How is transformation of function space to a vector field space gradient?

Question

From S.L Linear Algebra book:

Let $U$ be an open subset of $R^3$ , and let $V$ be the vector space of differentiable functions on $U$. Let $V'$ be the vector space of vector fields on $U$. Then grad: $V \rightarrow V'$ is a mapping. Is it linear? (For this part we assume you know some calculus).

The question above, does not seem quite familiar to me, but perhaps it is possible for me to find a certain connection.

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First, as we know, axioms for some $K$-linear mapping $F:S\rightarrow S'$ are:

1. $\forall x, y \in V,\,F(x+y)=F(x)+F(y)$

2. $\forall K \in R,\, \forall x \in V,\,F(cx)=cF(x)$

In the text above there is a following statement:

Let $V'$ be the vector space of vector fields on $U$.

This is where the confusion begins, from my understanding, vector fields map $n$-dimensional spaces to $n$-dimensional vectors ($n=3$ in this case).

Then, let $g:R^{3} \rightarrow R^{3}$ be a vector field mapping, then the basis for vector space $V'$ is $\{g_1, g_2, g_3\}$.

But then there is another statement:

Then $grad: V \rightarrow V′$ is a mapping. Is it linear?

Does $grad$ imply the gradient function (vector of all partial derivatives of a function)? If so, why? Also, wouldn't this be a composite mapping?

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In short, how can I apply the linear transformation axioms to the gradient mapping defined above?

Thank you!

Answer 1

(1) About your first question: note that the set of vector fields $V'$ on the open set $U$ is a vector space. If $V(x,y,z)=\bigl(V_1(x,y,z,),V_2(x,y,z),V_3(x,y,z)\bigr)$ and $W(x,y,z)=\bigl(W_1(x,y,z,),W_2(x,y,z),W_3(x,y,z)\bigr)$, their sum $(V+W)(x,y,z)$ is defined by $$ \bigl(V_1(x,y,z)+W_1(x,y,z), V_2(x,y,z)+W_2(x,y,z),V_3(x,y,z)+W_3(x,y,z)\bigr).$$ The notation is long, but note that we just sum every component of the vector field. Similarly, if $a\in \mathbb{R}$, the vector field $(aV)(x,y,z)$ is obtained by multiplying every component with $a$.

The basis "$\{g_1,g_2,g_3\}$" is not correct. The vector space $V'$ is infinite-dimensional.

(2) The map $\mathrm{grad} \colon V \to V'$ is indeed defined as taking the gradient of a function: $$ f(x,y,z) \mapsto \left(\frac{\partial f}{\partial x}(x,y,z),\frac{\partial f}{\partial y}(x,y,z),\frac{\partial f}{\partial z}(x,y,z)\right)$$

(3) For your last question: you should check/prove that $$\mathrm{grad}(f+g)=\mathrm{grad}(f)+\mathrm{grad}(g) \quad \text{and} \quad\mathrm{grad}(af)=a\mathrm{grad}(f)$$ where $f,g\in V$ and $a\in\mathbb{R}$.

I hope this clarifies things a bit.