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,+,.)\ \((n\cdot\alpha){\vec {v}}=n\alpha({\vec {v}})\)\ \((\beta+\gamma){\vec {v}}=\beta({\vec {v}})+\gamma({\vec {v}})\)

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:\ \(\widetilde{{v}^i}=\sum_{j=1}^{n}B_{ij}{v}^j\) \ \({v}^i=\sum_{j=1}^{n}F_{ij}\widetilde{{v}^j}\)