MATH 481: Vector and Tensor Analysis

This course has a limited number of seats. To find out about availability or to be put on a waitlist, please contact our office at


Introductory course in modern differential geometry focusing on examples, broadly aimed at students in mathematics, the sciences, and engineering. Emphasis is on rigorously presented concepts, tools and ideas rather than on proofs. Topics covered include differentiable manifolds, tangent spaces and orientability; vector and tensor fields; differential forms; integration on manifolds and Generalized Stokes' Theorem; Riemannian metrics, Riemannian connections and geodesics. Applications to configuration and phase spaces, Maxwell equations and relativity theory will be discussed.


Math 481 Syllabus.pdf


MATH 241 and one of Math 415 or Math 416 or equivalent.

Credit Hours


Undergraduate Students  
Graduate Students  
Courseware Cost None

Students must be able to view assignments online, write out solutions, then scan or take photos of their written work and upload it to Moodle. 

Students with a Bachelor's degree will be assessed graduate level tuition rate for this course. However, one cannot receive graduate level credit for courses numbered below 400 at the University of Illinois.


Exams: This course has two 90-minute midterm tests and a 3-hour final exam.

Proctorship Information: Exams in this course may be taken online.

Students with a course start date prior to January 1st, 2023 should use proctor information on this page to take their exams.

Students with a course start date on or after January 1st, 2023 should refer to this page for exam taking information.

Once registered, students will find relevant exam taking information and all other course requirements within their Nexus student dashboard.

Course Options

Please Note:

Students currently registered in a University of Illinois Graduate Degree program will be restricted from registering in 16-week Academic Year-term NetMath courses. Matriculating UIUC Grad students will be allowed to register in Summer Session II NetMath courses. 

This page has information regarding the self-paced, rolling enrollment course. If you are a UIUC student interested in taking a course during the summer, you may be interested in a Summer Session II course.


Individual students enrolled in this course are assigned to a course instructor. 

Course Timeline

Time in the course begins on the date your registration is processed. This course is 16 weeks long with the possibility of purchasing up to two 1-month extensions.

Students with course start dates prior to 1/1/2023:

Click here for information about extensions for this course. Click here to apply for an extension.

Students with course start dates on or after 1/1/2023:

Extension information is posted here. Extension applications must be made via the student Nexus Dashboard. Nexus access will be granted upon course registration.