Tensors for begineers

from ‘eigenchris’ on Youtube

Last modified: 20 November, 2022

1. Defination of tensor

Tensor : an object is invariant under a change of coordinates, and …
has componets that change in a special, predictable way under a change of coordinates.
Tensor : a collection of vectors and covectors combined together using the tensor product.
Tensor : as partial derivatives and gradients that transform with jacobian Matrix

Forward transforms: $\widetilde{\vec {e}_i}=\sum_{j=1}^{n}F_{ji}\vec{e}_j$

Backward transforms: $\vec{e}_i=\sum_{j=1}^{n}B_{ji}\widetilde{\vec {e}_j}$

$\sum_{j}F_{kj}B_{ji}=\delta_{ik}$ (Kronecker delta)

Vector : a member of vector space–> (V,S,+,.)
v : Set of Vectors
S : Set of Scalars
+ : Vector addition rule
. : Vector scaling
Covector :
- Fucntions $\alpha:V-\mathbb{R}$ that map a vector to a number and also obey the following rules:
  $\alpha({\vec {v}}+{\vec {w}})=\alpha({\vec {v}})+\alpha({\vec {w}})$
  $\alpha(n{\vec {v}})=n\alpha({\vec {v}})$
- Elements of $V^{*}$ dual vector space–> ($V^{*}$,S,+,.)
  $(n\cdot\alpha){\vec {v}}=n\alpha({\vec {v}})$
  $(\beta+\gamma){\vec {v}}=\beta({\vec {v}})+\gamma({\vec {v}})$
  3.2. Transforms Rules

Vector : ${\vec {v}}=\sum_{i=1}^{n}v^{i}\vec{e}_i=\sum_{i=1}^{n}\widetilde{v^{i}}\vec{e}_i$

Covector : ${{\alpha}}=\sum_{i=1}^{n}\alpha_{i}{\epsilon}^i=\sum_{i=1}^{n}\widetilde{\alpha_{i}}\widetilde{{\epsilon}^i}$

Covarient
- Basic vectors:
  $\widetilde{\vec {e}_j}=\sum_{i=1}^{n}F_{ij}\vec{e}_i$
  $\vec{e}_j=\sum_{i=1}^{n}B_{ij}\widetilde{\vec {e}_i}$
- Covector components:
  $\widetilde{{\alpha}_j}=\sum_{j=1}^{n}F_{ij}{\alpha}_i$
  ${\alpha}_j=\sum_{j=1}^{n}B_{ij}\widetilde{{\alpha}_i}$
Contravarient
- Basic covectors:
  $\widetilde{\epsilon^i}=\sum_{i=1}^{n}B_{ij}\epsilon^j$
  $\epsilon^i=\sum_{i=1}^{n}F_{ij}\widetilde{\epsilon^j}$
- Vector components:
  $\widetilde{{v}^i}=\sum_{j=1}^{n}B_{ij}{v}^j$
  ${v}^i=\sum_{j=1}^{n}F_{ij}\widetilde{{v}^j}$