Discrete Differential Geometry 5
DDG Lecture 5
Differential Forms in
A vector field is an assignment of vectors to each point in space.
A differential k-form is an assignment of a k-form to each point.
A common abbreviation is to shorten "differential k-form" to just "k-form".
A differential 0-form is a just a scalar field.
A differential 1-form is like a vector field.
(But they are 1-forms at each point, so it is how to take a measurement at each point)
A differential 2-form is an area measurement at each point (x1,x2,x3).
Pointwise operations on Differential k-forms
Apply operator over each k-form pointwise.
Notation: omit _p.
Differential k-forms in Coordinates
Basis Vector Fields
The coeffecients of the linear combination of basis fields can vary pointwise.
For differential 1-forms, call the basis 1-forms (dual bases) .
Ex. Hodge star of differential 1-form.
Consider the differential 1-form:
Q: what is the hodge star of ?
Ex. wedge of two 1-forms.
Note that in all wedges of 1-forms are scalar multiples of .
The same information is contained in the hodge star.
Ex. Applying a Differential 1-form to a vector field
apply to .