Department of Mathematics

Van Vleck Hall, 480 Lincoln Drive, Madison, WI

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

 
Student Body: 

Honors and advanced students. Open to freshmen.

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.

Vector spaces and linear transformations, differential calculus of scalar and vector fields, determinants, eigenvalues and eigenvectors, multiple integrals, line integrals, and surface integrals.

Enroll Info: Freshmen students are invited to enroll by the Department of Mathematics. They may choose to defer until their sophomore year.

Alternatives: 

Students who are not interested in theoretical approach can take Math 234 and Math 340 instead.

 
Subsequent Courses: 

Math 376.

 
Course Content: 
  • 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
credits: 
5. (N-A)
semester: 
Fall
prereqs: 
Consent of Instructor

UW-Madison Department of Mathematics
Van Vleck Hall
480 Lincoln Drive
Madison, Wi  53706

(608) 263-3054

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