## Definition **Algebraic perspective** A tensor is a collection of [[Vector]] and [[Co-Vector]] combined together using the [[Tensor Product]] operation. ## Change of coordinate transforms A tensor is invariant under change of coordinates. A tensor's components, like a [[Vector|vector's]], is NOT invariant, since the components measure things with respect to a [[Vector Space Basis|basis]]. A tensor's components change in a predictable way under change of basis coordinates: $ \begin{align} \widetilde{T_{xyz\dots}^{abc\dots}} &= (B_i^aB_j^bB_k^c\dots) T_{rst\dots}^{ijk\dots} (F_x^rF_y^sF_z^t\dots) \\ T_{rst\dots}^{ijk\dots} &= (F_a^iF_b^jF_c^k\dots) \widetilde{T_{xyz\dots}^{abc\dots}} (B_r^xB_s^yB_t^z\dots) \end{align} $ ```ad-warning I'm pretty sure the indices above are wrong. TODO re-work it out from $e\epsilon$ bases and cancel out the kronecker deltas ``` Where $F$ is the forward transform and $B=F^{-1}$ is the backward transform and the subscripts are covariant indices and superscripts are contravariant indices. Technically, anything that follows these rules is a tensor. ## Types We say that a tensor $T_{rst\dots}^{ijk\dots}$ with $m$ contravariant indices and $n$ covariant indices is an (m,n)-tensor where $(m,n)$ is the tensor's type ^[Similar to a [[Model Theory Signature|signature]] in [[Model Theory]]]. ## Raw notes - For any tensor we want to establish the properties that define it - how it behaves under a change of base - What's the multiplication formula for 2 tensors $Q(D)$. There's no single way to formulate how Q acts on D. It depends on how how many $\vec{e_i}$ and $\epsilon^j$ bases are in each tensor and how they're arranged. For higher typed tensors, there's several ways to apply the multiplication rules. - it's array shape - With higher type tensors, the array representation becomes less useful (because there's multiple ways to multiply it out. It's better to stick with the [[Einstein Component Notation]]. ## Resources --- - Links: - Created at: 2023-06-07