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##0##-form ##f =## function ##f##

##1##-form ##\alpha^{1} =## covariant expression for a vector ##\bf{A}##

Then consider the following dictionary of symbolic identifications of expressionsexpressed in the language of differential forms on a manifoldand expressionsexpressed in the language of vector calculus in ##\mathbb{R}^{n}##.

##i_{\bf{v}}(\alpha^{1}) \iff \bf{v}\cdot{\bf{A}}##

##df \iff \text{grad}f##

Now, with ##\alpha^{1} = a_{i}dx^{i}##,

why is ##\textbf{d}i_{\bf{v}}(a_{i}dx^{i})\iff\text{grad}({\bf{v\cdot{A}}})\cdot{d\bf{x}}##

and not ##\textbf{d}i_{\bf{v}}(a_{i}dx^{i})\iff\text{grad}({\bf{v\cdot{A}}})##?

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# A Differential forms and vector calculus

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