List of topics for each video lecture
Lecture
# |
Chapter # |
Section # |
Topic |
1 |
|
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Introduction to the course |
1 |
1 |
Begin Chapter 1: ÒEuclidean Geometry in 3-dimensional SpaceÓ. Review Cartesian Coordinates in the Plane; Begin
study of Cartesian coordinates in 3-dimensional space. |
|
2 |
1 |
2 |
Begin study of Vectors in two and three dimensions |
3 |
1 |
2 |
More on vectors; Positions vector representation of
a vector |
4 |
|
--- |
Brief discussion of Maple |
1 |
3 |
Begin study of the dot product of 2 vectors A and B |
|
5 |
1 |
3 |
More on the dot product |
1 |
4 |
Begin study of the cross product of 2 vectors A and B |
|
6 |
1 |
4 |
More on the cross product |
7 |
1 |
4 |
Example: orthogonal decomposition of vector A with respect to vector B |
1 |
5 |
Equations of Lines and Planes |
|
8 |
1 |
5 |
More on equations of Lines and Planes |
9 |
2 |
1 |
Begin Chapter 2: ÒThe Geometry of Curves in SpaceÓ Vector-valued functions,
limits and continuity |
10 |
2 |
1 |
Differentiating vector-valued functions,
rules of differentiation, definite integrals
of vector-valued functions |
11 |
2 |
1 |
Sketching curves in 3-d using Maple Anti-derivatives of
vector-valued function, parameterized curves |
2 |
Parametrized curves in
space, Introduction to the idea of curvature, reparametrizing a curve in terms
of arc
length, Initial-value problems, Application: NewtonÕs
second law and motion under
gravity |
||
12 |
2 |
2 |
Example of motion under gravity |
3 |
Begin section 3: ÒFundamental Quantities Associated with a CurveÓ, Smooth curve,
principal tangent vector to a curve, unit tangent vector, velocity, speed and acceleration
of a point particle |
||
13 |
2 |
3 |
Arc length of a curve, arc length re-parametrization
of a curve, Curvature
of a curve |
14 |
2 |
2 |
Exercise #23, section 2.2, |
3 |
The curvature of a circle of radius R is K = 1/R,
curvature of a curve in terms of an arbitrary parameter |
||
15 |
2 |
4 |
The Unit Normal Vector
N, and the osculating
plane, Theorem:
Acceleration vector is always parallel to the osculating plane. The Unit Binormal vector B |
16 |
2 |
4 |
The osculating circle
of a curve, instantaneous linear and instantaneous centripetal
acceleration, simple
formulas for the tangential and
normal components of acceleration, and for curvature. |
17 |
|
|
Review for Test #1 |
Test #1 on chapters 1 and 2 |
|||
18 |
3 |
1 |
Begin Chapter 2: ÒDifferential Multivariable CalculusÓ Definition and graphical
representation; Level curves of f(x,y); Quadric surfaces; |
19 |
3 |
1 |
Review level curves of f(x,y); Level surfaces of
f(x,y,z), examples of level
surfaces of functions of 3 variables. |
20 |
3 |
1 |
Examples of: Ellipsoids,
paraboloids, cones, planes and cylinders; |
2 |
Limits of multivariable functions;
theorems on limits |
||
21 |
3 |
2 |
Continuity of multivariable functions;
theorems on continuity |
3 |
Begin Directional Derivatives of multivariable functions |
||
22 |
3 |
3 |
x-, y- and z-partial derivatives; higher partial derivatives; new notation for partial derivatives |
23 |
3 |
3 |
Geometrical interpretation of
x- and y-partial derivatives of f(x,y). Tangent planes to the graph of f(x,y); Linearization of a function near a point |
24 |
3 |
4 |
Differentiability of multivariable functions;
examples; Theorem: a multivariable function is continuous at
P if it is differentiable at P. |
25 |
3 |
4 |
Theorem: A multivariable function is differentiable
at P if its partial derivatives are continuous at P. Examples:
Use the theorem to prove that a function is differentiable at a point. |
5 |
The directional derivative and the Gradient of a function. Examples. |
||
26 |
3 |
5 |
What does the gradient of the function f tell us
about f? Tangent planes to level surfaces
of a function f(x,y,z). |
27 |
3 |
5 |
The chain rules for multivariable functions;
Parametrizing functions that are graphs of functions; Implicit
differentiation |
28 |
3 |
6 |
Optimization of function of two
variables: finding local extreme
values of f(x,y); Examples |
29 |
3 |
6 |
The second derivative test; examples |
30 |
3 |
6 |
Global extreme values of f(x,y) on
closed and bounded sets |
31 |
|
|
Review of chapter 3 for
test #2 |
Test #2 on chapter 3 |
|||
32 |
4 |
1 |
Begin Chapter 4: ÒDouble and Triple Integrals in Cartesian
CoordinatesÓ. Start
with double
integrals over rectangles. |
33 |
4 |
1 |
FubiniÕs Theorem for rectangles. Examples. |
34 |
4 |
1 |
Type I regions. FubiniÕs theorem for Type I regions. Examples. |
35 |
4 |
1 |
Type II regions. FubiniÕs theorem for Type II regions.
Examples. |
36 |
4 |
1 |
Reversing the order of integration; Numerous
examples |
37 |
4 |
2 |
Applications of double integrals:
area; average values; volume between two surfaces; moments and center of mass
of plane laminas, |
38 |
4 |
2 |
More on applications of double integrals |
39 |
4 |
3 |
Triple integrals in Cartesian Coordinates |
40 |
4 |
3 |
More on triple integrals: z-simple, x-simple, and y-simple
regions; |
41 |
4 |
3 |
More examples of triple integrals |
42 |
4 |
3 |
Applications of triple integrals:
Volume; average value; mass and charge density; energy density; moments of
inertia |
43 |
5 |
1 |
Begin Chapter 5: Double and Triple Integrals in Curvilinear
Coordinates. Begin
with double
integrals in polar coordinates |
44 |
5 |
1 |
Examples of double integrals in polar coordinates |
45 |
5 |
2 |
Triple integrals in cylindrical coordinates |
46 |
5 |
3 |
Triple integrals in spherical coordinates |
47 |
5 |
3 |
More on triple integrals in spherical coordinates; Examples |
48 |
|
|
Review of chapters 4 and 5
for test #3 |
Test #3 on chapters 4 & 5 |
|||
49 |
6 |
1 |
Begin Chapter 4: ÒLine and Surface IntegralsÓ. Start with vector fields in space. Numerous examples. |
50 |
6 |
2 |
Line integrals of functions. |
51 |
6 |
2 |
Solutions to selected exercises in Section 6.2.1 on
Line integrals of functions |
52 |
6 |
3 |
Line integrals of vector fields. Work
and other examples |
53 |
6 |
3 |
The Fundamental Theorem for Line Integrals; Line
integrals of vector fields that are independent of path. |
54 |
6 |
3 |
Characterizing conservative vector fields. Finding potential functions for
conservative vector fields |
55 |
6 |
3 |
More on finding potential functions; Conservation
of total energy forNewtonian motion under a conservative force. |
4 |
Parametric surfaces in space |
||
56 |
6 |
4 |
Parametrizing surfaces that are graphs of functions
of two variables |
57 |
6 |
5 |
Surface integrals; surface area of a parametrized
surface;
example: surface
area of a sphere |
58 |
6 |
5 |
Surface integrals of functions;
examples |
59 |
6 |
5 |
Surface integrals of vector fields; examples |
60 |
6 |
5 |
More examples of flux integrals |
62 |
|
|
Review of chapter 6 for
test #4 |
Test #4 on chapter 6 |
|||
61 |
7 |
1 |
Begin Chapter 7: ÒVector AnalysisÓ. Start with integral curves of vector fields in
space. |
2 |
Differentiating vector fields: The divergence and curl. |
||
63 |
7 |
2 |
The concept of Òflux densityÓ and its
relationship to divergence |
64 |
7 |
2 |
Divergence and curl in terms of the ÒdelÓ
operator. Geometrical
interpretation of the divergence and curl of a vector field. A
vector field is conservative if and only if its curl is
zero. |
65 |
7 |
2 |
More on the geometrical interpretation of curl and
divergence |
3 |
GreenÕs theorems for both circulation and
flux. Examples |
||
66 |
7 |
4 |
StokesÕ Theorem. Examples |
67 |
7 |
5 |
The Divergence Theorem of Gauss Ð begin with
review of GreenÕs theorem for flux. |
68 |
7 |
6,7 |
Semester Review |
Comprehensive Final Exam |