====== 2020-02-21, Friday ======

$$\gdef\ot\otimes$$

===== Tensor power of a vector space =====
Let $V$ be a finite dimensional vector space. 

We denote the $k$-th tensor power of $V$ as 
$$ V^{\otimes k} = \underbrace{V\ot \cdots \ot V}_{\text{ $k$ times} } $$
Its elements are linear combinations of terms like $v_1 \otimes \cdots \ot v_k$, subject to the usual linearity relations. 

It is sometimes useful to consider the tensor algebra (we only mention it here, but do not use it later in this course).

** Definition (Tensor Algebra $T(V)$ ) **
$$T(V) = \R \oplus V \oplus V^{\ot 2} \oplus \cdots \oplus V^{\ot 3} \oplus \cdots $$
Given two elements $T = w_1 \ot \cdots \ot w_k$ and $T' = v_1 \ot \cdots \ot v_l$, their products is defined by juxtapostion.  
$$ T \ot T' = w_1 \ot \cdots \ot w_k \ot v_1 \ot \cdots \ot v_l $$


===== Exterior power of a vector space =====
** Definition (Exterior product $\wedge^k(V)$)**
The $k$-th exterior product $\wedge^k(V)$ is the vector space consisting of linear combinations of the following terms $v_1 \wedge \cdots \wedge v_k$, where the expression is linear in each slot, 
$$ c \cdot (v_1 \wedge \cdots \wedge v_k) = (c v_1) \wedge v_2 \wedge \cdots \wedge v_k $$
$$ (v_1+v_1') \wedge \cdots \wedge v_k = v_1 \wedge \cdots \wedge v_k + v_1'\wedge \cdots \wedge v_k$$
and the expression changes signs if we swap any two slots 
$$ v_1 \wedge \cdots \wedge v_i \wedge \cdots \wedge v_j\wedge \cdots \wedge v_k = -  v_1 \wedge \cdots \wedge v_j \wedge \cdots \wedge v_i\wedge  \cdots \wedge v_k, \forall 1 \leq i < j \leq k. $$

If $k=0$, we set $\wedge^0 V = \R$. If $k=1$, then $\wedge^1 V =V$. 

** Proposition ** If we choose a basis $e_1, \cdots, e_n$ of $V$, then for $0 \leq k \leq n$, the space $\wedge^k(V)$ has a basis consisting of the following vectors
$$ e_{i_1} \wedge \cdots \wedge e_{i_k}, \quad 1 \leq i_1 < i_2 < \cdots < i_k \leq n. $$

**Corrollary**
  * $\dim \wedge^k(V) = {n \choose k}$. 
  * If $k > n$, then $\wedge^k V = 0$. 

Just as we defined tensor algebra $T(V)$, we may define the exterior algebra $\wedge^* V$. This turns out to be very useful. 

** Definition(Exterior algebra $\wedge^* V$) ** 
$$ \wedge^* V := \bigoplus_{k=0}^{n} \wedge^k V, \quad \text{ where } \wedge^0 V:= \R.$$
The product between two elements is given by juxtaposition, more precisely, if $A = v_1 \wedge \cdots \wedge v_k \in \wedge^k V$, $B = w_1 \wedge \cdots \wedge w_l \in \wedge^l V$, then 
$$ A \wedge B :=  v_1 \wedge \cdots \wedge v_k \wedge w_1 \wedge \cdots \wedge w_l \in \wedge^{k+1} V.$$ 

==== Relationship between quotient algebra and tensor algebra ====

  - As quotient of tensor algebra
  - As subalgebra of tensor algebra

===== Differential Forms =====

$\Omega^k(M) = \wedge^k T^*M.$

One can do integration of $k$-forms on $k$-submanifold. 

===== exterior differentiation =====
In local coordinates $(x_1, \cdots, x_n)$, given a differential $k$-form, we have 
$$ d (\sum_I f_I dx^I) = \sum_I d(f_I) \wedge dx^I = \sum_I \sum_{i=1}^n \frac{\d f_I}{\d x^i} dx^i \wedge dx^I $$