^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] << [[Schutz - 2.20 - Basis One-Forms and Components of One-Forms|2.20 Basis One-Forms and Components of One-Forms]] | [[Schutz - 2.22 - Tensors and Tensor Fields| 2.22 Tensors and Tensor Fields]]>> # Index Notation Components of vectors have the index written as a superscript, e.g. $V^i$, while components of one-forms have their indices written as subscripts, e.g. $q_j$. The bases follow the opposite convention, members of a vector basis are labeled by a subscript $\{e_i\}$ while members of a one-form basis are written with a superscript $\{\tilde{\omega}^j\}$. For coordinate bases, one-forms have their index up, as they should (e.g. $\tilde{d}x^j$), while vectors have (e.g. $\partial/\partial x^i$) are considered to have their index down since it is in the denominator. These conventions are adopted for a good reason. They allow us to use the Einstein summation convention, where a sum over the same raised and lowered index is omitted, e.g. the contraction: $ \tilde{\omega}(\pmb{V}) = \sum_j V^j \omega_j \equiv V^j \omega_j $ Other examples include the contractions: $ \begin{align} \tilde{\omega} &= \omega_j \tilde{d}x^j,\\ \pmb{V} = V^j \frac{\partial}{\partial x^j} \end{align} $