Math 375 - Topics in Multi-Variable Calculus and Linear Algebra

• Prerequisites: Math 276 or consent of instructor.
• Frequency: Fall (I), Spring (II)
• Student Body: Honors and advanced students. Open to freshmen.
• Credits: 5. Students may not receive credit for both Math 375 and any of Math 234, 320, and 340. (N-A)
• Recent Texts: Calculus, Volume II, 2nd edition, by T. M. Apostol.
• Course Coordinator: Andreas Seeger
• Background and Goals: This course is the third semester of the Calculus Honors sequence developed by the Mathematics Department at the UW. The object of the course is to present the subjects of linear algebra and multivariable calculus and the interrelation between their mathematical ideas. This course is followed by a fourth semester Math 376, where multivariable calculus is further developed and where the students get an introduction to differential equations.
• Alternatives: Students who are not interested in theoretical approach can take Math 234 and Math 340 instead.
• Subsequent Courses: Math 376.

Content coverage:

• Linear spaces
  • Definitions and examples of vector spaces
  • Subspaces
  • Dependence and linear independence
  • Bases and dimension
  • Inner products and orthogonality
• Linear transformations and matrices
  • Linear transformations
  • null space and range
  • Inverses
  • Matrix representation of a linear transformation
  • Matrix multiplication
  • Inverse linear transformations and matrices
• Differential calculus of scalar and vector valued functions
  • Scalar and vector functions
  • Limits and continuity
  • The derivative as a linear transformation
  • Partial derivatives
  • The gradient
  • The chain rule
  • Implicit differentiation
• Determinants
  • A set of axioms for the determinant function
  • Proof of existence and uniqueness
  • Determinants of products of matrices
  • Transposed matrices
  • Minors and cofactors
  • Cramer's rule
• Eigenvalues and Eigenvectors
  • Eigenvalues and Eigenvectors of a linear transformation
  • Calculation of eigenvectors and eigenvalues
  • Diagonalization
  • Similar matrices
  • Spectral theorem for symmetric and Hermitian linear transformations
  • Quadratic forms
  • Unitary transformations
• Application of the differential calculus
  • Maxima, minima and saddle points
  • Lagrange multipliers