TAM 541: Mathematical Methods I

Course Descriptions:

Vector and tensor algebra and complex-variable methods; ordinary differential equations, qualitative questions of existence and uniqueness; analytic solution methods, numerical methods, power-series solution and special functions; eigenvalue problems, Green's functions, Laplace transforms, stability of solutions; engineering applications drawn from mechanics.

Announcements:

The current plan is that this course will be taught in person.The situation around Covid-related restrictions on campus can of course change at any moment, and I will keep the class updated of any developments.

Schedule: Mon Wed 11:00am~12:50pm, 2035 Campus Instructional Facility

Prerequisites: MATH 241 or MATH 380; one of MATH 385, MATH 386, MATH 441

Office hours: Fridays 5-6pm in CSL 359 (may be moved in online).

TA: Jin Kim, jkim684@illinois.edu

TA Office hours: Tuesdays and Thursdays from 4-6pm. Location: TBD

Supplementary texts:

• Michael Greenberg, Advanced Engineering Mathematics, Prentice-Hall

Assignments and grading policy:

Your grades will be based on homework and exams. Details will be explained in class.

Lecture notes:

• Week 1: Complex numbers

• Week 2: Functions on complex numbers

• Week 3: Cauchy-Riemann condition

• Week 4: Complex integration; Cauchy-Goursat theorem

• Week 5: Cauchy integral formula; Taylor and Laurent series

• Week 6: Laurent series and its application; Residue theorem

• Week 7: Vector spaces and change of basis

• Week 8: Inner product space

• Week 9: Gram-Schmidt; Least square approximation; Linear operators

• Week 10: Range and null spaces; Eigenvectors

• Week 11: Tensor representation of symmetric operators

• Week 12: Ordinary differential equations

• Week 13: Variation of constants formula

• Week 14: Linear systems