MATH 231 Fall 2011 Homepage
Lecture Notes for Fall 2011:
I have a pdf with all of these together, but the file size was too large. Thus, I post Chapters of my 2011 notes. Notice Chapter 6 is unfinished whereas the other chapters are closer to what I intend. Of course all of these are works in progress and I appreciate comments and corrections/suggestions to improve in the next edit:
- Chapter 1
cartesian coordinates, vectors, dot and cross products, lines, planes, curves, surfaces. Parametric and implicit viewpoints are contrasted. We discuss how the concept of a graph is in some sense a middle ground. Index notation is introduced and used to streamline difficult vector identities.
- Chapter 2
calculus of vector-valued functions of a real variable (a.k.a space curves) is given, properties including product rules for scalar multiplication, dot and cross products are given. The concept of arclength is studied with some care, although the derivation from the piecewise linear approximation is left to the standard text. We introduce tangent, normal and binormal vector fields for non-stop, non-linear curves. Curvature and torsion describe the geometry of the curve and in turn the evolution of the Frenet frame as elegantly summarized by the Frenet Serret Equations. The osculating plane is studied for a plane curve. Finally we use the Frenet Frame to study three dimensional physical motion.
- Chapter 3
Open and closed sets in n-dimensional euclidean space are described. This is the natural generalization of the open and closed intervals of the real line. The basic theme is that we must replace absolute value (the distance function for one-dimension) with the norm which measures distance in several dimensions. Limits of functions from Rm to Rn are defined and I present about 5 pages of advanced calculus to explain why the natural calculations for functions of several variable are in fact justified. Having worked with some care to show how the simplest of multivariate limits are calculated we turn to the less obvious, perhaps non-continuous examples. Even in two dimensions there are infinitely many ways to approach a limit point, if even two of these paths give differing values then the limit fails to exist. However, perhaps non-intuitively, even if all possible linear paths yield the same limit it is possible for the limit to diverge. Why? Because it may happen that parabolic paths yield distinct results for distinct parabolas. Examples exist (but are not given) where all lines, and parabolas yield a particular limit and yet cubic paths yield inconsistent limits. In other words, the business of transitioning from limits along curves to two-dimensional limits is difficult. Squeeze theorems, algebra and subsitutions are used in the conclusion of this chapter to calculate some indeterminant limits.
- Chapter 4
This chapter is the heart of this course. We study the partial derivative, the tangent plane, the directional derivative and spend considerable effort connecting these concepts. The chain-rule also finds a natural extension to the multivariate case. In contrast to most popular texts of this generation we take a serious look at the concept of the derivative. This leads us to the Jacobian and ultimately the ability to see chain rules as arising from simple matrix multiplications. As is typical in any calculus course, some of the deeper results are relegated to advanced calculus, but I do not try to hide these things here. I would like you to understand that the derivative is the best linear approximation to the change in a mapping at a point. Computationally the methods of partial differentiation in this chapter are crucial to success in this course. I try very hard here to make partial differentiation more than just symbol pushing.
- Chapter 5
questions of optimization arise naturally in many applications of calculus. Analogies with single variable calculus abound here, we have to understand where critical points are found, focus on continuous functions on connected domains, local extrema are found at critical points, for a closed set we have to look at the interior and the boundary separately in much the same way as we did with the closed interval test of single-variable calculus. However, the first and second derivative tests do not generalize quite as simply. The infinity of directions we can approach a critical point renders the first derivative sign-chart approach untenable. It turns out that the second derivative test does find generalization, but to really understand it we need some linear algebraic techniques to unravel cross-terms. If cross terms happen to absent from the example then extrema of functions of any number of variables are not too hard to understand. We attempt to show how this is understood from the multivariate Taylor series. Extrema on the boundary are dealt with via the method of Lagrange multipliers.
- Chapter 6 (part a) (part b) (part c)
I follow the standard texts fairly closely in these handwritten notes. Nothing terribly profound in here, except of course the things which are implicitly assumed about the existence of integrals (you'll have to find a good analysis text) or the change of variables theorem (perhaps see Analysis on Manifolds by Munkrese). What I do show is how to perform the standard calculations here. These computational skills are another essential skill you must master before you leave this course. I include some homework solutions, many examples are given. These are based on some now extinct edition of Stewart.
- Chapter 7
we study the calculus of vector fields on this chapter.Integrals along curves and surfaces are studied. The parametric viewpoint is emphasized. The construction of line and surface integrals are both based in parametric analysis. We take some time to show why the definition is indpendent of the parameterization. This is crucial logically as there exist an infinite number of parameterization for a given geometric object. The line and surface integrals, like the definite integral the generalize, yield a particular number once the vector field and curve or surface is given. Orientation must be specified to unambiguously define these integrals and we discuss why and how this is accomplished. The fundamental theorem of calculus is seen to generalize to four seemingly distinct theorems: the fundamental theorem of calculus for line integrals, Greene's theorem, Stoke's theorem and Gauss' theorem. We also study the concept of a conservative vector field. We study how the topology of the domain cannot be neglected if we are to completely understand the concept. I discuss electrostatics in two dimensions and more generally the role singularities play in potential theory. We conclude by working out a few of Green's integral identities. We find an inversion of the Laplacian which has interesting implications for properties of the voltage function. You can also read Susan Colley's Vector Calculus, she also discusses these theorems. Some of the heavy lifting in this chapter is intended to help you think more deeply about the structure of electromagnetic fields. It may be that you have not had a course in this topic, if that is the case then focus on the math and keep it in mind as you continue your education.
Problem Sets and Solutions:
Useful Materials and Links:
- THE ZOO.
This page contains the animations I have created for this course. I mention these at times in the lecture notes. Be patient, it may take a few moments for the animals to wake up.
- Multivariable Calculus Archive
here I have collected the many solutions, reviews, old tests and Lecture Notes from previous calculus III courses I've taught. Take a look.
- Additional Comments on the basics of "Einstein" notation.
This notation hides summations by a sneaky convention. It saves a lot of writing without sacrificing much detail.
- A note on how to compute determinants.
We use these patterns to calculate cross-products for now, Jacobians a little later on.
- My Physics 231 Page.
Many additional examples offered. Lots of good vector examples. Worth some surfing. Sorry lacks some organization, I hope to add more descriptions soon.
- Fall 2011 Syllabus.
I'm not entirely sure this belongs here, but for our convenience I'll give it a space.
Bonus Point Policy:
It is possible to earn bonus points by asking particularly good questions or suggesting corrections to errors in notes and materials on the course website. This does not include spelling or grammatical errors, those are provided for your amusement. Once I notify the class of the error you may no longer ask for that point.
I also provide a little bonus project from time to time. These are not required. It is entirely possible to earn an A without completing these. I will usually be able to take these as late as the final exam day, just ask.
- Differential Forms Project (10pts)
- If you are interested in some topic involving calculus III then suggest and alternate project( I could draft something in variational calculus if you're interested)
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Last Modified: 5-28-12