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

 

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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