Introduction of Tensor Algebra



Introduction


Even though tensor algebra does not belong to linear algebra I decided to incorporate this topic in here since it can be also viewed as an extension of the knowledge required in linear algebra. Concepts like inner product, vectors, matrices and metrics are further generalized in tensor algebra.



Change of basis and coordinate transformation


To understand some properties of covariant and contravariant vectors let us start with the basis transformation matrix and its corresponding coordinate transformation matrix. These two operators exhibit a somehow reversed behaviour. When we though analyse how they are constructed it becomes obvious:


Let \(\mathcal V (\mathbb{K})\) be a vectorspace over a fied \(\mathbb{K}\) and let \(\mathcal B_{1} = \{b_{1}, ..., b_{n} \}\) and \(\mathcal B_{2} = \{\hat{b_{1}}, ..., \hat{b_{n}} \}\) be two basis.
\(\forall \ v \in \mathcal V\) with \(v = \sum_{i=1}^{\\n} b_{i} \mu_{i} = \sum_{i=1}^{\\n} \hat{b}_{i} \lambda_{i}\) we have \(\mathcal T_{B} = (\mathcal T_{C}^{-1})^\top\) where:


\[\text{1.} \quad \mathcal T_{B} (b_{1}, ..., b_{n}) = (\hat{b_{1}}, ..., \hat{b_{n}})\]


\[\text{2.} \quad \mathcal T_{C} \begin{bmatrix} \mu_{1} \\ \mu_2 \\ \vdots \\ \mu_n \end{bmatrix} \ = \begin{bmatrix} \lambda_{1} \\ \lambda_{2} \\ \vdots \\ \lambda_{n} \end{bmatrix}\]


From the chapter, Matrix Representation, we know that the coordinate transformation from basis $\mathcal{B}_1$ to basis $\mathcal{B}_2$ has the following form: $$ \Phi_{\mathcal{B}_2} = [\mathcal{I}d_{\mathcal{V}}]_{\mathcal{B}_1, \mathcal{B}_2} \ \circ \ \Phi_{\mathcal{B}_1} $$
$$ (\mathcal{T}_{C}^{-1})^\top = (([\mathcal{I}d_{\mathcal{V}}]_{\mathcal{B}_1, \mathcal{B}_2} \ \circ \ \Phi_{\mathcal{B}_1})^{-1})^\top = ( \Phi_{\mathcal{B}_1}^{-1} \ \circ \ [\mathcal{I}d_{\mathcal{V}}]_{\mathcal{B}_1, \mathcal{B}_2}^{-1})^\top = ( \Phi_{\mathcal{B}_1}^{-1} \ \circ \ [\mathcal{I}d_{\mathcal{V}}]_{\mathcal{B}_2, \mathcal{B}_1})^\top $$

Now we need to ... (dual map)


On the left plot is the coordinate transformation and on the right the basis transformation. In the center of both plots you can see the unit disc: $$\{ v \in \mathbb{R}^{2} : \| v \|_2 \leq 1 \}$$ As we can see these transformations are acting dual to each other.




Vectorfields




Covariant Vectors


Due to the duality …



Contravariant Vectors




Transformation of a Gradient