Read sections 12.1 and 12.2 in the Edwards & Penney text. Homework exercises and the 1st problem set will be posted shortly.

Problem Set #1 (due Thurs, Sept 15)

Detailed Problem Set #1

References:
D. Determinants (4 pages)
Exercises for Vectors and Matrices (1A to 1K, 12 pages)

P-set 1 due Sept 15.
Read sections 10.4 (Parametric Curves), 12.5 (Curves and Motions in Space), 12.6 (Curvature and Acceleration), and appropriate sections from the Supplementary Notes (1F, 1G, 1H, 1J).

Problem Set #2 (due Thurs, Sept 22)

Detailed Problem Set #2

References:
D. Determinants (4 pages)
Exercises for Vectors and Matrices (1A to 1K, 12 pages)
Parameterized Curves (3 pages - NEW)

Parameterized curves in R² and R³ (vector-valued functions); velocity vectors; unit tangent vector, acceleration vector; arclength, acceleration vector, curvature. Calculus of vector-valued functions.

Here's a Java applet that may help you to understand Part II, Problem 3 in Problem Set #2: Wheel (choose the "Trace" option) and click on the double arrows to activate.

Parameterized surfaces vs. surfaces described by equations; cylinders, spheres; cylindrical and spherical coordinates; spheres, hyperboloids, ellipsoids, etc.

P-set 2 due Sept 22. Read section 12.7 (Quadric Surfaces)

Practice Exam #1 Questions
Exam topics: basics of vectors and vector algebra, application to lines, planes, areas, volumes, angles, etc.; parameterized curves, velocity vectors, unit tangent and normal vectors, and related topics. [Solutions to Practice Exam Questions]

Functions of several variables and their level sets. Graphs and contours (level sets) of a function. Exam #1 [Solutions]

Read sections 13.1-13.2 (Functions of Several Variables);
13.3 (Limits and Continuity);
13.4 (Partial Derivatives).

Problem Set #3 (due Thurs, Oct 6)

Detailed Problem Set #3

P-set 3 due Oct 6.

Read sections 13.7 (Chain Rule and Implicit Differentiation), 13.8 (Directional Derivatives and the Gradient); the Supplement on the Chain Rule and Implicit Differentiation; and Supplementary Notes TA (Tangent Approximation) and do the following problems:

Problem Set #4 (due Thurs, Oct 13)

Detailed Problem Set #4

Extrema of functions, stationary points and the 2nd Derivative test; local maxima, local minima, and saddle points. Unconstrained optimization.

P-set 4 due Oct 13. Read sections 13.5 (Multivariable Optimization Problems), 13.9 (Lagrange Multipliers and Constrained Optimization), 13.10 (Critical Points of Functions of Two Variables and the Second Derivative Test); and Supplementary Notes LS (Least Squares Interpolation).

Problem Set #5 (due Mon, Oct 24)
You may use any method unless a particular method is specified.

Detailed Problem Set #5

A set of Practice Exam Questions is now posted. The exam will cover partial derivatives, linear approximation, differentials, gradient vector, normal vectors to curves and surfaces, directional derivative, the Chain Rule, implicit differentiation, stationary points, second derivative test, optimization, Method of Lagrange Multipliers. Solutions

Integration of a function f (x, y) over a region in R²; applications to volume, mass, population, etc. Average value of a function; centroid (geometric center) of a region. Calculation of integrals over regions R² via iterated single integrals (successive slicing) in Cartesian coordinates; The Fubini Theorem; interchanging the order of integration in an iterated double integral.

Calculation of integrals over regions R² via iterated single integrals in Cartesian coordinates and polar coordinates. Exam #2 [Solutions]

Read sections:
14.1 (Double Integrals)
14.2 (Double Integrals Over More General Regions)
14.3 (Area and Volume by Double Integration)
14.4 (Double Integrals in Polar Coordinates)
14.5 (Applications of Double Integrals)
14.6 (Triple Integrals); and
Supplementary Notes I (Limits in Iterated Integrals).

Problem Set #6 (due Thurs, Nov 3)

Detailed Problem Set #6

P-set 6 due Nov 3. Read sections:
14.7 (Integration in Cylindrical and Spherical Coordinates)
14.8 (Surface Area);
14.9 (Change of Variables in Multiple Integrals)
Supplementary Notes I (Limits in Iterated Integrals)
Supplementary Notes CV (Changing Variables in Multiple Integrals)
Supplementary Notes G (Gravitational Attraction)

Problem Set #7 (due Mon, Nov 14)

Detailed Problem Set #7

P-set 7 due Monday, Nov 14.

Practice Questions for Exam #3    Solutions

Calculation of line integrals. Independence of path and conservative (gradient) vector fields. Fundamental Theorem of Line Integrals.

Exam #3 [Solutions]

Read sections:
15.1 (Vector Fields, Divergence and Curl)
Notes V1 (Plane Vector Fields)
Notes V8 (Vector Fields in Space)

15.2 (Line Integrals)
Notes V11 (Line Integrals in Space)

15.3 (Fundamental Theorem and Independence of Path)
Notes V2 (Gradient Fields and Exact Differentials)
Notes V12 (Gradient Fields in Space)

15.4 (Green's Theorem)
Notes V3 (Two-dimensional Flux)
Notes V4 (Green's Theorem in Normal Form)

Do the following problems:

Problem Set #8 (due Tues, Nov 29)

Detailed Problem Set #8

Problem Set #9:

Read sections:
14.8 (Surface Area)
Notes V7 (Laplace's Equation and Harmonic Functions)
15.5 (Surface Integrals)
Notes V9 (Surface Integrals)
15.6 (The Divergence Theorem)
Notes V10 (The Divergence Theorem)

Problem Set #9 (due Tues, Dec 6)

Detailed Problem Set #9

Stokes' Theorem and examples. Geometric (coordinate-free) definition of the curl of a vector field. Proof of Stokes' Theorem from the geometric definition of curl.

Practice Questions for Exam #4    Solutions

P-set 9 due.
P-set 10 should be done but is not to be turned in.
Read sections:
15.7 (Stokes' Theorem)
Notes V13 (Stokes' Theorem)
Notes V14 (Some Topological Questions)
Notes V15 (Relations to Physics)

Problem Set #10 (not to be turned in)

Detailed Problem Set #10

More examples; proof of Green's Theorem as a special case of Stokes' Theorem, proof that curl F = 0 implies F is conservative (given appropriate conditions).

Exam #4 [Solutions]