Math H53: Honors Multivariable Calculus

UC Berkeley, Fall 2025

Instructor

Michael Hutchings
hutching@math.berkeley.edu.
Office: 923 Evans.
Tentative office hours: Wednesday 9:00-10:00 and 11:00-12:00.

GSI

Ryan Martinez
ryan_martinez@berkeley.edu.
Office: 1045 Evans.
Tentative office hours: Tuesday 11:00-12:00 and Friday 4:00-5:00.

Course Goals

This course will give an intense and (as much as feasible) mathematically rigorous introduction to multivariable calculus, with a bit of background from linear algebra and real analysis. It is intended for prospective math majors, or students with a serious interest in mathematics. There will be definitions! There will be proofs! There will be differential forms! And hopefully it will be fun.

Course outline

The following outline is tentative, and some adjustments might be needed as we go along.

1. Crash course in linear algebra (about 2 weeks)
- Vectors in R^n, dot product, cross product in R^3
- Linear maps and matrices
- Vector spaces, subspaces of R^n, kernel and image of a linear map
- Determinants, eigenvalues
2. Crash course in real analysis (about 2 weeks)
- Metric spaces, limits, continuous functions, delta-epsilon proofs.
- Cauchy sequences, completeness, axioms of the real numbers
- Compact sets
3. Differentiation in multiple variables (about 4 weeks)
- Partial derivatives
- Differentiable functions of multiple variables
- The chain rule
- Inverse function theorem and implicit function theorem.
- Level sets of functions and tangent spaces.
- Optimization: critical points, second derivative test, Lagrange multipliers
4. Integration in multiple variables (about 2 weeks)
- Outline of Riemann integration, Fubini's theorem
- Computations of integerals over regions in R^2 and R^3
- Diffeomorphisms, change of variables formula
5. Fundamental theorems of calculus (about 4 weeks)
- Curves, line integrals, conservative functions, Green's theorem in two dimensions
- Integrals over surfaces, Stokes theorem and divergence theorem in three dimensions
- Differential forms and Stokes theorem in n dimensions

Textbooks and resources

This course will not be closely following any single textbook. However I did list the book by Munkres as a recommended text for the course. The following is a longer list of books and resources that you might find useful. There are many other books out there on these topics, and you can explore to see what suits your background and learning style. Beware that notation and conventions can vary a bit from one book to another.

Writing proofs

This handout explains the basic mechanics of writing mathematical proofs. (There are whole books which expound on this a lot more. But once you learn the basic mechanics, writing proofs is a matter of practice, practice, and more practice.)

Linear algebra

Henry Helson, Linear algebra. This is a good concise explanation of basic linear algebra (we will cover some fraction of this). Legal download here from the UCB computer nextwork.
Vineet Gupta, David Nadler, and Alexander Paulin, Linear algebra. This is the textbook for Math 56, and it has abundant examples and pictures. There is a free download link here.

Real analysis

Charles Pugh, Real Mathematical Analysis. This is a good reference for real analysis (we will cover a small fraction of this material) with lots of useful exercises. It is currently the textbook for Math H104, which covers chapters 1 through 4. Chapter 5 is a long chapter on multivariable calculus. Legal download here from the UCB computer nextwork.

Multivariable calculus

James Munkres, Analysis on Manifolds. This book covers a lot of what we will do in the course (and more) and seems to have good reviews from students for its pedantic approach.
Michael Spivak, Calculus on Manifolds. This classic text is like a much more concise version of Munkres. Some students love it, others find it difficult.
Peter Lax and Maria Terrell, Multivariable Calculus with Applications. This is a nice multivariable calculus book which is at a slightly easier level than Munkres, with lots of computations. Legal download here from the UCB computer network.
If you would like to see how I used to teach Math 53 without the H, you can view my videos here. That course uses Stewart's calculus book, which has lots of good computational exercises.

Homework

Homework assignments will be posted here periodically. Homework will not be graded. However it is strongly recommended that you do as much of the homework as you can. Trying to learn mathematics without doing homework exercises is like trying to get in shape by watching sports on TV. Working on the homework in groups is encouraged.

HW#1, suggested due date 9/15.
HW#2, suggested due date 9/26.
HW#3, suggested due date 10/2.
HW#4, suggested due date 11/3.
HW#5, suggested due date 11/24.
HW#6, suggested due date 12/12.

Exams and grades

Course grades will be determined as follows:

Discussion section: 25% (based on weekly quizzes with some extra credit for participation)
Midterm #1: 25%
Midterm #2: 25%
Final exam: 50%
Lowest exam score: -25%

Each of the above four components of the course grade will be curved to a uniform scale before dropping the lowest exam score and averaging. Incomplete grades can be given only if both (1) an unanticipated event such as illness prevents you from completing the course, and (2) you are otherwise passing the course with a grade of C or above.

Electronic devices and AI

Electronic devices should not be used in class except as needed for learning the class material and when this does not distract other students. Appropriate uses of devices include taking lecture notes electronically, and looking up relevant mathematics. However if you have a question, it might be better to simply ask the question out loud, as other students may be wondering the same thing.

Use of ChatGPT and similar tools for graded work is not allowed (but this should not be possible anyway because the only graded work will be paper and pencil quizzes and exams). You can use these tools for studying, although I generally discourage this, as these AI tools currently make many errors (some blatant and some subtle), and when they answer questions correctly they often spare you from doing the work that you need to do in order to learn. There are also various ethical concerns with their use (e.g. they may be stealing human work or have an excessive impact on the environment). However AI will probably play some nontrivial role in mathematical work in the future. For now it can be fun (after you have learned the material) to test AI on math questions and see how it does.

DSP accommodations

Students requiring DSP accommodations should have a letter sent from the DSP office to the instructor, and should contact the instructor and/or GSI to make any necessary arrangements.

Academic honesty

Graded work (quizzes and exams) is expected to be done within the time limits of the exam, and without aid from other people, books or notes, or the internet, unless explicitly allowed by the rules of the exam. The code of student conduct may be found here.

Lecture summaries and references

After each lecture, brief summaries and references will be posted here.

(Thursday 8/28) Vectors in R^n. Definition of dot product, length, perpendicularity, orthogonal projection, and angle between two vectors. Proof of the Pythagorean theorem, the Cauchy-Schwarz inequality, and the geometric interpretation of dot product. (We completed this discussion with the triangle inequality in the following lecture.)
(Tuesday 9/2) Linear independence and span of vectors in R^n. Given k vectors in R^n, if they are linearly independent then k <= n, and if they span all of R^n then k >= n.
(Thursday 9/4) Subspaces of R^n. Every subspace has a basis, and all bases of a subspace have the same number of elements (the dimension of the subspace). Introduction to matrices and linear maps. (We didn't have much time for matrices and linear maps, but every linear algebra book discusses these extensively.)
(Tuesday 9/9) No class (I have a medical procedure)
(Thursday 9/11) Kernel and image of a linear map. "Rank-nullity theorem". Invertible linear maps. Introduction to determinants.
(Tuesday 9/16) Metric spaces. Limits of sequences in a metric space. Some simple proofs regarding limits. Definition of a complete metric space.
(Thursday 9/18) Remarks on the real numbers: you should know that they are complete and have the least upper bound property. Open sets, closed sets, and compact sets in metric spaces. A subset of R^n is compact if and only if it is closed and bounded. (Only sketched the "if" part of the proof.)
(Tuesday 9/23) Limits of functions between metric spaces. Some delta-epsilon proofs. Continuous functions between metric spaces.
(Thursday 9/25) Equivalence of two definitions of continuous functions between metric spaces. Proof that a continuous function on a nonempty compact set has a maximum and a minimum.
(Tuesday 9/30) Differentiable maps from R^n to R^m. Partial derivatives. Directional derivatives. Example (using polar coordinates) of a non-differentiable function whose partial derivatives are defined.
(Thursday 10/2) Midterm #1, in class. Will cover the material up to 9/25.
(Tuesday 10/7) Gradient and level sets. First version of the chain rule, for the derivative of a function along a parametrized curve.
(Thursday 10/9) General chain rule with proof. Computing partial derivatives in different coordinate systems. Implicit partial differentiation and statement of the implicit function theorem.
(Tuesday 10/14) Statement and proof of the contraction mapping theorem. Statement and most of the proof of the inverse function theorem.
(Thursday 10/16) Proof of the implicit function: a level set of a continuously differentiable function with nonzero gradient is locally a graph. The tangent space to a graph or a level set.
(Tuesday 10/21) Optimization. Critical points. Introduction to Lagrange multipliers.
(Thursday 10/23) More about Lagrange multipliers and optimization.
(Tuesday 10/28) Eigenvalues of symmetric matrices. The second derivative test.
(Thursday 10/30) Definition of the Riemann integral (no proofs), statement of Fubini's theorem. Calculation of some double integrals.
(Tuesday 11/4) Midterm #2, in class. Will cover the material up to 10/28, with emphasis on the material after 9/25.
(Thursday 11/6) Change of variables in multiple integrals. Integrals in polar and spherical coordinates.
(Tuesday 11/11) Academic and Administrative Holiday, no class
(Thursday 11/13) Integral of a function over a curve with respect to arc length. Integral of a function over a surface with respect to surface area.
(Tuesday 11/18) Integral of a vector field along an oriented curve. Fundamental theorem of line integrals. Started discussing conservative vector fields.
(Thursday 11/20) Criteria for vector fields to be conservative. Green's theorem.
(Tuesday 11/25) TBA
(Thursday 11/27) Academic and Administrative Holiday, no class
(Tuesday 12/2) TBA
(Thursday 12/4) TBA
(Tuesday 12/9) Reading/Review/Recitation week, optional review session
(Thursday 12/11) Reading/Review/Recitation week, optional review session
(Tuesday 12/16, 8-11am, in the usual classroom, 223 Dwinelle) Final exam. Will cover the entire class, with somewhat more emphasis on the last third.