findƒ(x,y) up to an arbitrary function
ψ(x), and then impose the condition P=ƒx
to determine ψ(y)];
Example 4(a): finding a function ƒ:R2→R
such that the conservative vector field
F(x,y)=(3+2xy)i+(x2−3y2)j is equal to ∇ƒ(x,y)
[pages 1099-1104 of Sec. 16.3]
Reading assignment:
Example 4(b): using the function ƒ(x,y) such that
F(x,y)=∇ƒ(x,y)
to compute an integral of F(x,y) over a curve;
Example 5: finding a function ƒ:R3→R
such that the conservative vector field
F(x,y,z)=y2i+(2xy+e3z)j+3ye3zk
is equal to ∇ƒ(x,y,z)
[pages 1103-1105 of Sec. 16.3]
Homework:
Exercises
16.2 / 49, 50;
16.3 / 2, 5,
11,
14, 28, 30.
FFT:
Exercises
16.3 / 1, 21, 22,
25,
29.
The complete Homework 10 (problems given on April 5, 6, 7) is due on April 12 (Tuesday).
-
Lecture 34 (Tue, Apr 12):
The Fundamental Theorem for line integrals (cont.):
proof that path independence of a vector field over an open
connected region implies that the vector field is conservative
(Theorem 4, with detailed proof);
conservation of mechanical energy for conservative forces,
i.e., forces satisfying
F(r)=−∇U(r);
an illustration that the resistance force
F=−kv where k=const>0
always does negative work, thus leading to a loss of mechanical energy
[pages 1101, 1102, 1105, 1106 of Section 16.3]
Homework:
additional problems.
-
Lecture 35 (Wed, Apr 13):
The Fundamental Theorem for line integrals (cont.):
simple curve, simply-connected region;
necessary and sufficient condition for a vector field
in a simply-connected region in R2
to be conservative (Theorem 6);
an example illustrating the importance of simply-connectedness: the vector field
F(x,y)=(−yi+xj)/(x2+x2)
satisfies Py=Qx
but is non-conservative because integral of it over the unit circle
in counterclockwise direction is 2π (checked by a direct calculation)
- therefore, there exists a closed curve C the integral of F
over which is non-zero, which by Theorem 3 implies that
the integral of F is not path-independent, so by the
Fundamental Theorem for line integrals F is not conservative;
this example does not contradict Theorem 6 because F is not
defined at (0,0), so that the region D of F
is not simply-connected
- see Exercise
16.3/35
[pages 1102, 1103 of Sec. 16.3]
Green's Theorem:
orientation of the boundary of a planar region;
statement of Green's Theorem;
derivation of formulas for areas of planar regions,
Examples 1, 2
[pages 1108, 1110, 1111 of Section 16.4]
Reading assignment:
Study Example 5 on pages 1104, 1105 of Sec. 16.3.
Food for thought: for a given vector field F(x,y)=P(x,y)i+Q(x,y)j, the recipe for finding a function ƒ(x,y)
such that F(x,y)=∇ƒ(x,y) by integrating
would not work if Py≠Qx
- for example, consider the vector field
F(x,y)=(ey+2x)i+(xey+y+x2sin(y))j
which does not satisfy Py=Qx,
and try to integrate in order to obtain a function ƒ(x,y),
and you will see that the practical recipe will lead to a non-solvable equation.
Homework:
Exercises
16.3 / 18, 20, 32, 34;
16.4 / 1,
7,
13 (clockwise=negative!).
FFT:
Exercises
16.3 / 31, 33.
-
Lecture 36 (Thu, Apr 14):
Green's Theorem (cont.):
proof of Green's Theorem for a simple region (i.e., region that is both type-I and type-II);
proof that the theorem works for regions that are not type-I and type-II;
generalization of Green's Theorem for regions that are not simply-connected
(i.e., for regions "with holes");
examples of using Green's Theorem for practical calculations
(Examples 1 and 2)
[pages 1109-1111 of Sec. 16.4]
Curl and divergence:
dot and cross product of vectors in R3;
methods for memorizing the expression
for a×b, cyclic permutations;
the gradient operator as a vector:
∇=i∂x+j∂y+k∂z;
definition of curl(F)=∇×F
of a vector field F(x,y,z) in R3;
definition of divergence div(F)=∇⋅F of a vector field F;
a proof that the curl of any conservative vector field is identically zero, i.e.,
curl(grad(f))=0 for any function f:R3→R
(Theorem 3);
div(curl(F))=0 for any vector field F(x,y,z)
(Theorem 11 - prove it yourself!)
[pages 1115-1118 of Sec. 16.5]
Reading assignment (mandatory):
An example when the contour can be deformed to some extent but
the value of the integral of a vector field does not change
(because Py=Qx)
- Example 5 (recall Exercise 16.3/35 considered in detail in Lecture 35)
[pages 1112, 1113 of Sec. 16.4]
Reading assignment (optional):
Read the sketch of the proof of Theorem 16.3.6
[page 1113 of Sec. 16.4]
Homework:
Exercises
16.4 / 19, 22;
16.5 / 3,
19,
21
("irrotational"="with zero curl"),
22 ("incompressible"="with zero divergence"),
30(a,b),
31.
FFT:
Exercises
16.4 /
9,
29;
16.5 / 9,
11,
12.
The complete Homework 11 (problems given on April 12, 13, 14) is due on April 19 (Tuesday).
-
Lecture 37 (Tue, Apr 19):
Curl and divergence (cont.):
condition for conservativeness of a vector field F
defined on all of R3 in terms of curl(F) (Theorem 4);
an example of computing a function ƒ(r) such that
such that the conservative vector field F(r) equals ∇ƒ(r);
Laplacian Δ=∇2 of a scalar function;
conservative (gradient, potential), incompressible, and irrotational vector fields;
an example: proving the vector identity
∇⋅(uF)=∇u⋅F+u∇⋅F (Exercise 16.5/25);
Maxwell's equations
[pages 1117-1119 of Sec. 16.5]
Parametric surfaces and their areas:
recall: different ways to represent surfaces in R3 studied so far:
(1) defined implicitly by an equation Φ(x,y,z)=0
(every surface can be written in this way),
or (2) defined as a graph of a function of two variables, z=ƒ(x,y)
(not every surface can be written in this way);
recall: parametric curves in R3 written as the image
of a function R:[a,b]→R3 with
R(t)=X(t)i+Y(t)j+Z(t)k;
parametric curves and parametric surfaces in R3
defined by a function R:D→R3 with
R(u,v)=X(u,v)i+Y(u,v)j+Z(u,v)k;
an example; writing the Earth's surface as a parametric surface
using the latitude u∈[−π/2,π/2]
and the longitude v∈[0,2π),
as
X(u,v)=REsin(π/2−u)cos(v),
Y(u,v)=REsin(π/2−u)sin(v),
Z(u,v)=REcos(π/2−u)
(see also Example 4)
[pages 1123-1126 of Sec. 16.6]
Homework:
Exercises
16.5 / 16, 23, 26, 29;
16.6 / 4, 6,
19,
23,
59(a)
(for the equation of an ellipsoid see Table 1 on page 854).
FFT:
Exercises
16.5 /
13;
16.6 /
3
(see Example 3),
4 (cylinder with elliptical cross-section), 6 (paraboloid),
13-18,
26.
-
Lecture 38 (Wed, Apr 20):
Parametric surfaces and their areas (cont.):
writing the graph z=g(x,y)
of a function g:D→R3
(where D is a region in R2)
as a parametric surface:
R(u,v)=ui+vj+g(y,v)k;
surfaces of revolution;
grid curves in a parameterized surface;
the vectors Ru(u0,v0)
and Rv(u0,v0)
are tangent to the surface R(u,v) at the point
R(u0,v0);
the vector
Ru(u0,v0)×Rv(u0,v0)
is normal to the tangent plane of the surface R(u,v)
at the point R(u0,v0);
the equation of the tangent plane is
Ru(u0,v0)×Rv(u0,v0)⋅[r−R(u0,v0)]=0;
the area of the parallelogram spanned by the vectors a and b is
|a×b|;
the area ΔSij of a surface element is
ΔSij≈|Ru(ui,vj)×Rv(ui,vj)|ΔAij,
where
ΔAij=(Δui)(Δvj);
area of a parametric surface as a limit of Riemann sums, i.e., as an integral
∫∫D|Ru(u,v)×Rv(u,v)|dA
where dA=dudv
[pages 1127-1130 of Sec. 16.6]
Thinking assignment 1 (mandatory):
Recall that the equation of the tangent plane to a surface defined parametrically is
∇Φ(r0)⋅(r−r0)=0,
and the equation of the tangent plane to the graph z=g(x,y) of a function is
z=gx(x0,y0)(x−x0)+gy(x0,y0)(y−y0).
Thinking assignment 2 (optional):
Look at Equation 4 on page 926 and Figure 1 on page 927,
and reread "Interpretations of partial derivatives"
on pages 927-928 of Sec. 14.3;
look at Fig. 12 on page 1127 and Equations 4 and 5 on pages 1127-1128,
and think about the connection with the definition
and interpretation of partial derivatives from Sec. 14.3.
Homework:
Exercises
16.6 / 2, 30 (skip the graphing), 34,
49,
51 (use property 15.3/11 on page 1018 to give bounds on A(S)).
-
Lecture 39 (Thu, Apr 21):
Parametric surfaces and their areas (cont.):
surface area of a graph of a function;
analogy between the formula for the surface area of a graph of a function of two variables
and the formula for the length of a graph of a function of one variable
[page 563 of Sec. 8.1, page 1130 of Sec. 16.6]
Surface integrals:
integral of a scalar function over a surface S defined parametrically
by R(u,v), (u,v)∈D -
do not forget the area scaling factor
|Ru×Rv|,
so that
∫∫SƒdS=∫∫Dƒ(R(u,v))|Ru(u,v)×Rv(u,v)|dA,
where dA=dudv;
orientable surfaces; an example of a non-orientable surface - the Möbius strip;
unit normal vector n to an orientable parameterized surface:
n is equal to
±Ru×Rv/|Ru×Rv|;
the positive orientation of a closed orientable surface
is given by the outward unit normal vector to the surface;
surface integrals of vector fields;
flux of a vector field;
examples from physics: for fluid flow through a surface
the flux of the vector field equal to the velocity v of the fluid
is equal to the volume of the liquid flowing through the surface in one unit of time;
"natural" orientation of a parametric surface S
defined by a vector equation R(u,v)
- given by the unit normal vector
n=Ru×Rv/|Ru×Rv|
(Equation 6 on page 1139);
alternative expressions for surface integrals
∫∫SF⋅dS=∫∫SF⋅ndS=∫∫SF(R(u,v))⋅n(R(u,v))|Ru(u,v)×Rv(u,v)|dA=±∫∫SF(R(u,v))⋅Ru(u,v)×Rv(u,v)dA
[Sec. 16.7]
Homework:
Exercises
16.7 /
9,
23,
39
(in Exercise 39 you may use the fact obtained in Example 10 on pages 1129-1130 of Sec. 16.6 that
for a sphere of radius a, |Rφ×Rθ|=a2sin(φ),
without deriving it).
FFT:
Exercise
16.7 / 38;
Chapter 16 Concept Check on page 1160 / 1-13;
Chapter 16 True-False Quiz on page 1060 / 1-10, 12
(the answers are given on page A130).
The complete Homework 12 (problems given on April 19, 20, 21) is due on April 26 (Tuesday).
-
Lecture 40 (Tue, Apr 26)
Stokes' Theorem:
orientation of the boundary of an oriented surface
consistent with the orientation of the surface;
Stokes' Theorem;
circulation of a vector field over a simple closed curve
(i.e., the integral ∫CF⋅dr);
physical meaning of the curl of a vector field;
Green's Theorem is a particular case of the Stokes' Theorem
for a flat surface D that lies entirely in the (x,y) plane
(the unit normal vector to which is n=k=⟨0,0,1⟩) and a vector field of the form
F(x,y,z)=P(x,y)i+Q(x,y)j+0k
[Sec. 16.8]
Homework (not to be turned in):
Exercises
16.8 /
1,
16, 20.
FFT:
Exercise
16.8 /
5,
19.
-
Lecture 41 (Wed, Apr 27):
Exam 3
[on Sec. 15.8-15.10, 16.1-16.6 covered in Lectures 25, 27-39]
-
Lecture 42 (Thu, Apr 28):
The Divergence Theorem:
the Divergence Theorem; examples;
physical meaning of the divergence of a vector field;
summary of all "integral theorems" covered:
the Fundamentl Theorem of Calculus (which is the foundation of all such theorems),
the FTC for line integrals, Stokes' Theorem (and its particular case,
Greene's Theorem), and the Divergence Theorem
as particular cases of a general theorem
(called the Stokes' Theorem)
that can be written as
∫Σdω=∫∂Σω;
[Sec. 16.9, the Summary table on page 1159]
Homework (not to be turned in):
Exercises
16.9 /
7,
25,
27, 28, 29, 30.
FFT:
Exercise
16.9 /
19;
Chapter 16 Concept Check on page 1160 / 14-16;
Chapter 16 True-False Quiz on page 1060 / 11
(the answer is given on page A130).
-
Lecture 43 (Tue, May 3):
Maxwell's equations and the heat equation:
derivation of the wave equation for the electric field and the magnetic field
(see Exercise 16.5/38);
derivation of the
heat/diffusion
equation from the law of
conservation
of energy,
Fourier's
law j=−k∇T of heat conduction, and the
Divergence Theorem (the diffusion equation is the same, but instead of Fourier's law it relies on
Fick's law of diffusion).
On the side: Brilliant blunders:
Lord Kelvin's
miscalculation of the
age of the Earth.
On the side: Physical applications of integral theorems:
Maxwell's equations of electrodynamics
as a theory unifying theory of electricity, magnetism, and optics
(see its interesting history);
the next great unifications in physics - the theory of
electroweak interactions
(for which Sheldon Glashow,
Abdus Salam,
and Steven Weinberg
received the 1979 Nobel Prize in Physics),
predicting the existence and properties of the
W and Z bosons
(discovered experimentally in 1983 at CERN
by a team led by Carlo Rubbia
and Simon van der Meer
who received the 1984 Nobel Prize in Physics).
On the side: History of black holes:
Laplace
and the
early history
of
black holes
(see also in the FAQ
of the Hubble telescope).
Final remarks:
last words of wisdom and advice about math, science, and life in general.
-
Final exam:
1:30-3:30 p.m. on Thu, May 12, in 356 PHSC.
Good to know: