Math 530: Multivariable Mathematics Resource Page:
This page collects many solutions and links which aid in the study of multivariate calculus and advanced calculus. In particular, we collect resources to add examples and insights for the students of Math 530. Naturally you should first consult announcements and the Syllabus provided by your instructor before weighing advice offered here.
YouTube Playlists from Calculus III (Math 231):
The video resources below were originally created for Math 231 at Liberty University. They are helpful background for Math 530. Enjoy:
Useful Visualizations and Calculations:
Calculus III Lecture Notes:
I have a pdf with all of these together (you should save a local copy for future reference and ease of use, this is about 15 Mb),
Below is a brief overview of each of the chapters:
- 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. (Problems 1 to 45 given at end of chapter)
- 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. (Problems 46 to 69 given at end of chapter)
- 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. (Problems 70 to 75 given at end of chapter)
- 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. (Problems 76 to 123 given at end of chapter)
- 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. (Problems 124 to 150 given at end of chapter)
- Chapter 6: definition of double and triple and $n$-fold integral is given as natural extension of single integral; the integral is just a continuous sum. Naturally, the questions of existence and convergence are too subtle for this first course, but we do attempt to make the reader aware of the question. Salas and Hille probably has more insight on this point. All the basic computational methods are illustrated in depth. Simple iterated integrals over boxlike regions initiate the study then we go on to study TYPE I and II regions which require a good grasp of graphing in the plane. Triple integrals with nontrivial bounds likewise require a good grasph of graphs and level surfaces in three dimensional space. The last half of this chapter is devoted to change of variable for multiple integrals. We make use of the general derivative introduced in Chapter 4 to appreciate the structure of the change of variables theorem. Jacobian notation is introduced and a swath of examples using the standard polar, cylindrical and spherical coordinate systems are given. Finally, breaking from mainstream treatments, we pause to introduce the wedge product and show how it reproduces the Jacobian calculation. (thanks to Bailu Zhang for helping type this chapter in Summer 2014) (Problems 151 to 177 given at end of chapter)
- 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. (Problems 178 to 250 given at end of chapter)
Solution to end of chapter problems in Fall 2014 Lecture Notes:
These are the solutions written for Fall 2011 and partly Fall 2013 of Math 231. In the Fall 2014 version of my notes for this course these problems appear at the end of each Chapter as appropriate. There are doubtless some errors in here, sorry for those, I am human, keep this in mind:
Assignments and Solutions of Fall 2014:
These are the required homework for my section of Math 231,
- 2014 Course Planner
- Mission 1: and its solution
(vectors, angles, dot and cross products)
- Mission 2: and its solution
(lines, planes, surfaces, coordinates, calculus of paths)
- Quiz 1: and its solution
- Test 1: and its solution
- Mission 3: and its solution
(limits, partial derivatives, directional derivatives)
- Mission 4: and its solution
(chain rules, normal to level or parametrized surface, differentials)
- Quiz 2: and its solution
- Test 2: and its solution
- Mission 5: and its solution
(critical points, Lagrange multipliers, multivariate Taylor)
- Mission 6: and its solution
(double and triple integrals)
- Quiz 3: and its solution
- Test 3: and its solution
- Mission 7: and its solution
(line integrals, conservative vector fields, Green's Theorem)
- Mission 8: and its solution
(surface integrals, Gauss' and Stokes' Theorems)
- Quiz 4: and its solution
- Test 4: and its solution
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Last Modified: 7-29-2020