MAC 2313 (sec 13294): Calculus - Analytic Geometry III
The course provides an introduction to standard techniques from multivariable calculus. The main focus is on 2- and 3-dimensional real space. In particular, after completion of the course, you should be acquainted with the basic concepts of three-dimensional analytic geometry. You should know how to compute derivatives and integrals of vector-valued functions, and you should be able to apply basic concepts of multivariable calculus. After completion of the course, you should be acquainted with multiple integrals and vector fields, and you should be able to explain the similarities between the Fundamental Theorem for line integrals, Green's Theorem, Stokes' Theorem and the Divergence Theorem. Finally, you should be able to solve simple differential equations of second order.
The lectures cover Chapter 12-Chapter 17 of the book Multivariable Calculus, Early Transcendentals (James Stewart, 5th edition, Thomson Brooks/Cole, 2003), subsequently referred to as [Ste03]. The material will be presented in the same order as in the textbook:
- Vectors and the geometry of space
- Vector functions
- Partial derivatives
- Multiple integrals
- Vector calculus
- Second-order differential equations
More information on the course is available in the syllabus, and comments are welcome.
If you are interested, there is also tutoring for this class available.
Topics discussed in class
- 01/07/08: 3-dimensional coordinate systems
Literature: [Ste03, Ch. 12.1]
- 01/09/08: vectors, dot product of vectors
Literature: [Ste03, Ch. 12.2, Ch. 12.3]
- 01/10/08: cross product of vectors
Literature: [Ste03, Ch. 12.4]
- 01/11/08: properties of the cross product
Literature: [Ste03, Ch. 12.4]
- 01/14/08: equations of lines and planes
Literature: [Ste03, Ch. 12.5]
- 01/16/08: cylindrical and spherical coordinates; vector functions and space curves
Literature: [Ste03, Ch. 12.6-Ch. 12.7, Ch. 13.1]
- 01/17/08: derivation rules for vector functions
Literature: [Ste03, Ch. 13.1-Ch. 13.2]
- 01/18/08: derivation rules for vector functions; Homework 1 is available
Literature: [Ste03, Ch. 13.2]
- 01/23/08: arc length
Literature: [Ste03, Ch. 13.3]
- 01/24/08: curvature, normal and binormal vectors
Literature: [Ste03, Ch. 13.3]
- 01/25/08: motion in space: velocity and acceleration
Literature: [Ste03, Ch. 13.4]
- 01/28/08: functions of several variables: limits and continuity
Literature: [Ste03, Ch. 14.1, Ch. 14.2]
- 01/30/08: partial derivatives
Literature: [Ste03, Ch. 14.3]
- 01/31/08: partial differential equations, tangent plane approximation
Literature: [Ste03, Ch. 14.3, Ch. 14.4]
]
- 02/01/08: tangent plane approximation, differentiability
Literature: [Ste03, Ch. 14.4]
- 02/04/08: solution of Homework 1
- 02/06/08: chain rule
Literature: [Ste03, Ch. 14.5]
- 02/07/08: directional derivatives and the gradient vector
Literature: [Ste03, Ch. 14.6]
- 02/08/08: properties of the gradient vector
Literature: [Ste03, Ch. 14.6]
- 02/11/08 maximum and minimum values; Homework 2 is available
Literature: [Ste03, Ch. 14.7]
- 02/13/08 second derivative test
Literature: [Ste03, Ch. 14.7]
- 02/14/08 finding maximum and minimum values: examples
Literature: [Ste03, Ch. 14.7]
- 02/15/08: Lagrange multipliers
Literature: [Ste03, Ch. 14.8]
- 02/18/08: Lagrange multipliers with two contraints, double integrals over rectangles
Literature: [Ste03, Ch. 14.8, Ch. 15.1]
- 02/20/08: double integrals and iterated integrals
Literature: [Ste03, Ch. 15.1, Ch. 15.2]
- 02/21/08: Fubini's Theorem, double integrals over general regions
Literature: [Ste03, Ch. 15.2, Ch. 15.3]
- 02/22/08: double integrals over general regions: examples
Literature: [Ste03, Ch. 15.3]
- 02/25/08: double integrals over general regions: examples
Literature: [Ste03, Ch. 15.3]
- 02/27/08: solution of Homework 2
- 02/28/08: double integrals in polar coordinates
Literature: [Ste03, Ch. 15.4]
- 02/29/08: applications of double integrals
Literature: [Ste03, Ch. 15.5]
- 03/10/08: computing surface areas
Literature: [Ste03, Ch. 15.6]
- 03/12/08: triple integrals; Exam 1 is available
Literature: [Ste03, Ch. 15.7]
- 03/13/08: triple integrals
Literature: [Ste03, Ch. 15. 7, Ch. 15.8]
- 03/14/08: change of variables in double integrals
Literature: [Ste03, Ch. 15.9]
- 03/17/08: change of variables in double integrals
Literature: [Ste03, Ch. 15.9]
- 03/19/08: change of variables in double integrals
Literature: [Ste03, Ch. 15.9]
- 03/20/08: change of variables in triple integrals; spherical coordinates
Literature: [Ste03, Ch. 15.8, Ch. 15.9]
- 03/21/08: introduction vector fields; line integrals
Literature: [Ste03, Ch. 16.1, Ch. 16.2]
- 03/24/08: line integrals: examples
Literature: [Ste03, Ch. 16.2]
- 03/26/08: solution of Exam 1
- 03/27/08: line integrals of vector fields
Literature: [Ste03, Ch. 16.2]
- 03/28/08: fundamental theorem for line integrals
Literature: [Ste03, Ch. 16.3]
- 03/31/08: Green's Theorem
Literature: [Ste03, Ch. 16.4]
- 04/02/08: Green's Theorem
Literature: [Ste03, Ch. 16.4]
- 04/03/08: curl of a vector field
Literature: [Ste03, Ch. 16.5]
- 04/04/08: curl: examples
Literature: [Ste03, Ch. 16.5]
- 04/07/08: divergence of a vector field; Homework 3 is available
Literature: [Ste03, Ch. 16.5]
- 04/09/08: parametric surfaces and their area
Literature: [Ste03, Ch. 16.6]
- 04/10/08: surface integrals
Literature: [Ste03, Ch. 16.7]
- 04/11/08: Stokes' Theorem
Literature: [Ste03, Ch. 16.8]
- 04/14/08: divergence theorem
Literature: [Ste03, Ch. 16.9, Ch. 17.3]
- 04/16/08: homogeneous linear differential equations of second order
Literature: [Ste03, Ch. 17.1]
- 04/17/08: nonhomogeneous linear differential equations of second order
Literature: [Ste03, Ch. 17.2]
- 04/18/08: variation of parameters
Literature: [Ste03, Ch. 17.2]
- 04/21/08: solving differential equations with series; exercises for the final exam
Literature: [Ste03, Ch. 17.4]
- 04/23/08: exercises for the final exam
For questions or comments, please feel free to contact me anytime
(see my homepage for email, phone number, etc.).
Apr 28, 2008