Class timings: MWF 3-4

Tutorial hour: Wednesdays 5:30-6:30pm (link on Teams channel)

TA for the course: Ajay Nair (ajaynair@)

Recommended books: M. Spivak, Calculus on Manifolds, Reprint 2018, CRC Press.

J.R. Munkres, Analysis on Manifolds, Reprint 2018, CRC Press.

W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill International edition.

J. H. Hubbard and B.B. Hubbard, Vector Calculus, Linear algebra and differential forms, Matrix Editions, 2002.

Grading policy: 50% Homework , 20% Midterm exam, 30% Final exam.

Announcements

The final exam is December 6th (Monday) from 2-5pm.

There will be no class on Nov 8th (Monday).

There will be no classes on Oct 18th (Monday) and Oct 22nd (Friday).

The midterm exam will be on Friday, October 1st from 3-4:30pm. There will be no classes during the midterm week (Sep 27 - Oct 1).

There will be no class on Wednesday, September 8.

Classes will be online, on Microsoft Teams. The Team code for IISc students is: gfm1qro. The first meeting link has now expired.

Assignments

Weekly assignments will be posted on the Teams channel, and the problems will be available here.

Homework 1 - due on 16 August

Homework 2 - due on 23 August

Homework 3 -- due on 30 August

Homework 4 - due on 6 September

Homework 5 - due on 13 September

Homework 6 - due on 20 September

Homework 7 - due on 28 September

Midterm exam (corrected version!)

Homework 8 - due on 19 October

Homework 9 - due on 1st November (deadline extended!)

Homework 10 - due on 8th November

Homework 11 - due on 16th November

Homework 12 - due on 24th November

Final exam

Lecture notes

The recorded videos will be available on the Teams channel.

Lecture 1 About the course, and a review of linear maps

Lecture 2 More review: inner products, Cauchy-Schwarz inequality, Frobenius norm, Continuity, open and closed sets

Lecture 3 Partial derivatives, Total derivative

Lecture 4 More examples, Theorem: partial derivatives exist and are continuous implies differentiability

Lecture 5 Review of the last discussion, Basic facts about derivatives, Proof of the Chain Rule

Lecture 6 Finishing the proof of the Chain Rule, the Product Rule, the Mean-Value theorem, C^1 implies Lipschitz

Lecture 7 A few more applications of the Chain Rule and MVT, statement of the Inverse Function Theorem (to be proved next)

Lecture 8 An updated statement of IFT, a bit of motivation, and Step 1 of the proof

Lecture 9 Proof of Step 2 of the proof (continuity of the local inverse)

Lecture 10 An alternative proof of Step 2, started the final step (differentiability of the local inverse)

Lecture 11 Finished the proof of IFT, some remarks, and started discussion on the Implicit Function Theorem

Lecture 12 Proof of the Implicit Function theorem, an application of IFT to square-roots of matrices

Lecture 13 Examples related to the Implicit Function Theorem, the Rank Theorem as an application of IFT, setup for Taylor's theorem

Lecture 14 Taylor polynomials of degree k, proof of Taylor's Theorem

Lecture 15 The Hessian of a function, and the Second Derivative Test

Lecture 16 Wrapped up discussion of the Second Derivative Test, Manifolds, some examples and an application of the Rank Theorem

Lecture 17 Derivative of the determinant function, Integration over a closed rectangle in R^n.

Lecture 18 Examples and non-examples of manifolds, Measure zero sets, Proof that an integrable funtion is continuous almost everywhere

Lecture 19 Examples with various sets of discontinuities, Proof that if the discontinuity set has measure zero, the function is integrable.

Lecture 20 Discussion of midterm (to be continued next class), Rectifiable sets, statement of Fubini's theorem

Lecture 21 Discussion of midterm (continued), proof of Fubini's theorem

Lecture 22 Statement of the Change of Variables theorem, warmup to the Existence of a Partition of Unity

Lecture 23 Proof of the Existence of a Partition of Unity, defining integrals over open subsets of R^n

Lecture 24 (Most of the) proof of the Change of Variables Theorem

Lecture 25 Finished the proof of Change of Variables, started with Differential Forms

Lecture 26 Examples of k-forms/alternating k-tensors on R^n, elementary forms, dimension. Defined k-form fields, integrating an n-form field

Lecture 27 The tensor product and wedge product, various examples.

Lecture 28 Pullback of differential forms and some properties, Integrating k-form fields on parametrized subsets, some worked-out examples.

Lecture 29 Another example involving pullback, Exterior derivatives of differential forms, some properties

Lecture 30 Exterior derivatives commute with pullbacks, Volume of a k-dimensional box in R^n, Volume of a parametrized k-manifold

Lecture 31 Independence of reparametrizations, integrating functions on a k-manifold using a partition of unity.

Lecture 32 Orientable manifolds, Integrating k-form fields on an oriented k-dimensional manifold.

Lecture 33 More on orientation, defining opposite orientation, orientation for 1-manifolds and 2-manifolds

Lecture 34 Manifolds with boundary, the induced orientation on the boundary manifold

Lecture 35 Stokes' theorem, first part of the proof

Lecture 36 Rest of the proof of Stokes' theorem, closed forms and exact forms, a glimpse of de Rham cohomology.