Exterior Calculus - solsarratea/discrete-differential-geom GitHub Wiki

• Exterior Algebra: Little Volumes.
• Exterior Calculus:the study of how “little volumes” change over space and time

exterior calculus is to exterior algebra what vector calculus is to linear algebra

Measurement of little-volumes

Linear algebra: covectors are objects that “measure”, vectors are objects that “get measured".

Primal and Dual Spaces

note: every vector encodes a linear transformation

covectors linear maps from vector to scalars.

sharp and flat operators:

Basic idea: applying the flat of a vector is the same as taking the inner product; taking the inner product w/ the sharp is same as applying the original 1-form.

** covectors are to vectors what k-vectors are to k-forms**

• k-form is a fully antisymmetric, multilinear measurement of a k-vector. It “projects” onto k-dimensional space and take determinant there -differential k-form is an assignement of a k-form to each point. I t is what a vector field is an assignement of a vector to each point.

Differential 0-form: assigns scalars to each point Differential 1-form: assigns a 1-form to each point

vector field vs differential 1-form : ** dual vector space is 1-form **(?) DIFFERENTIAL K-FORMS MEASURE K-VECTOR FIELDS

Hodge star: provides a sort of orthogonal complement for k-vectors. In particular, if we have a k-vector v in Rn then ?v will be an (n − k)-vector that is in some sense “complementary.” Operators over k-form :

• apply the Hodge star to the individual k forms at each point p;
• to take the wedge of two differential k-forms we just wedge their values at each point.

Differential Forms and the Hodge Star.

resource Gradient vs. Differential gradient depends on inner product; differential doesn’t