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07-01-05, 09:31
Given a function f: M → R, define a special element of Tp*M. Call it ωf. Define ωf by the following:
ωf = (∂f/∂xβ) dxβ. In other words, the coefficients ωβ = ∂f/∂xβ.
Now, let's see what happens if we act with ωf on a vector vp in TpM:
ωf[ vα∂/∂xα ] = (∂f/∂xβ) vβ = vp(f).
ωf[ v ] is the directional derivative of f in the direction of v. It exactly gives us the same result we would get if we acted on f with v as a directional derivative.
We give ωf a new name: ωf = df = (∂f/∂xβ) dxβ.
Now, df[ v ] = v(f) is the directional derivative of f in the v-direction.
Now, we can see that our notation for "df" connects with our notation for "dxβ ". We have used boldface to distinguish them thus far, but soon that will not be necessary. As a special case, let f be the coordinate function f = xβ. Then,
df = d(xβ) = (∂xβ/∂xα) dxα, where "dxα " is still our dual vector notation.
Now, ∂xβ/∂xα = δαβ is just our friendly Kronecker Delta again, which is fairly easy to see, since our coordinates are independent of each other. Therefore, we have the relation
d(xβ) = dxβ. Our notation is consistent.
So, in a fairly unorthodox manner, the notation has lead us to a map d from functions into forms. d is known as the exterior derivative (http://www.everything2.com/index.pl?node=exterior%20derivative).
df=(∂f/∂xβ) dxβ 原来是一个微分形式 dx 不能看作传统意义下的微小增量
这个符号让我迷惑了几乎一年了，天, 转变观念实在太费劲了。
ωf = (∂f/∂xβ) dxβ. In other words, the coefficients ωβ = ∂f/∂xβ.
Now, let's see what happens if we act with ωf on a vector vp in TpM:
ωf[ vα∂/∂xα ] = (∂f/∂xβ) vβ = vp(f).
ωf[ v ] is the directional derivative of f in the direction of v. It exactly gives us the same result we would get if we acted on f with v as a directional derivative.
We give ωf a new name: ωf = df = (∂f/∂xβ) dxβ.
Now, df[ v ] = v(f) is the directional derivative of f in the v-direction.
Now, we can see that our notation for "df" connects with our notation for "dxβ ". We have used boldface to distinguish them thus far, but soon that will not be necessary. As a special case, let f be the coordinate function f = xβ. Then,
df = d(xβ) = (∂xβ/∂xα) dxα, where "dxα " is still our dual vector notation.
Now, ∂xβ/∂xα = δαβ is just our friendly Kronecker Delta again, which is fairly easy to see, since our coordinates are independent of each other. Therefore, we have the relation
d(xβ) = dxβ. Our notation is consistent.
So, in a fairly unorthodox manner, the notation has lead us to a map d from functions into forms. d is known as the exterior derivative (http://www.everything2.com/index.pl?node=exterior%20derivative).
df=(∂f/∂xβ) dxβ 原来是一个微分形式 dx 不能看作传统意义下的微小增量
这个符号让我迷惑了几乎一年了，天, 转变观念实在太费劲了。