Honors Mathematics III
Linear Algebra and Functions of Multiple Variables

Volume: 30 lectures @ 90 minutes

Syllabus: Download Course Description (PDF)

Literature: The first part of this course gives some necessary background in linear algebra. There are several books that might be helpful:

• J. Hefferon, Linear Algebra can be obtained here.
  This is very much a US-style textbook for a first course in linear algebra, with emphasis on calculations. Abstract concepts are omitted or introduced only when flowing from previous calculations.

• K. Jänich, Linear Algebra, Springer 1994. Free download here from within the SJTU network.
  This is the translation of a classic German book that is used as a reference in nearly every course on linear algebra in Germany. Like Spivak's Calculus, it is written in a conversational style, eschewing the definition-theorem-proof-example chain that is so prevalent in serious textbooks.

• S. Axler, Linear Algebra Done Right, (2nd Edition), Springer 1997. Free download here from within the SJTU network.
  This very readable book by a US-American author is the reaction to the prevalent style of linear algebra textbook in the US (see Hefferon above for an example). It is very much in the continental-European tradition of putting structures and relationships before calculations. (You will notice similarities with Jänich's book above.)

• S. Lang, Introduction to Linear Algebra, Springer 1986. Free download here from within the SJTU network.
  Another serious book on linear algebra, written by an eminent french-born mathematician (now deceased). Although first published in 1986 and in much the same vein as Axler's book, it never became widely popular as an undergraduate textbook in the US, perhaps because it was considered too difficult. But it is actually very readable.

• S. Lang, Linear Algebra, Springer 1987. Free download here from within the SJTU network.
  A more advanced version of the above book, treating many topics that we do not have time for in this course. Read this for a deeper understanding of linear algebra.

• R. Shakarchi, Solutions Manual for Lang’s Linear Algebra, Springer 1996. Free download here from within the SJTU network.
  You may find this useful :-)

For the remainder of the course, focussing on multidimensional calculus, there a several books that can be read alongside the lecture notes. However, none of these books covers the course material in exactly the same way as we do and most of them also include much additional material. I am teaching the material in the style of German textbooks, few of which have been translated into English. In the anglo-saxon literature, one is often faced either with (many!) easy calculus books that ignore general concepts or graduate books that are much too advanced.

• S. Lang, Calculus of Several Variables, 3rd Edition, Springer 1987. Free download here from within the SJTU network.
  Yes, Serge Lang wrote a lot of undergraduate and graduate textbooks. This one overlaps with some of the course, but takes a different approach to some topics. It is perhaps less abstract than what we do.

• W. Fleming, Functions of Several Variables, 2nd Edition, Springer 2010. Free download here from within the SJTU network.
  Like Lang's book, this one also overlaps with part of our course. However, the part that does not overlap is more abstract than what we do, so it is useful for further reading.

• J. J. Duistermaat, J. A. C. Kolk and J. P. van Braam Houckgeest, Multidimensional Real Analysis I, II, Cambridge University Press 2004. Free download here from within the SJTU network. Volume I and Volume II.
  This is for the ambitious student. Like Lang, Duistermaat was a very well-known and respected mathematician, and he does everything rigorously and proves everything.

After you have completed this course, you may be interested in discovering more about surfaces and Stokes's theorem in higher dimensions. For this, I recommend

• K. Jänich, Vector Analysis, Springer 2001. Free download here from within the SJTU network.
  Like the Linear Algebra quoted above, this book is written in a light, conversational style. You now have the prerequisites to read it, but it is not as easy as it looks!

• M. Spivak, Calculus on Manifolds, Westview Press 1971. Link to Publisher's Description.
  Spivak's second book formally does not require any prerequisites apart from some linear algebra and the Calculus we used in Vv186. However, the book is not very easy to read, so be warned! It's probably a good idea to have a look at it only after finishing Vv285.

Prerequisites: Vv186.

Background and Goals: The sequence Honors Math Vv186-285-286 is an introduction to calculus at the honors level. It differs from the Applied Calculus sequence in that new concepts are often introduced in an abstract context, so that they can be applied in more general settings later. Most theorems are proven and new ideas are shown to evolve from previously established theory. Initially, there are fewer applications, as the emphasis is on first establishing a solid mathematical background before proceeding to the analysis of complex models.

The present course includes an overview of linear algebra followed by the calculus of functions in multidimensional euclidean space.

Key Words: Linear systems of equations and the Gauss-Jordan algorithm; finite-dimensional vector spaces, linear independence and bases; scalar products and Gram-Schmidt orthonormalization; linear maps and matrices; determinants; topology of normed spaces; the derivative and applications; curves, potentials and vector fields; higher derivatives and applications; the Riemann integral in n-dimensional space; integration on curves and surfaces; the classical theorems of vector analysis in three dimensions (Green, Gauss and Stokes); the inverse and implicit function theorems (as time permits).

Detailed Content: The course starts with an introduction to linear algebra, featuring the Gauß-Jordan algorithm for solving systems of equations, the theory of finite dimensional vector spaces, linear maps, matrices and determinants. These concepts are not only essential tools for the following calculus in ${ℝ}^n$ but (together with material to be presented in Vv286) comprise the content of an independent course in linear algebra.

The second part of the course starts with a discussion of convergence and continuity before embarking on differential calculus in ${ℝ}^n$ and, more generally, in finite-dimensional vector spaces. Derivatives are defined as multilinear maps between vector spaces, and the Jacobian is introduced as their representing matrix. Curves in ${ℝ}^n$ and potential functions serve as specific examples and starting points for various applications to engineering and physics. Curve integrals of scalar and vectorial functions are introduced. The second and higher derivatives are defined as multilinear maps, with the Hessian of a potential function a concrete example. Based on the properties of the Hessian, extrema (with and without constraints) are discussed.

The third part focuses on vector fields, surfaces and integration. First, the concepts of divergence and rotation (curl) are introduced together with the corresponding flux and circulation integrals in ${ℝ}^2$. A more general discussion of integration of scalar functions on Jordan-measurable sets as well as scalar functions and vector fields on surfaces in ${ℝ}^{\mathrm{n}}$ follows. For brevity and simplicity, only parametrized surfaces are treated, i.e., in contradistinction to the theory of curves discussed previously, no definition of surfaces independent of parametrizations is made. The classical theorems of Green in ${ℝ}^2$ and Stokes (after a rigorous definition of surfaces with boundary) in ${ℝ}^3$ are introduced and proven in simplified cases. The theorem of Gauß (a.k.a. Ostrogradskii or divergence theorem) is proven in ${ℝ}^{\mathrm{n}}$ and Green's formulas round off the course.

Alternatives: Vv255 (Applied Calculus III) is an applications-oriented course, which covers much of the same material.

Subsequent Courses: Vv286 (Honors Mathematics IV) is the natural sequel.

Overview:

Lecture	Lecture Subject
1	Systems of Linear Equations
2	Finite-Dimensional Space
3	Geometry of Angles
4	Linear Transformations
5	Calculus of Linear Transformations
6	Calculus of Linear Transformations
7	Theory of Systems of Linear Equations
8	Volumes and more Geometry in ${ℝ}^3$
9	Volumes and more Geometry in ${ℝ}^{\mathrm{n}}$
10	Further Properties and Applications of the Determinant
11	First Midterm Exam
12	Convergence and Continuity
13	Functions and the Derivative
14	Properties and Applications of the Derivative
15	Properties and Applications of the Derivative
16	Curves in ${ℝ}^{\mathrm{n}}$
17	Curves in ${ℝ}^{\mathrm{n}}$
18	Potential Functions
19	The Second Derivative
20	Free Extrema of Potential Functions
21	Constrained Extrema of Potential Functions
22	Second Midterm Exam
23	Vector Fields and Line Integrals
24	Circulation and Flux
25	The Riemann Integral and Measurable Sets
26	Integration in Practice
27	Surfaces and Surface Integrals
28	Surfaces and Surface Integrals
29	Theorems of Gauß and Stokes
30	Final Exam

The exact location of the exam classes may vary according to the term schedule. The exams in the list above are placed so that the exam topics are precisely those of the preceding lectures.

Lecture Slides (2017 version; PDF):

• Summer 2017

(There will always be minor modifications from one iteration of the course to the next; if you are presently taking the course, it is not advisable to print out all these slides at the beginning of the term.)