Math 321: Advanced Calculus for the Physical Sciences
Prereq:
Math 222, Math 234 .
If you did not get a B or higher, you are probably in over your head. If you did get a B or higher, don't coast, this class is not easy.
The 4 stages of learning: are you at stage 1 or at stage 2? Our goal is to get to stage 3 but we've got to go through 2 first...
Current Textbook None!!
Course notes are posted below.
Some of the material is covered in your Calculus book (Math 222, 234), so you'll use
that book again, hopefully you didn't throw it away!
Common Latin Abbreviations used in scientific and technical literature.
Quick Outline (Detailed Outline)
- Vector, Matrix and Tensor Algebra (in 3D Euclidean space)
- Vector Calculus
- Complex Calculus
Lecture notes
(i.e. this is what you should work on mostly
)
[To save paper, do not print too quickly as these notes are still under development, changes and additions occur during the semester]
If you have trouble with the lecture notes, use the resources at your disposal: ask questions in class, go to office hours to ask questions, go to the discussions, talk to other students and struggle on issues together, but it all needs to come from you.
You want to be mad if someone just tells you how to do it, you want to be able to read the notes and do the problems on your own. You want to develop the ability to figure things out independently and have the confidence to know when you did it right. This does not happen without effort from you.
Supplementary information
There are all sorts of good things available in other books and on the web... although don't believe everything you read!
-
Basics of vectors in 3D space and generalizations of the vector concept (scroll down to `Representation of a vector' first. Unfortunately that page uses the old fashioned
i, j, k notation and refers too much to that orthogonal basis [not anymore, I replaced i, j, k by e1, e2, e3 (2007/01/23)],
but all these things have geometric interpretations that are important, so focus on the pictures not the formula.)
The Summation convention page explains why
you should forget about i, j, k and replace them in your mind with
e1, e2, e3, as we always do in Math 321, and beyond.
These vector concepts and the boldface notation are due to
Josiah Willard Gibbs , yep, the same fellow that you
have heard about in Chemistry, Physics, Thermodynamics, Statistical Mechanics,...
Here's a link to the paper where
Einstein introduced the summation convention . This is his paper on the foundation of general relativity. The summation convention is introduced at the bottom of page 158.
- Index notation, Summation convention
- Chapter 3 of the book by R. Panton (a UW Mech E undergrad!!) on `Incompressible Flows' (Wiley 1996), this is a well-known fluid mechanics book in the engineering community.
Chapter 3 reviews the index (or indicial) notation and vector calculus as needed for fluid dynamics.
-
The Primer on Index Notation is for an undergrad course on ElectroMagnetism at MIT.
- Here's an
appendix to a course in solid mechanics at Brown U, also nice and compact.
- If you prefer West to East, here's a Primer on index notation from Colorado for a course in
Environmental Fluid Mechanics in the dept of
Civil, Environmental and Architectural Engineering at U of Colorado.
- Then of course there is Wikipedia which is generally quite good and has special pages on
the Summation convention (recommended reading), Levi-Civita (recommended), Kronecker delta,
cross-product etc. (you know how to click and follow links, don't you?)
Hopefully, this will help, or at least convince you that this is not just for the sake of torturing you. All of these concepts are vital to mechanics, electromagnetism, fluid mechanics, solid mechanics, general relativity, geometry, algebra, etc...