AMS 510: Analytical Methods for Applied Mathematics and Statistics

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Course description

Review of techniques of multivariate calculus, convergence and limits, matrix analysis, vector space basics, and Lagrange multipliers.

Linear Algebra and Its Applications by Gilbert Strang, 4th edition, 2006, Cengage Learning
Advanced Calculus: Theory and Practice by John Srdjan Petrovic, 2nd edition, 2020, Chapman & Hall/CRC

How to Prove It: A Structured Approach by Daniel J. Velleman, Cambridge University Press
Introduction to Linear Algebra by Gilbert Strang, 5th edition, 2016, Wellesley-Cambridge Press
Calculus by Ron Larson & Bruce Edwards, 11th edition, 2017, Cengage Learning

Demonstrate mastery of topics in linear algebra:
- Review fundamentals of linear algebra, Cauchy–Schwarz;
- Echelon form, pivot and free variables, existence and uniqueness;
- Linear independence, basis, space and dimension;
- Homogeneous and nonhomogeneous equations, column and null spaces;
- Linear mapping, kernel and range.
Demonstrate mastery of differentiation in calculus:
- Function, limit, continuity and derivative;
- Product rule, quotient rule, and chain rule;
- Mean value theorem and L’Hôpital’s rule;
- Maximum and minimum.
Demonstrate mastery of integration in calculus:
- Antiderivative, Riemann sum and Newton-Leibniz formula;
- Integration techniques: substitution method, integration by parts, partial fraction;
- Area and volume by revolution, improper integral.
Demonstrate mastery of multivariable calculus:
- Multivariable function, limit, and partial derivatives;
- Transformation, Jacobian, and Lagrange multiplier;
- Double and triple integrals;
- Applications, volume, mass, moment of inertia;
- Transformation to polar, cylindrical and spherical coordinates.
Advanced topics:
- Vector functions, gradient, divergence and curl;
- Surface and line integrals;
- Green’s theorem and Stokes’ theorem.