MAC 2313 (sec 82629): 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 syllabus, and comments are welcome.
If you are interested, there is also tutoring for this class available.
Topics discussed in class
- 08/25/08: three-dimensional coordinate systems, elementary properties of vectors
Literature: [Tho08, Ch. 12.1-12.2]
- 08/27/08: dot product of vectors
Literature: [Tho08, Ch. 12.3]
- 08/28/08: cross product of vectors
Literature: [Tho08, Ch. 12.4]
- 08/29/08: lines in space, computing the distance from a point to a line
Literature: [Tho08, Ch. 12.5]
- 09/03/08: describing planes in space
Literature: [Tho08, Ch. 12.5]
- 09/04/08: intersection of planes
Literature: [Tho08, Ch. 12.5]
- 09/05/08: angles between planes, distance from a point to a plane, cylinders and quadric surfaces
Literature: [Tho08, Ch. 12.5, Ch. 12.6]
- 09/08/08: vector-valued functions, derivatives of vector-valued functions
Literature: [Tho08, Ch. 13.1]
- 09/10/08: integrals of vector functions, modeling projectile motion
Literature: [Tho08, Ch. 13.1, Ch. 13.2]
- 09/11/08: modeling projectile motion, arc length
Literature: [Tho08, Ch. 13.2, Ch. 13.3]
- 09/12/08: unit tangent vector T, curvature; Homework 1 is available
Literature: [Tho08, Ch. 13.3, Ch. 13.4]
- 09/15/08: normal vector N, binormal vector B
Literature: [Tho08, Ch. 13.5]
- 09/17/08: introduction to functions in several variables, limits of functions in several variables
Literature: [Tho08, Ch. 14.1, Ch. 14.2]
- 09/18/08: examples for limits of functions with two variables
Literature: [Tho08, Ch. 14.2]
- 09/19/08: partial derivatives
Literature: [Tho08, Ch. 14.3]
- 09/22/08: derivatives of higher order, differentiability
Literature: [Tho08, Ch. 14.3]
- 09/24/08: chain rule for differentiation
Literature: [Tho08, Ch. 14.4]
- 09/25/08: directional derivatives and gradient vectors
Literature: [Tho08, Ch. 14.5]
- 09/26/08: gradients and tangents to level curves
Literature: [Tho08, Ch. 14.5]
- 09/29/08: tangent planes and normal lines; Homework 2 is available
Literature: [Tho08, Ch. 14.6]
- 10/01/08:standard linear approximation of a function
Literature: [Tho08, Ch. 14.6]
- 10/02/08: first and second derivative test for local extreme values
Literature: [Tho08, Ch. 14.7]
- 10/03/08: finding absolute/global extrema
Literature: [Tho08, Ch. 14.7]
- 10/06/08:Langrange multipliers
Literature: [Tho08, Ch. 14.8]
- 10/08/08:Lagrange multipliers: examples
Literature: [Tho08, Ch. 14.8]
- 10/09/08: partial derivatives with constrained variables, Taylor's formula for two variables
Literature: [Tho08, Ch. 14.9, Ch. 14.10]
- 10/10/08: solution of Homework 2; Exam 1 is available
- 10/13/08: introduction to double integrals
Literature: [Tho08, Ch. 15.1]
- 10/15/08: evaluating double integrals: examples
Literature: [Tho08, Ch. 15.1]
- 10/16/08: area, moments, and centers of mass
Literature: [Tho08, Ch. 15.2]
- 10/17/08: double integrals in polar form
Literature: [Tho08, Ch. 15.3]
- 10/20/08: introduction to triple integrals
Literature: [Tho08, Ch. 15.4]
- 10/22/08: evaluating triple integrals: examples
Literature: [Tho08, Ch. 15.4]
- 10/23/08: triple integrals in cylindrical and sherical integrals, Jacobian determinant
Literature: [Tho08, Ch. 15.6]
- 10/24/08: triple integrals in cylindrical and spherical coordinates, Jacobian determinant
Literature: [Tho08, Ch. 15. 6, Ch. 15. 7]
- 10/27/08: solution of Homework 3
- 10/29/08: substitutions in double integals
Literature: [Tho08. Ch. 15.7]
- 10/30/08: substitutions in triple integrals
Literature: [Tho08, Ch. 15.7]
- 10/31/08: using triple integrals to compute the mass and the center mass, introduction to line integrals
Literature: [Tho08, Ch. 15.5, Ch. 16.1]
- 10/31/08: line integrals, introduction to vector fields; Homework 3 is available
Literature: [Tho08, Ch. 16.1, Ch. 16.2]
- 11/03/08: work done by a force over a curve in space, flow integral
Literature: [Tho08, Ch. 16.2]
- 11/05/08: flux across a plane curve
Literature: [Tho08, Ch. 16.2]
- 11/06/08: path independence, fundamental theorem for line integrals
Literature: [Tho08, Ch. 16.3]
- 11/07/08: finding a potential function, introduction to divergence
Literature: [Tho08, Ch. 16.3, Ch. 16.4]
- 11/10/08: divergence and k-component of the curl; Green's Theorem
Literature: [Tho08, Ch. 16.4]
- 11/12/08: applying Green's Theorem
Literature: [Tho08, Ch. 16.4]
- 11/13/08: formula for surface area
Literature: [Tho08, Ch. 16.5]
- 11/14/08: finding surface area; surface integrals
Literature: [Tho08, Ch. 16.5]
- 11/17/08: parametrized surfaces
Literature: [Tho08, Ch. 16.6]
- 11/19/08: parametric surface integrals
Literature: [Tho08, Ch. 16.6]
- 11/21/08: Stokes' Theorem, divergence theorem
Literature: [Tho08, Ch. 16.7, Ch. 16.8]
- 11/24/08: divergence theorem
Literature: [Tho08, Ch. 16.8]
- 11/26/08-12/03/08: exercises for the final exam
For questions or comments, please feel free to contact me anytime
(see my homepage for email, phone number, etc.).
Dec 11, 2008