DDG Week 4
Week 3: Exterior algebra (k-vectors)
Week 4: Exterior calculus (k-forms)
Exterior Calculus, "how do lengths, areas, volumes, change over curved surfaces?
Measurement devices have the same dimensionality as the thing we're measuring.
Study exterior calculus in flat spaces ()
May seem redundant, but will help when space starts curving.
Vectors & Covectors
covectors do measurements
vectors get measured
Wedging covectors get k-forms, which are duals to k-vectors.
Similar to row/column vectors.
A row vector is a linear map which maps the column vector to a real value.
let be a unit covector, be a vector of any magnitude.
tells us how long is in the direction
Let be a ceal vector space. Its dual space is the collection of all linear functions together with the operations of addition.
and scalar multiplication
An element of a dual vector space is called a dual vector or covector
(unrelated to the Hodge Star/Dual)
e.g and (Dirac delta)
Let be column vectors, then the transpose converts between row/column.
Likewise, let be vectors, then .
let be covectors, then .
let the inner product be instead of the normal inner product.
k-forms will be multilinear maps from a k-vector.
Measurement of 2-vectors
The "shadow" of a parallelogram onto another parallelogram.
Given , pick orthonormal basis for plane .
- project onto plane
- calculate area via cross product
Can apply same expression when not orthonormal.
This is the definiton of the application of a 2-form to two vectors.
Note: result scales with "area" of
Note that the application of the 2-form is antisymettric
Geometrically, this shows that the orientation of the face is different.
The arguments to the wedge are also antisymmetric.
this has the same meaning, as the 2-form's orientation is flipped.
Measurement of 3-vectors
in , all 3-vectors have the same "direction". Can only measure magnitude.
Let be the edges of a parallelopiped. Let be any orthonormal basis.
- project onto the basis
- apply standard formula for volume (det)
Generally, a k-form is a fully antisymmetric, multilinear measurement of a vector.
Typically, think of a k-form as a map from a k-vector to a scalar.
For historical reasons, we do not write
and instead use the notation we have been using above.
note: a 0-form is a scalar
k-forms in coordinates
For vectors in a basis , write
The scalar values are the coordinates of v.
For covectors in a so-called dual basis , write
where is the Kronecker delta.
This is sometimes called the musical isomorphism