AMS 510: Analytical Methods for Applied Mathematics and Statistics

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

Required Textbooks:

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

Supplementary Textbooks:

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

Learning Outcomes

1. 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;
   • Homogegeous and nonhomogeneous equations, column and null spaces;
   • Linear mapping, kernel and range.
2. Demonstrate mastery of differentiation in calculus:
   • Function, limit, continuity and derivative;
   • Product rule, quotient rule, and chain rule;
   • Mean value theorem and L’Hospital’s rule;
   • Maximum and minimum.
3. Demonstrate mastery of integration in calculus:
   • Antiderivative, Riemann sum and Newton-Leibniz formula;
   • Integration techniques: substitution method, integration by part, partial fraction;
   • Area and volume by revolution, improper integral.
4. Demonstrate mastery of multivariable calculus:
   • Multivariable function, limit, and partial derivatives;
   • Transformation, Jacobian, and Lagrangian multiplier;
   • Double and triple integrals;
   • Applications, volume, mass, moment of inertial;
   • Transformation to polar, cylindrical and spherical coordinates.
5. Advanced topics:
   • Vector functions, gradient, divergence and curl;
   • Surface and line integrals;
   • Green’s theorem and Stokes theorem.