Definition talk:Gradient Operator/Real Cartesian Space

Isn't the gradient supposed to be a vector-valued function rather than a column matrix? Also, as for now, the gradient has only been defined at points $\mathbf x$. --Lord_Farin 10:14, 2 April 2012 (EDT)
 * ooh, I got confused about our discussion in my sandbox page. Let me try to fix it. --GFauxPas 10:17, 2 April 2012 (EDT)
 * Okay so looking for closely at Larson he says $\nabla$ is an operator in the same sense $\frac {\mathrm d}{\mathrm dx}$ is an operator. He says that $\nabla$ operates on $f(\mathbf x)$ and produces the vector $\nabla f(\mathbf x)$. I think he means $\nabla: \R^\R \to \R^n, f \mapsto \nabla f$. In any event what I put down is what Larson has as a definition, he has this stuff as a footnote --GFauxPas 10:26, 2 April 2012 (EDT)


 * I would say we have $\nabla: [\R^n\to\R] \to [\R^n\to\R^n]$ where $[X\to Y]$ is intended to be $Y^X$, the set of all mappings (of course, the domain has to be suitably restricted). So, it transforms functions into functions. But $\nabla$ surely does not operate on $f\left({\mathbf x}\right)$ because $f(x) = g(x)$ does not imply $\nabla f(x) = \nabla g(x)$, evidently. Thus, $\nabla$ needs to be seen as taking functions as input, giving again functions as output. --Lord_Farin 10:33, 2 April 2012 (EDT)


 * I don't understand your explanation as to how we know $\nabla$ doesn't operate on $f(\mathbf x)$ but I think you're right. I'm having a hard time getting clarity because too many sources use $f$ to mean $f(\mathbf x)$ and vice versa which in general is annoying but here is particularly annoying. I think what's happening here is indeed that $f \mapsto \nabla f$. But we were able to define "derivative" without coming on to $f \mapsto (D_xf: y \mapsto \frac {\mathrm dy}{\mathrm dx})$. I'm guessing many people are more comfortable with "functions on numbers/vectors" than "functions on functions". But then again, It is standard fare to learn $f + g$, $f \circ g$ etc. in high school... --GFauxPas 10:54, 2 April 2012 (EDT)