MATH 481: Vector and Tensor Analysis


Introductory course in modern differential geometry focusing on examples, broadly aimed at students in mathematics, the sciences, and engineering. Emphasis on rigorously presented concepts, tools and ideas rather than on proofs. The 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

Please note that the above tuition rates include a $50 per credit hour non-refundable administrative fee for course registration.

Students must be able to print out assignments, write out solutions, then scan their written work and upload it to Moodle. 


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

Proctorship Information: This course requires that you obtain a proctor. Please see our Proctor Information for further instructions.

Course Options

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 mentor

Course Timeline

Your time in the course begins on the date your registration is processed. This course is 16 weeks long with the possibility of purchasing an extension. Eligible students may purchase up to two 1-month extensions for $300 each. Click here to apply.