so basically, treat dx, dy as formal symbols? you can multiply them by scalar, add them up
NOTE in agda it would be just n-form with ordered requirement? or just a subset?
a 1-diff form is an expression F(x, y) dx + G(x, y) dy
total differential is an example of diff. form:
df = pf/px dx + pf/py dy
a diff form is similar to vector field
a form is exact if it’s a total differential of a scalar function
closed – if pF/py = pG/px
exact impllies closed, but not in reverse
exact means that integral is path independent!
this is sort of similar to complex analysis?
wedge space: given vector space V:
wedge rules: vu = -uv; u ^ u = 0; and linearity
build the space: \Wedge2 V = {Sumi ui ^ vi | ui ^ vi | ui, vi in V} are imposed
2-form: an expression, built using wedges on pairs of 1-forms
hmm. are all formal experessions looking like F dxdy + G dxdz + H dydz looking like that??
rules for derivative:
linearity
d(f alpha) = df ^ alpha + f d alpha
d (dx) = d (dy) = d(dz) = 0 #TODO hmm what does that one mean??
orientation: for a curve, direction; for a surface – normal direction
basically, continuous normal field. If it exists, there are two of them: n and minus n
parameterise the surface by two coordinates u, v so that du ^ dv is the direction of normal
so basically, by definition: integral of 2-form would be
int F dxdy + G dydz + H dz ^dx = int int [ F p(x, y)/p(u, v) + G p(y, z)/p(u, v) + H p(z, x)/p(u, v)] du dv
what would it mean to eval surface integral of dx ^ dy + dx ^ dz + dy ^ dz over a sphere?
ah! so it’s actually a flux: int intS F \dot dS = ∫ intS F1 dydz + F2 dz ^ dx + F3 dx ^ dy
NOTE arclength is NOT a differential form (always > 0)
but what we can do is to define a metric tensor
the manifold has to be parameterised for defining diff forms (or at least, for defining integrals?)
page 50: Maxwell’s equations. EM field is a 2-form