The following are pdf notes compiled by A. Havens as supplementary course materials for math 233. They contain additional exercises and problems for practice and offer some challenges to those who are interested.
- Notes on Vector Algebra
http://people.math.umass.edu/~havens/VectorAlgebra.pdf”
Contains a discussion of vector arithmetic and elementary properties of vectors (which are stated sufficiently generally to work as axioms of a vector space), the dot product, cross product, lines and planes. The exercises and problems are purely optional, for your practice, amusement, and enrichment. In the 233 of yesteryear I used them as extra credit problems to allow students to make up for lost quiz points.
2. The Suspended Banana Problem
http://people.math.umass.edu/~havens/Suspended_banana_problem.pdf
A handwritten example computing tensions using Newton’s second law, and describing how one can think about component decomposition using dot products. Obligatory apologies for the handwriting.
3. Notes on Vector Valued Functions and Curves
http://people.math.umass.edu/~havens/VVFsandCurves.pdf
Contains a discussion of calculus with vector valued functions including arc-length, as well as projectile motion. These notes do not discuss curvature (as curvature is not in the 233 curriculum). The optional notes below on curvature and natural frames cover topics skipped in our coverage of chapter 13, and then some.
4. Notes on Curves, Curvature, and Natural Frames
http://people.math.umass.edu/~havens/SpaceCurvesandFrames.pdf
These optional notes contain additional material on differential geometry of curves, including curvature, torsion, and acceleration expressed in natural frames (such as tangential and normal components). There is also an introduction to cylindrical and spherical coordinates (the spherical coordinates in these notes use a different convention than we will later in the course; the effect is exchanging sin phi and cosine phi). Some desired pictures and figures are missing (until I have time to scan some drawings and update.) The final sections discuss the Keplerian problem of orbital mechanics and include some challenge problems leading up to an interesting proof of Kepler’s laws via vector algebra and calculus of curves. These are provided as is, for the benefit of curious math, physics, and astronomy majors, or anyone else with an interest in these topics.
5. Notes on Partial Derivatives
http://people.math.umass.edu/~havens/Partials.pdf
These notes cover partial derivatives, the multivariable chain rule, tangent planes, linear approximations, differentiability, directional derivatives, and the gradient. They contain examples and exercises throughout.
6. Slides on limits and continuity for multivariate functions
http://people.math.umass.edu/~havens/LimContBivar.pdf
This slideshow covers the definitions of limits and continuity for multivariate functions, and goes through some case studies in limits of two variable functions. It briefly notes how these ideas generalize for functions of 3 or more variables.
7. Notes on Double integrals
The double integrals notes have been updated and split into parts. The file sizes are somewhat large; if I find a simple way to reduce them I’ll update accordingly. Future notes will likely appear in moodle rather than here.
http://people.math.umass.edu/~havens/DoubleIntegralsRectangles.pdf
http://people.math.umass.edu/~havens/DoubleIntegralsOverGeneralRegions.pdf
http://people.math.umass.edu/~havens/DoubleIntegralsPolar.pdf
8. Notes on Vector Calculus
http://people.math.umass.edu/~havens/BriefVectorCalculus.pdf
These notes review gradients from the perspective of vector fields, and provide an introduction to vector fields in 2 and 3 dimensions, and develop the theory of line integrals, culminating in the fundamental theorem of line integrals and Green’s Theorem, flux, surface integrals, Stokes-Kelvin Theorem and the divergence theorem. They contain many examples and difficult problems, and are still under construction. The problems aren’t assigned, but are there for you to sharpen your abilities and delve more intensely into the concepts if desired (if you are a mathematics major, then the delving is recommended). A word of caution: as with the notes on curvature, these notes use a different spherical coordinate convention than the standard.