Vector and Tensor Calculus
A geometrically oriented introduction to the calculus of vector and tensor fields on three-dimensional Euclidean point space, with applications to the kinematics of point masses, rigid bodies, and deformable bodies. Aside from conventional approaches based on working with Cartesian and curvilinear components, coordinate-free treatments of differentiation and integration will be presented. Connections with the classical differential geometry of curves and surfaces in three-dimensional Euclidean point space will also be established and discussed.
1. Euclidean point and vector spaces
2. Geometry and algebra of vectors and tensors
3. Cartesian and curvilinear bases
4. Vector and tensor fields
5. Differentiation and integration
6. Covariant, contraviant, and physical components
7. Basis-free descriptions
8. Kinematics of point masses
9. Kinematics of rigid bodies
10. Kinematics of deformable bodies
weekly problem sets, a midterm examination, and a final examination
multivariate calculus and linear (or, alternatively, matrix) algebra
none, working from personal notes
Alternate years course, odd years alternates with A112