303CRS Calculus 2

CMI, Aug-Nov 2025, Krishna Hanumanthu
Ramadas (n.d.), Spivak (1965) (errata), Rudin (1976), Apostol (1985), Pugh (2015), Zorich (2016)

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Lectures

Integration in ${\mathbb{R}}^n$

LEC CAL2 2 ✅ Review
LEC CAL2 3, 4 ✅ Integration on rectangles
LEC CAL2 5 ✅ Integration on Jordan measurable sets
LEC CAL2 6 ✅ Fubini’s theorem

Intermezzo: CAL 1 review

LEC CAL2 7 ✅ Partitions of unity
LEC CAL2 8 Change of variables
LEC CAL2 9 Tensor products
LEC CAL2 10 Differential forms
LEC CAL2 11

Tutorials

TUT CAL2 1 $S^1$ has measure zero in ${\mathbb{R}}^2$
TUT CAL2 2 Sard’s Theorem
TUT CAL2 3 The Rank Theorem
TUT CAL2 4

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If $F:[0,1]\times U\to \mathbb{R}$ is continuous, and $G(x)={\int }_0^{1}F(t,x)dx$, then

$$\frac{\partial G(x)}{\partial x_i}={\int }_0^1\frac{\partial F(t,x)}{\partial x_i}dt$$

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References

Apostol, T. M. (1985). Mathematical Analysis (2d ed). Narosa.

Pugh, C. C. (2015). Real Mathematical Analysis. Springer International Publishing. https://doi.org/10.1007/978-3-319-17771-7

Ramadas, T. R. (n.d.). (MULTIDIMENSIONAL INTEGRAL) CALCULUS.

Rudin, W. (1976). Principles of Mathematical Analysis (3d ed). McGraw-Hill.

Spivak, M. (1965). Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. Addison-Wesley publ.

Zorich, V. A. (2016). Mathematical Analysis 2 (R. Cooke & O. Paniagua, Trans.; Second edition). Springer.