# Definition ## Manifolds Consider [[vector|vectors]] and [[Tensor|tensors]] defined on some point $P \in M$, for some [[manifold]] $M$. Suppose we begin with a vector [[basis]] $\{\pmb{e}_i\}$ and wish to transform to a new [[basis]] $\{\pmb{e}_{j'}\}$. Primes on the indices denote a new basis. Then, in the [[tangent space]] $T_P$ there is a linear transformation $\Lambda$ from the old [[basis]] to the new: $ \pmb{e}_{j'} = \Lambda^i_{\,j'} \pmb{e}_i $ The matrix $\Lambda^i_{\,j'}$ is [[Non-singular Matrix|non-singular]] to guarantee [[linear independence]] of the new-basis, but otherwise arbitrary. It is not the collection of components of a [[Tensor]] since the [[Index Notation|indices]] refer to two different [[basis|bases]]. Moreover, the old [[one-form]] satisfies: $ \tilde{\omega}^i(\pmb{e}_k) = \delta^i_{\, k} $ Thus, we get: $ \tilde{\omega}^i(\pmb{e}_{j'}) = \tilde{\omega}^i(\Lambda^i_{\,j'} \pmb{e}_k) = \Lambda^k_{\,j'}\delta^i_{\, k} = \Lambda^i_{\,j'} $ We define the inverse of $\Lambda^i_{\,\,j'}$ to be $\Lambda^{k'}_{\,\,j}$, giving: $ \begin{align} \Lambda^{k'}_{\,\,i}\Lambda^i_{\,\,j'} &= \delta^{k'}_{\,j'}\\ \Lambda^{k'}_{\,\,j}\Lambda^i_{\,\,k'} &= \delta^{i}_{\,j} \end{align} $ Multiply above by $\Lambda^{k'}_{\,\,i}$: $ \Lambda^{k'}_{\,\,i}\tilde{\omega}^i(\pmb{e}_{j'}) = \Lambda^{k'}_{\,\,i}\Lambda^i_{\,\,j'} = \delta^{k'}_{\,\,j'} = \tilde{\omega}^{k'}(\pmb{e}_{j'}) $ Thus we get the transformation law: $ \tilde{\omega}^{j'} = \Lambda^{j'}_{\,\,i}\tilde{\omega}^i $ And we can now transform components simply as (using linearity and identities defining the components): $ \begin{align} V^{i'} &= \tilde{\omega}^{i'}(\pmb{V}) = \Lambda^{i'}_{\,\,j} \tilde{\omega}^{i}(\pmb{V}) = \Lambda^{i'}_{\,\,j} V^j, \\ q_{i'} &= \tilde{q}(\pmb{e}_{i'}) = \Lambda^{j}_{\,\,i'} \tilde{q}(\pmb{e}_j) = \Lambda^{j}_{\,\,i'} q_j \end{align} $ And similarly for tensors of higher type. > **Transformation Laws for Tensors** > Bases transform as: >$ \begin{align} \pmb{e}_{j'} &= \Lambda^i_{\,j'} \pmb{e}_i, \\ \tilde{\omega}^{j'} &=\Lambda^{j'}_{\,\,i}\tilde{\omega}^i, \end{align} > $ > While components transform the opposite way: > $ \begin{align} V^{i'} &= \Lambda^{i'}_{\,\,j} V^j, \\ q_{i'} &= \Lambda^{j}_{\,\,i'} q_j \end{align} >$ > Tensors transform similarly. For an $(N, N')$ tensor: >$ T^{i_{1}' \ldots i_N'}_{\quad \quad \,\,\,j_{1}'\ldots j_{N'}'} = \Lambda^{i'_1}_{\,\,k_1} \ldots \Lambda^{i'_N}_{\,\,k_N} \Lambda^{l_1}_{\,\,j'_1} \ldots \Lambda^{l_N}_{\,\,j'_{N'}} T^{k_{1} \ldots k_N}_{\quad \quad \,\,\,l_{1}\ldots l_{N'}} >$