How should one define the cross product for two vector valued functions?

Question

In many textbooks regarding vector analysis, the cross product is mention in the section about the properties of the derivative of vector functions, but there seems to be no explanation what that actually is. Perhaps, we can use the same definition as in linear algebra?

Answer 1

$\newcommand{\Reals}{\mathbf{R}}$The cross product of functions is defined pointwise: If $U$ is a non-empty open set in $\Reals^{n}$ for some positive integer $n$, and if $f = (f_{1}, f_{2}, f_{3})$ and $g = (g_{1}, g_{2}, g_{3})$ are functions from $U$ to $\Reals^{3}$, then $f \times g:U \to \Reals^{3}$ is defined by $$ f \times g = (f_{2}g_{3} - f_{3}g_{2}, f_{3}g_{1} - f_{1}g_{3}, f_{1}g_{2} - f_{2}g_{1}). $$