Vector and Tensor Calculus
To develop a geometric and coordinate-free understanding of vector and tensor calculus in three-dimensional Euclidean space, with applications to the kinematics of physical systems and connections to classical differential geometry.
By the end of this course, students will be able to:
Explain the structure and properties of Euclidean point and vector spaces, and apply them to physical and geometric contexts.
Perform calculus operations on vector and tensor fields, including differentiation and integration in both component-based and coordinate-free frameworks.
Analyze and transform vector and tensor quantities using Cartesian and curvilinear bases, and interpret covariant, contravariant, and physical components.
Apply vector and tensor calculus to model the kinematics of point masses, rigid bodies, and deformable bodies.
Establish connections between vector/tensor calculus and classical differential geometry, particularly in the study of curves and surfaces in three-dimensional space.
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