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.
The lectures cover Chapter 12-Chapter 16 of the book Thomas' Calculus, Early Transcendentals (11th edition, Pearson Education, 2008, ISBN-13: 978-0-321-49575-4, ISBN-10: 0-321-49575-6), subsequently referred to as [Tho08]. The material will be presented in the same order as in the textbook:
- Vectors and the geometry of space
- Vector-valued functions and motion in space
- Partial derivatives
- Multiple integrals
- Integration in vector fields
More information on the course is available in the preliminary syllabus, and comments are welcome.
If you are interested, there is also tutoring for this class available.
Topics discussed in class
- 01/05/09: three-dimensional coordinate systems
Literature: [Tho08, Ch. 12.1]
- 01/07/09: distance between points, elementary properties of vectors
Literature: [Tho08, Ch. 12.1-12.2]
- 01/08/09: dot product of vectors
Literature: [Tho08, Ch. 12.3]
- 01/09/09: cross product of vectors
Literature: [Tho08, Ch. 12.4]
- 01/12/09: lines in space, computing the distance from a point to a line
Literature: [Tho08, Ch. 12.5]
- 01/14/09: describing planes in space, intersection of planes
Literature: [Tho08, Ch. 12.5]
- 01/15/09:intersection of a line and a plane, quadric surfaces, vector-valued functions
Literature: [Tho08, Ch. 12.5, Ch. 12.6, Ch. 13.1]
- 01/16/09: derivatives and integrals of vector-valued functions; Homework 1 is available
Literature: [Tho08, Ch. 13.1]
- 01/19/09: modeling projectile motion
Literature: [Tho08, Ch. 13.2]
- 01/21/09: arc length and unit tangent vector
Literature: [Tho08, Ch. 13.3]
- 01/22/09: principal unit normal vector and curvature
Literature: [Tho08, Ch. 13.4]
- 01/23/09: binormal vector and torsion
Literature: [Tho08, Ch. 13.5]
- 01/26/09: introduction to functions of several variables
Literature: [Tho08, Ch. 14.1]
- 01/28/09: limits and continuity in higher dimensions
Literature: [Tho08, Ch. 14.2]
- 01/28/09: limits in several variables, partial derivatives
Literature: [Tho08, Ch. 14.2, Ch. 14.3]
- 01/28/09: differentiability of functions is several variables, partial derivatives of higher order
Literature: [Tho08, Ch. 14.3]
- 01/30/09: chain rule for functions in several variables
Literature: [Tho08, Ch. 14.4]
- 02/02/09: solution of Homeork 1
- 02/04/09: directional derivatives and gradient vectors
Literature: [Tho08, Ch. 14.5]
- 02/05/09: tangent planes
Literature: [Tho08, Ch. 14.6]
- 02/06/09: standard linear approximation; Homework 2 is available
Literature: [Tho08, Ch. 14.6]
- 02/09/09: extreme values and saddle points
Literature: [Tho08, Ch. 14.7]
- 02/11/09: finding extrema on closed bounded regions
Literature: [Tho08, Ch. 14.7]
- 02/12/09: Lagrange multipliers
Literature: [Tho08, Ch. 14.8]
- 02/13/09: Lagrange multipliers with 2 contstraints
Literature: [Tho08, Ch. 14.8]
- 02/16/09: partial derivatives with constrained variables; Taylor's formula for two variables
Literature: [Tho08, Ch. 14.9, Ch. 14.10]
- 02/18/09: solution of Homework 2
- 02/19/09: double integrals
Literature: [Tho08, Ch. 15.1]
- 02/20/09: double integrals: examples
Literature: [Tho08, Ch. 15.1]
- 02/23/09: moments and centers of mass for thin flat plates
Literature: [Tho08, Ch. 15.2]
- 02/25/09: double integrals in polar form; Exam 1 is available
Literature: [Tho08, Ch. 15.3]
- 02/26/09: triple integrals
Literature: [Tho08, Ch. 15.4]
- 02/27/09: triple integrals: examples
Literature: [Tho08, Ch. 15.4]
- 03/09/09: solution to Exam 1
- 03/11/09: substitution rule in in multiple integrals
Literature: [Tho08, Ch. 15.7]
- 03/12/09: triple integrals in cylindrical coordinates
Literature: [Tho08, Ch. 15.6]
- 03/13/09: substitutions in multiple integrals: examples
Literature: [Tho08, Ch. 15.7]
- 03/16/09: triple integrals in spherical coordinates
Literature: [Tho08, Ch. 15.6, Ch. 15.7]
- 03/18/09: introduction to line integrals vector fields
Literature: [Tho08, Ch. 16.1, Ch. 16.2]
- 03/19/09: flow integral, flux across a closed curve in the plane
Literature: [Tho08, Ch. 16.2]
- 03/20/09: path independence, potential functions, fundamental theorem for line integrals; Homework 3 is available
Literature: [Tho08, Ch. 16.3]
- 03/23/09: divergence and k-component of curl
Literature: [Tho08, Ch. 16.4]
- 03/25/09: Green's Theorem in the plane
Literature: [Tho08, Ch. 16.4]
- 03/26/09: formula for surface area
Literature: [Tho08, Ch. 16.5]
- 03/27/09: computing surface areas: examples
Literature: [Tho08, Ch. 16.5]
- 03/30/09: surface integrals, flux of three-dimensional vector fields
- 04/01/09: parametrized surfaces
Literature: [Tho08, Ch. 16.6]
- 04/02/09: parametric surface integrals, Stokes' Theorem
Literature: [Tho08, Ch. 16.6, Ch. 16.7]
- 04/03/09: Solution to Homework 3
- 04/06/09: using Stokes' Theorem
Literature: [Tho08, Ch. 16.7]
- 04/08/09: Divergence Theorem
Literature: [Tho08, Ch. 16.8]
- 04/09/09: using the Divergence Theorem
Literature: [Tho08, Ch. 16.8]
- 04/10/09: unifying the integral theorems
Literature: [Tho08, Ch. 16.8]
- 04/13/09-04/22/09: exercises for the final exam
The final exam will take place on Sunday, April 26, 2009, 4:00-6:30 pm, in room ED 123. You can bring any notes, printouts, and books you like. Calculators can be used, but no equipment with the possibility to connect to the internet is allowed.
For questions or comments, please feel free to contact me anytime
(see my homepage for email, phone number, etc.).
Apr 18, 2009