<A Geometric Approach to Differential Forms > (by Bachman, David ) Notes

<A Geometric Approach to Differential Forms > (by Bachman, David ) Note

A differential form is precisely a linear function which eats vectors, spits out numbers and is used in integration. The strength of differential forms lies in the fact that their integrals do not depend

on a choice of parameterization.
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In general, an n-form is a linear function which takes n vectors and returns a real number.Their strength is that the value of such an integral does not depend on the choice of parameterization.
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∇($_V$)f(x0, y0): directional derivative of f, at the point (x0, y0), in the direction V
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<1, 2>: a vector with x=1, y=2
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The vector <∂f/∂x, ∂f/∂y> is called the gradient of f and is denoted ∇f.
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∇($_V$)f(x0, y0) = ∇($_{<a,b>}$) f(x0, y0) 
= a*∂f/∂x + b*∂f/∂y 
= <∂f/∂x, ∂f/∂y> * <a, b> 
= ∇f(x0, y0) * V
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if U = ∇f/|∇f|, then ∇($_U$)f = ... = |∇f|
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Given a curve C in Rn, a 【parameterization】 for C is a (one-to-one, onto, differentiable) function of the form φ:R1→Rn whose image is C.
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CH3
The line can be thought of as the set of all tangent vectors at the point p. We denote that line as TpC, the tangent space to C at the point p.
TpC is the set of all vectors tangents to C at p.
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In other words, any point of TpP can be written as dx<0, 1> + dy<1, 0>, where dx, dy ∈ R.
Hence, “dx” and “dy” are 【coordinate functions】 for TpP
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<2, 3>p : the vector <2, 3> in TpP
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Evaluating a 1-form on a vector is the same as projecting onto some line(向量<∂f/∂x,∂f/∂y>) and then multiplying by some constant(vector length).
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Evaluating a 1-form on a vector is the same as projecting onto each coordinate axis (∂f/∂x,∂f/∂y), scaling each by some constant(坐标值dx,dy) and adding the results.
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3.3 Multiplying 1-forms
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If ω and ν are 1-forms, can’t we just define ω · ν(V) = ω(V) · ν(V)?” Well, of course we can, but then ω · ν is not a linear function, so we have left the world of forms.（我们希望把2-form定义为线性的）
The trick is to define the product of ω and ν to be a 2-form. So as not to confuse this with the product just mentioned, we will use the symbol “∧”。
(pronounced “wedge”) to denote multiplication（e.g. ω ∧ ν）.
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In other words, ω and ν together give a way of taking each vector and returning a pair of numbers. How do we visualize pairs of numbers? In the plane, of course! Let’s define a new plane with one axis as the ω-axis and the other as the ν-axis. So, the coordinates of V1 in this plane are [ω(V1), ν(V1)] and the coordinates of V2 are [ω(V2), ν(V2)].
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(1, 2) : a vector in the xy-plane,
<1, 2> : a vector in the dxdy-plane 。（dxdy-plane是抽象意义上的平面）
[1, 2] : a vector in the ων-plane.  ω and ν are 1-forms. （ων-plane是抽象意义上的平面）
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Let’s not forget our goal now. We wanted to use ω and ν to take the pair of vectors (V1, V2) and return a number.
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ω∧ν (V1, V2) =
| ω(V1) ν(V1) |
| ω(V2) ν(V2) |
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ω ∧ ν(V1, V2) = −ω∧ν(V2, V1) (ω ∧ ν is skew-symmetric) (the 2-form ω ∧ ν, is a skew-symmetric operator on pairs of vectors) （由2-form的定义得到这个性质）
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ω ∧ ν(V, V) = 0
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ω ∧ ν(V1 + V2, V3) = ω ∧ ν(V1, V3) + ω ∧ ν(V2, V3) and 
ω ∧ ν(cV1, V2) = ω ∧ ν(V1, cV2) = c ω∧ν(V1, V2), 
where c is any real number 
(ω ∧ ν is 【bilinear】).
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ω ∧ ν(V1, V2) = −ν ∧ ω(V1, V2) (It says that ∧ can be thought of as a skew-symmetric operator on 1-forms.)
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ω ∧ ω(V1, V2) = 0
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(ω + ν) ∧ ψ = ω ∧ ψ + ν ∧ ψ     (∧ is distributive)
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There is another way to interpret the action of ω ∧ ν which is much more geometric. First, let ω = a dx + b dy be a 1-form on TpR2 . Then we let <ω> be the vector <a, b>.
ω ∧ ν(V1, V2) is the area of the parallelogram spanned by V1 and V2 , times the area of the parallelogram spanned by <ω> and <ν>.
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ν($_ω$) ≡ ν − xω, x is a scalar value
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Evaluating ω ∧ ν on the pair of vectors (V1, V2) gives the area
of parallelogram spanned by V1 and V2 projected onto the plane
containing the vectors <ω> and <ν>, and multiplied by the area of
the parallelogram spanned by <ω> and <ν>
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CAUTION: While every 1-form can be thought of as projected length, not every 2-form can be thought of as projected area. The only 2-forms for which this  interpretation is valid are those that are the product of 1-forms. See Problem 3.18.
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An m-form on TpRn can be thought of as a function which takes
the parallelepiped spanned by m vectors, projects it onto each
of the m-dimensional coordinate planes, computes the resulting
areas, multiplies each by some constant(坐标值), and adds the results.
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CH4 Differential Forms
If ω = f (x, y)dx ∧ dy, then ʃ($_R$) ω = ʃ($_R$) f dxdy.
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[2-form和differential 2-form ω的区别]
Do not confuse this 2-form with the differential 2-form, ω, of Equation 4.2.
The 2-form ν is only defined at the single tangent space T p R 3 , whereas ω is defined everywhere.
4.7 Summary: How to integrate a differential form
4.7.1 The steps
4.7.2 Integrating 2-forms
Geometrically, the value of dx ∧ dy is the area of the parallelogram
 spanned by the vectors ∂Ψ/∂u and ∂Ψ/∂v (tangent vectors to S), projected onto
the dxdy-plane (see Figure 4.9).
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We will see in Chapter 6 that the thing that makes (linear) differential forms so useful is the Generalized Stokes Theorem. We do not have anything like this for nonlinear forms, but that is not to say that they do not have their uses.
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Equation 4.5. Classically, this is called a 【surface integral】。It might be a little clearer how to compute such an integral if we write it as follows:

4.8.2 Arc length

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CH5  Differentiation of Differential Forms
5.1 The derivative of a differential 1-form（结果是2-form）
∇($_W$) f(p) = ∇f (p) · W =  <∂f/∂x, ∂f/∂y> · W
∇($_W$) ω(V) = ∇ω(V) · W = <∂ω/∂x, ∂ω/∂y> · W
We want dω, the derivative of ω, to be a 2-form. Hence, dω(V, W ) should equal −dω(W, V ).(因为这是2-form 的一个性质：ω∧ν(V1, V2) = −ω∧ν(V2, V1) ) How can we use the variations above to define dω so this is true? Simple. We just define it to be the difference in these variations:
dω(V, W) = ∇($_V$) ω(W) − ∇($_W$) ω(V)              (5.1)
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设一般形式ω = f (x, y, z)dx + g(x, y, z)dy + h(x, y, z)dz
dω ≡ dω(<1, 0, 0>, <0, 1, 0>) dx ∧ dy 
      + dω(<0, 1, 0>, <0, 0, 1>) dy ∧ dz
      + dω(<1, 0, 0>, <0, 0, 1>) dx ∧ dz
因为：
dω(<1, 0, 0>, <0, 1, 0>) = ∂g/∂x - ∂f/∂y
dω(<0, 1, 0>, <0, 0, 1>) = ∂h/∂y - ∂g/∂z
dω(<1, 0, 0>, <0, 0, 1>) = ∂h/∂x - ∂f/∂z
所以：

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5.2 Derivatives of n-forms
the derivative of a 2-form（结果是3-form）： dω(U, V, W)
为了满足交换率，所以不能定义2-form的结果为∇($_U$) ω(V, W)（因为∇($_U$) ω(V,W)  != −∇($_V$) ω(U,W)）
所以dω(U, V, W)定义为：
dω(U, V, W)≡∇($_U$) ω(V, W) − ∇($_V$) ω(U, W) + ∇($_W$) ω(U, V)
这个定义满足alternating 和 multilinear（线性代数的作用！！）
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In other words, dω, applied to n + 1 vectors, is the alternating sum of the variations of ω applied to n of those vectors in the direction of the remaining one.
Note that we can think of d as an operator（微分算子） which takes n-forms to (n+1)-forms.
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5.3 Interlude: 0-Forms
由df = ∂f/∂x dx + ∂f/∂y dy(右边是1-form)，得：f 是0-form
After all, the input to a differential n-form on Rm is a point and n vectors based at that point. So, the input to a differential 0-form should be a point of Rm and no vectors. 
In other words, a 【0-form on Rm】 is just another word for a 【real-valued function on Rm】
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Suppose f is a 0-form and ω is an n-form
f ∧ ω = f*ω
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0-form的积分：
积分范围是R0 (即某个点p)
ʃ($_p$) f ≡ +/- f(p)
- f(p) ≡ f(-p)
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Suppose f(x, y) dx is a 1-form on R2. 
d(f dx) = ∂f∂x dx∧dx + ∂f∂y dy∧dx = ∂f∂y dy∧dx
df ∧ dx = (∂f∂x dx + ∂f∂y dy) ∧ dx = ∂f∂y dy∧dx = d(f dx)
一般化得：d(f dx1 ∧ dx2 ∧ · · · ∧ dxn ) = df ∧ dx1 ∧ dx2 ∧· · · ∧ dxn
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d(dω) = 0
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If ω is an n-form and μ is an m-form, then show that d(ω ∧ μ) =dω ∧ μ + (−1)^n ω ∧ dμ
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5.4 Algebraic computation of derivatives
d(ω + μ) = dω + dμ
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5.6 Application: Foliations and contact structures
a 【foliation】 is when some region of space has been “filled up” with lower-dimensional surfaces
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Πp是由向量∂φt/∂x(p), ∂φt/∂y(p)张成的切平面。
In other words, if U is foliated, then at every point p of U , we get a plane Πp in TpR3
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The family {Πp} is an example of a 【plane field】.In general, a 【plane field】 is just a choice of a plane in each tangent space which varies smoothly from point to point in R3
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We say a 【plane field is integrable】 if it consists of the tangent planes to a foliation。类比，可积函数在每个点处有切线
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Suppose {Π p } is a plane field. At each point p, we can define a line in TpR3 (i.e., a line field) by looking at the set of all vectors that are perpendicular to Πp . We can then define a 1-form ω by projecting vectors onto these lines. 
So, if Vp is a vector in Πp, then ω(Vp) = 0  （切平面上的向量投影到法线上）
the plane Πp is the 【set 】of all vectors which yield zero when plugged into ω. In shorthand, we write 【this set】 as 【Ker ω】，即：Πp = Ker ω。 （用1-form表示 plane field）
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Theorem 2. If Ker ω is an integrable plane field, then ω ∧ dω = 0 at every point of R3 .
ω ∧ dω ! = 0, Such a (用ω表示的)plane field is called a 【contact structure】
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CH 6 Stokes’ Theorem
Definition 1. Let I = [0, 1]. An 【n-cell】, σ, is the image of a differentiable map, φ : In → Rm , with a specified orientation. We denote the same cell with opposite orientation as −σ. We define a 0-cell to be an oriented point of R m
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Definition 2. An 【n-chain】 is a formal linear combination of n-cells.
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【the boundary of an n-chain】. If C = Σ ni σi , then we define ∂C = Σ ni∂σi
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6.2 The generalized Stokes’ Theorem

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F ≡ <Fx , Fy , Fz>
↔ ω1($_F$) = Fx dx + Fy dy + Fz dz
↔ ω2($_F$)= Fx dy ∧ dz − Fy dx ∧ dz + Fz dx ∧ dy
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f (x, y, z) 
↔ ω3($_f$) = f dx ∧ dy ∧ dz
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∇≡<∂/∂x, ∂/∂y, ∂/∂z>
∇f = <∂f/∂x, ∂f/∂y, ∂f/∂z>
df = ∂f/∂x*dx + ∂f/∂y*dy +  ∂f/∂z*dz = ∇f * ω1 = ω1(∇f ) ≡ ω1($_{∇f}$) 
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dω1($_F$) ≡ ω2($_{∇×F}$)
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div F≡ ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z ≡ ∇·F 
dω2($_F$) = d( Fx dx∧dy + Fy dy∧dz + Fz dx∧dz ) 
= (∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z) dx∧dy∧dz ≡ ω3($_{∇·F}$)
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d(dω) = ddω= 0
∇ × (∇f ) = 0
∇ · (∇ × F) =0

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Recall that at each point, a differential from is simply an alternating, multilinear map on a tangent plane.
All we need to define a form on a space other than Rm is some notion of a tangent space at every point. We call such a space a 【manifold】.
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For the purposes of detecting global properties, certain forms are interesting and certain forms are completely uninteresting.
The interesting forms are the ones whose derivative is zero，e.g. dω0=0, we say ω0 is a【closed form】.
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In a sufficiently small region of every manifold, every closed 1-chain bounds a disk. So integrating closed 1-forms on “small” 1-chains gives us no information. 【In other words, closed 1-forms give no local information.】
Suppose now that we have a closed 1-form ω 0 and a closed 1-chain C such that ʃ($_C$) ω0 != 0. Then we know C does not bound a disk. The fact that there exists such a 1-chain is global information. This is why we say that the closed forms are the ones that are interesting, from the point of view of detecting
only global information.
suppose ω1 = df, We say ω1 is 【exact】。Integrating an exact form over a closed 1-chain always gives zero. This is why we say the exact forms are completely uninteresting.
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CH 8  Differential Geometry via Differential Forms
W is really a function from R2 to TR2 , so we can write W =<w1(x, y), w2(x, y)>. The parameterization α:R1→R2 can be written as α(t) = (α1(t), α2(t)).

The previous example illustrates an important point. Our final answer doesn’t depend on the curve α, only the tangent vector α'(0). V≡ α'(t) = <α1'(t), α2'(t)> ≡ <v1(t), v2(t)>
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covariant derivative of W in the V direction : ∇v W ≡ [∇W] V
用微分形式来表示：
∇v W ≡ [∇W] V = <dw1(V), dw2(V)>
为了形式上一致（类比方向导数∇v f = df(V)），
定义 ∇vW ≡ dW(V)，得： dW(V) = <dw1(V), dw2(V)>
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covariant differentiation求导法则：
∇v (X · Y) = (∇v X) · Y + X · (∇v Y),  (X and Y are vector fields. Then X · Y is a real-valued function)
∇v(f W) = (∇v f)W + f (∇v W).
d(V · W) = dV · W + V · dW
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dW(V) · U = dW · U (V)   (W is a vector field and U and V are vectors in TpRn)
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8.2 Frame fields and Gaussian curvature
Definition 3. Let S be a surface in R3 . A 【frame field】 on S is a choice of vector fields {E1, E2}  such that at each point p of S, E1(p) and E2(p) form an orthonormal basis for TpS.
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Ω(V) = (∇v E1) · E2    （E1， E2是切平面上的单位正交基）
≡ dE1(V) · E2 
= dE1 · E2(V)  （由性质dW(V) · U = dW · U (V)得）
简记为：Ω = dE1 · E2
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Gaussian curvature K = −dΩ(E1, E2)

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P125 Example 56. 如何用微分形式计算球面S的曲率K
S：Ψ (θ, φ) = (R sin φ cos θ, R sin φ sin θ, R cos φ)

dΩ(E1, E2) = (sinφ dφ∧dθ)(E1, E2) = sinφ (dφ∧dθ(E1, E2)) 

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A 【geodesic】 is a parameterized curve on a surface whose tangent vector field is parallel.
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Raoul Bott and Loring W. Tu. 
《Differential Forms in Algebraic Topology,》volume 82 of Graduate Texts in Mathematics. 
Springer-Verlag, New York, 1982.
Victor Guillemin and Alan Pollack. 
《Differential Topology.》
 AMS Chelsea Publishing, Providence, RI, 2010. Reprint of the 1974 original.
John Hamal Hubbard and Barbara Burke Hubbard. 
《Vector Calculus, Linear Algebra, and Differential Forms.》 
Prentice Hall Inc., Upper Saddle River, NJ, 1999. A unified approach.
Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. 
《Gravitation. 》也讨论了微分形式
W. H. Freeman and Co., San Francisco, Calif., 1973.
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