Infinite Series
Infinite series are among the most powerful and useful tools that you've encountered in your introductory calculus course. It's easy to get the impression that they are simply a clever exercise in manipulating limits and in studying conver
Mathematical Tools for Physics
by James Nearing
Physics Department University of Miami
[email protected]
www.physics.miami.edu/nearing/mathmethods/
Copyright 2003, James Nearing Permission to copy for individual or classroom use is granted. QA 37.2 Rev.
Vector Spaces
The idea of vectors dates back to the middle 1800's, but our current understanding of the concept waited until Peano's work in 1888. Even then it took many years to understand the importance and generality of the ideas involved. This one und
Vector Calculus 2
There's more to the subject of vector calculus than the material in chapter nine. There are a couple of types of line integrals and there are some basic theorems that relate the integrals to the derivatives, sort of like the fundamental
Vector Calculus 1
The first rule in understanding vector calculus is draw lots of pictures. This subject can become rather abstract if you let it, but try to visualize all the manipulations. Try a lot of special cases and explore them. Keep relating the m
Calculus of Variations
The biggest step from derivatives with one variable to derivatives with many variables is from
one to two. After that, going from two to three was just more algebra and more complicated pictures.
Now the step will be from a nite num
Tensors
You can't walk across a room without using a tensor (the pressure tensor). You can't align the wheels on your car without using a tensor (the inertia tensor). You definitely can't understand Einstein's theory of gravity without using tensors (many
Partial Differential Equations
If the subject of ordinary differential equations is large, this is enormous. I am going to examine only one corner of it, and will develop only one tool to handle it: Separation of Variables. Another major tool is the metho
Operators and Matrices
You've been using operators for years even if you've never heard the term. Differentiation falls into this category; so does rotation; so does wheel-alignment. In the subject of quantum mechanics, familiar ideas such as energy and m
Differential Equations
The subject of ordinary differential equations encompasses such a large field that you can make a profession of it. There are however a small number of techniques in the subject that you have to know. These are the ones that come up
Numerical Analysis
You could say that some of the equations that you encounter in describing physical systems can't be solved in terms of familiar functions and that they require numerical calculations to solve. It would be misleading to say this however,
Multivariable Calculus
The world is not one-dimensional, and calculus doesn't stop with a single independent variable. The ideas of partial derivatives and multiple integrals are not too different from their single-variable counterparts, but some of the d
Fourier Analysis
Fourier series allow you to expand a function on a finite interval as an infinite series of trigonometric functions. What if the interval is infinite? That's the subject of this chapter. Instead of a sum over frequencies, you will have an
Fourier Series
Fourier series started life as a method to solve problems about the flow of heat through ordinary materials. It has grown so far that if you search our library's catalog for the keyword "Fourier" you will find 618 entries as of this date. I
Densities and Distributions
Back in section 12.1 I presented a careful and full definition of the word "function." This is useful even though you should already have a pretty good idea of what the word means. If you haven't read that section, now would be
Complex Variables
In the calculus of functions of a complex variable there are three fundamental tools, the same fundamental tools as for real variables. Differentiation, Integration, and Power Series. I'll first introduce all three in the context of comp
Complex Algebra
When the idea of negative numbers was broached a couple of thousand years ago, they were considered suspect, in some sense "real." Later, when probably one of the students of Pythagoras discovered not that numbers such as 2 are irrational
Basic Stuff
1.1 Trigonometry The common trigonometric functions are familiar to you, but do you know some of the tricks to remember (or to derive quickly) the common identities among them? Given the sine of an angle, what is its tangent? Given its tangent