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When hand-writing the gradient of $f$ as "$\nabla f$" or "grad $f$", is it necessary to indicate that it is a vector using the usual vector markings (cap, arrow, wavy line, etc.)?

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It should be considered obligatory to write, for example $\vec{a}$ or $\mathbf{a}$, when you're writing in a context in which vectors and vector-valued functions are generally written that way. But that is not always done. The style should be consistent throughout the document.

However, notice that the $f$ in $\nabla f$ is scalar-valued. The expression $\nabla f$ is vector-valued, and that is indicated by the meanings of the symbols.