# 外微分形式 | {Differential Forms} [1]

Chapter 1 Why forms?

Orientation of the integration, determined by the argument.

More simple for some calculation. I always encounter calculations hard to be solved only with tensor analysis. That is why I started to learn a little about exterior differential forms.

Chapter 2 Exterior Algebra

1. Set up the space we will work on.

✟\mathcal{R}✟: field of real numbers.

✟\mathcal{L}✟: n dimensional space over ✟\mathcal{R}✟ with elements ✟\alpha✟, ✟\beta✟, ✟\dots✟

✟\wedge^p \mathcal{L}✟: space of p-vectors on ✟\mathcal{L}✟. This will be made clear later.

2. Basic rules for basis.

Linear: ✟(a \alpha +b \beta)\wedge \alpha_2\wedge\cdots\wedge\alpha_p = a(\alpha\wedge\alpha_2\wedge\cdots\wedge\alpha_p)+b(\beta\wedge\alpha_2\wedge\cdots\wedge\alpha_p)✟

Any of the terms repeated in the expression ✟\alpha_1\wedge\alpha_2\wedge\cdots\wedge\alpha_p✟ leads the whole to 0.

Interchanging between two terms in ✟\alpha_1\wedge\alpha_2\wedge\cdots\wedge\alpha_p✟ changes the whole’s sign.

3. Properties of p-vector space

Function g can be defined on ✟\mathcal{L}✟ and mapping f on ✟\wedge^p \mathcal{L}✟, satisfying ✟f(\alpha_1\wedge\cdots\wedge\alpha_p)=g(\alpha_1, \cdots, \alpha_p)✟.

4. Determinants

Function ✟g_A✟: ✟\bigotimes^n \mathcal{L} \rightarrow \wedge^n \mathcal{L}✟ and ✟f_A✟: ✟\wedge^n\mathcal{L}\rightarrow\wedge^n\mathcal{L}✟ can be related by

f_A(\alpha_1\wedge\cdots\wedge\alpha_p)=g_A(\alpha_1,\cdots, \alpha_p)\equiv A\alpha_1\wedge\cdots\wedge A\alpha_p

Since only one kind of transformation the linear function ✟f_A✟ has, the only possible expression for ✟f_A✟ is ✟|A|\alpha_1\wedge\cdots\wedge\alpha_p✟, where ✟|A|✟ is a scalar. Define ✟\wedge^p(\alpha_1\wedge\cdots\wedge\alpha_p)\equiv A\alpha_1\wedge\cdots\wedge A\alpha_p✟.

Another way of looking at ✟g_A✟ is ✟A\alpha_1\wedge\cdots\wedge A\alpha_p = |a^i_j|\alpha_1\wedge\cdots\wedge\alpha_p✟, if we have ✟A\sigma^i=\Sigma a^i_j \sigma^j✟.

Comparing the two understandings, we find ✟|A|=|a^i_j|✟. Here ✟|A|✟ is the determinant.

5. Exterior products

Basic rules: (✟\mu✟ and ✟\lambda✟ are p vector and q vector respectively)

a. ✟\mu\wedge\lambda✟ distribuative

b. associative

c. ✟\mu\wedge\lambda=(-1)^{pq}\lambda\wedge\mu✟

6. Linear transformations

Linear spaces: ✟\mathcal{M},\mathcal{N}✟ with dimension m and n respectively. The basis of ✟\mathcal{M}✟ is ✟\sigma^i✟.

Define a linear transformation A : ✟\mathcal M \rightarrow \mathcal N✟, which is

(\alpha_1, \cdots, \alpha_p) \rightarrow A\alpha_1\wedge\cdots\wedge A\alpha_p .

This is a map from ✟\bigotimes^p\mathcal M✟ to ✟\wedge^p \mathcal N✟.

One would find a useful linear transformation can be denoted ✟\wedge^p A✟ : ✟\wedge^p \mathcal M\rightarrow \wedge^p\mathcal N✟.

Note that ✟\wedge^p A✟ has matrix representation.

7. Inner product spaces

Basic rules

a. Linear in each variable

b. Symmetric: ✟(\alpha, \beta)=(\beta, \alpha)✟

c. Nondegenerate: if for fiexed ✟\alpha✟, ✟(\alpha,\beta)=0✟ for all ✟\beta✟, then ✟\alpha=0✟.

Properties:

a. Each inner product space ✟\mathcal L✟ has an orthonormal basis.

b. ✟f✟: linear function on ✟\mathcal L✟. Then there is a vector ✟\beta✟ in ✟\mathcal L✟ such that ✟f(\alpha)=(\alpha, \beta)✟. (This property is very important.)

8. Inner product of p-vectors

Vector space ✟\mathcal L✟ has inner product ✟(\alpha,\beta)✟ on it. Define ✟(\lambda,\mu)=|(\alpha_i,\beta_j)|✟ on ✟(\wedge^p \mathcal L)\bigotimes (\wedge^p\mathcal L)✟, in which ✟\lambda=\alpha_1\wedge\cdots\wedge\alpha_p✟, ✟\mu=\beta_1\wedge\cdots\wedge\beta_p✟.

9. Hodge star

Linear transformation on ✟\wedge^p\mathcal L\rightarrow \wedge^{n-p}\mathcal L✟. (Depends on inner product and orientation. Orientation of ✟\mathcal L✟ which will remain fixed means we hake on basis for ✟\mathcal L✟ and only consider other basis which are expressed in terms of this one by a matrix with positive determinant.)

\lambda\wedge\mu=(*\lambda, \mu)\sigma (=f_\lambda (\mu)\sigma)

here * map is a linear one ✟\wedge^p\mathcal L\rightarrow \wedge^{n-p}\mathcal L✟.

Reference:

Differential forms with applications to the physical sciences
By By Harley Flanders