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MATH-431: Theory of stochastic calculus

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Lectures in this course (78)

Integrals in Higher Dimensions

Explores integrals in higher dimensions, emphasizing the versatility of integration methods and the significance of changing the order of integration.

Vector spaces and norms

Covers vector spaces, differentiable functions, scalar product, and norms in depth.

Metric spaces: topology

Covers metric spaces and topology, exploring properties of metrics, open/closed sets, and boundaries.

Sequences: convergence and limits

Covers sequences, convergence, and limits in Analysis II, including Cauchy sequences and closed sets.

Subsequences: bounded sequences and Bolzano-Weierstrass theorem

Introduces bounded sequences, subsequence definition, Bolzano-Weierstrass theorem, and applications to continuous functions.

Differentiating under the integral sign

Explores differentiating under the integral sign and continuity of functions in integrals.

Subsequences: bounded sequences and Bolzano-Weierstrass theorem

Introduces bounded sequences and subsequences, culminating in the Bolzano-Weierstrass theorem.

Bolzano-Bastra-Sterl: Applications and Uniform Continuity

Explores the Bolzano-Bastra-Sterl theorem and uniform continuity in sequences.

Boundary Analysis

Explores boundaries in sets, defining them as points not well separated from the set or its complement.

Chain Rules

Explores the extension of the chain rule to higher dimensions and compositions of functions.

Differentiability: Functions and Matrices

Explores partial differentiability, the chain rule, and invertibility of vector fields.

Applications of Theorems

Demonstrates the practical application of theorems in calculus through two clever examples.

Continuous Functions: Criteria and Equivalences

Explores criteria for continuity, convergence of sequences, and equivalence conditions for continuous functions.

Coordinate Change: Polar and Spherical Coordinates

Covers coordinate changes, focusing on polar and spherical coordinates and their transformations, determinants, and invertibility.

Curves: differentiability and smoothness

Explores the differentiability and smoothness of curves in R², covering criteria, singularity points, and length.

Equivalent norms: properties and proofs

Explores equivalent norms in a vector space and their continuity properties, including proofs of norm equivalence.

Change of Variables Theorem

Explores the change of variables formula in higher dimensions and its application through determinant algebra and a practical example.

Functions on R^n: Limits and Partial Differentiation

Delves into functions on R^n, covering limits and partial differentiation.

Hessian Matrix

Covers the concept of the Hessian matrix and its role in optimization problems.

Implicit Differentiation

Explores implicit differentiation and the Implicit Function Theorem for finding unique local functions.

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