- #1
sponsoredwalk
- 533
- 5
I think I finally understand the wedge product & think it explains things
in 2-forms that have been puzzling me for a long time.
My post consists of the way I see things regarding the wedge product & interspersed with
my thoughts are only 3 questions (in bold!) that I'm hoping for some clarification on.
The rest of the writing is just meant to be read & hopefully it's all right, if it's wrong do
please correct me
If
v = v₁e₁ + v₂e₂
w = w₁e₁ + w₂e₂
where e₁ = (1,0) & e₂ = (0,1) then
v ⋀ w = (v₁e₁ + v₂e₂) ⋀ (w₁e₁ + w₂e₂)
_____ = v₁w₁e₁⋀e₁ + v₁w₂e₁⋀e₂ + v₂w₁e₂⋀e₁ + v₂w₂e₂⋀e₂
_____ = v₁w₂e₁⋀e₂ + v₂w₁e₂⋀e₁
v ⋀ w= (v₁w₂ - v₂w₁)e₁⋀e₂
This is interpreted as the area contained in v & w.
My first question is based on the fact that this is a two dimensional calculation
that comes out with the exact same result as the cross product of
v'_= v₁e₁ +_v₂e₂_+ 0e₃
w' = w₁e₁ + w₂e₂ + 0e₃
Also the general x ⋀ y = (x₁e₁ + x₂e₂ + xe₃) ⋀ (y₁e₁ + y₂e₂ + ye₃)
comes out with the exact same result as the cross product.
In all cases the end result is a vector orthogonal to v & w, or to v'& w',
or to x'& y'. Is this true for every wedge product calculation in every
dimension? The wedge product of two vectors in ℝ³ gives the area
of parallelogram they enclose & it can be interpreted as a scaled up factor
of a basis vector orthogonal to the vectors. So (e₁⋀e₂) is an orthogonal
unit vector to v & w & (v₁w₂ - v₂w₁) is a scalar that also gives the area
enclosed in v & w.
Judging by this you'd take the wedge product of 3 vectors in ℝ⁴ &
get the volume they enclose, and 4 vectors in ℝ⁵ gives hypervolume or
whatever. If we ended up with β(e₁⋀e₂⋀e₃) this would be in ℝ⁴ where β is
the scalar representing the volume & β(e₁⋀e₂⋀e₃) is pointing off into
the foursth dimension whatever that looks like. If all of this holds I can
justify why e₁⋀e₂ = - e₂⋀e₁ both mentally & algebraically by taking dot
products & finding those orthogonal vectors so I'd like to hear if this
makes sense in the grand scheme of things!
I really despise taking things like e₁⋀e₂ = - e₂⋀e₁ as definitions unless I
can justify them. I can algebraically justify why e₁⋀e₂ = - e₂⋀e₁ by
thinking in terms of the cross product which itself is nothing more than
clever use of the inner product of two orthogonal vectors. Therefore I
think that e₁⋀e₂ literally represents the unit vector that is orthogonal to
the vectors v & w involved in my calculation. So if there are n - 1 vectors
then e₁⋀e₂⋀...⋀en lies in ℝⁿ and is the unit vector as part of some
new vector βe₁⋀e₂⋀...⋀en that is orthogonal to (n - 1) vectors.
I read a comment that the wedge product is in an "exterior square" so I
guess this generalizes to products of all arity (exterior volumes et al) &
from browsing I've seen that a "bivector" is a way to interpret this, like
this:
it's a 2 dimensional vector here for example. My second question is -
if I were to just think in terms of orthogonality as I have explained
in this thread is there any deficiency? As far as I can tell this 2-D
vector in the picture is just a visual representation of the area & as it is
explained via a scaled up orthogonal vector I think there is virtually no difference.
A lot of the wiki topics on "bivectors" and forms etc... were previously
unreadible to me & are only now slowly beginning to make sense (I hope!).
And finally, I'm hoping to use this knowledge above (assuming it's right) to try to
understand terms like
Adx + Bdy + Cdz
&
Adydz + Bdzdx + Cdxdy
in this context. I've seen calulation that specifically require dxdx = dydy = dzdz = 0 &
you're supposed to remember this magic but I don't buy it as just magic, I think there are
very good reasons why this is the case. My third question arises from the fact that I think
these algebraic rules, like dxdy = -dydx & dxdx = 0 etc... are just encoding within
them rules that logically follow from everything I've explained above & would probably be
more clearly delineated through vectors, are they encoding vector calculations dealing
with orthogonality?
Perhaps someone more knowledgeable could expand upon this, I'd greatly appreciate it.
in 2-forms that have been puzzling me for a long time.
My post consists of the way I see things regarding the wedge product & interspersed with
my thoughts are only 3 questions (in bold!) that I'm hoping for some clarification on.
The rest of the writing is just meant to be read & hopefully it's all right, if it's wrong do
please correct me
If
v = v₁e₁ + v₂e₂
w = w₁e₁ + w₂e₂
where e₁ = (1,0) & e₂ = (0,1) then
v ⋀ w = (v₁e₁ + v₂e₂) ⋀ (w₁e₁ + w₂e₂)
_____ = v₁w₁e₁⋀e₁ + v₁w₂e₁⋀e₂ + v₂w₁e₂⋀e₁ + v₂w₂e₂⋀e₂
_____ = v₁w₂e₁⋀e₂ + v₂w₁e₂⋀e₁
v ⋀ w= (v₁w₂ - v₂w₁)e₁⋀e₂
This is interpreted as the area contained in v & w.
My first question is based on the fact that this is a two dimensional calculation
that comes out with the exact same result as the cross product of
v'_= v₁e₁ +_v₂e₂_+ 0e₃
w' = w₁e₁ + w₂e₂ + 0e₃
Also the general x ⋀ y = (x₁e₁ + x₂e₂ + xe₃) ⋀ (y₁e₁ + y₂e₂ + ye₃)
comes out with the exact same result as the cross product.
In all cases the end result is a vector orthogonal to v & w, or to v'& w',
or to x'& y'. Is this true for every wedge product calculation in every
dimension? The wedge product of two vectors in ℝ³ gives the area
of parallelogram they enclose & it can be interpreted as a scaled up factor
of a basis vector orthogonal to the vectors. So (e₁⋀e₂) is an orthogonal
unit vector to v & w & (v₁w₂ - v₂w₁) is a scalar that also gives the area
enclosed in v & w.
Judging by this you'd take the wedge product of 3 vectors in ℝ⁴ &
get the volume they enclose, and 4 vectors in ℝ⁵ gives hypervolume or
whatever. If we ended up with β(e₁⋀e₂⋀e₃) this would be in ℝ⁴ where β is
the scalar representing the volume & β(e₁⋀e₂⋀e₃) is pointing off into
the foursth dimension whatever that looks like. If all of this holds I can
justify why e₁⋀e₂ = - e₂⋀e₁ both mentally & algebraically by taking dot
products & finding those orthogonal vectors so I'd like to hear if this
makes sense in the grand scheme of things!
I really despise taking things like e₁⋀e₂ = - e₂⋀e₁ as definitions unless I
can justify them. I can algebraically justify why e₁⋀e₂ = - e₂⋀e₁ by
thinking in terms of the cross product which itself is nothing more than
clever use of the inner product of two orthogonal vectors. Therefore I
think that e₁⋀e₂ literally represents the unit vector that is orthogonal to
the vectors v & w involved in my calculation. So if there are n - 1 vectors
then e₁⋀e₂⋀...⋀en lies in ℝⁿ and is the unit vector as part of some
new vector βe₁⋀e₂⋀...⋀en that is orthogonal to (n - 1) vectors.
I read a comment that the wedge product is in an "exterior square" so I
guess this generalizes to products of all arity (exterior volumes et al) &
from browsing I've seen that a "bivector" is a way to interpret this, like
this:
it's a 2 dimensional vector here for example. My second question is -
if I were to just think in terms of orthogonality as I have explained
in this thread is there any deficiency? As far as I can tell this 2-D
vector in the picture is just a visual representation of the area & as it is
explained via a scaled up orthogonal vector I think there is virtually no difference.
A lot of the wiki topics on "bivectors" and forms etc... were previously
unreadible to me & are only now slowly beginning to make sense (I hope!).
And finally, I'm hoping to use this knowledge above (assuming it's right) to try to
understand terms like
Adx + Bdy + Cdz
&
Adydz + Bdzdx + Cdxdy
in this context. I've seen calulation that specifically require dxdx = dydy = dzdz = 0 &
you're supposed to remember this magic but I don't buy it as just magic, I think there are
very good reasons why this is the case. My third question arises from the fact that I think
these algebraic rules, like dxdy = -dydx & dxdx = 0 etc... are just encoding within
them rules that logically follow from everything I've explained above & would probably be
more clearly delineated through vectors, are they encoding vector calculations dealing
with orthogonality?
Perhaps someone more knowledgeable could expand upon this, I'd greatly appreciate it.
Last edited: