Do these integral orthogonality relations imply pointwise relations?

Question

Let $\mathbb{D}^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball. Let $f,h:\mathbb{D}^n \to \mathbb{R}^{k}$ be smooth.

Suppose that for every smooth vector field $V$ on $\mathbb{D}^n$, $\langle h, df(V) \rangle_{L^2} =0$.

Is it true that $\langle h(x), df_x(v) \rangle_e=0$ for every $x \in \mathbb{D}^n$ and every $v \in \mathbb{R}^n$?

I am asking whether these integral orthogonality relations* imply pointwise relations.

I think that the answer is positive. My rough idea is to take "test fields" $V$ which are compactly supported on smaller and smaller neighbourhoods of $x$, but I am not sure if this approach actually works here.

*Just to clear, the assumption is $$ \int_{\mathbb{D}^n} \langle h, df(V) \rangle_{e} =0 $$ where $\langle , \rangle_{e}$ is the Euclidean inner product on $\mathbb{R}^k$.

Answer 1

For every smooth vector field $V$ on $\mathbb{D}^n$, $\int_{\mathbb{D}^n}{f’(x)(V(x))\cdot h(x)\,dx}=0$.

So, for every smooth vector field $V$ on $\mathbb{D}^n$, $\int_{\mathbb{D}^n}{f’(x)^*(h(x)) \cdot V(x)\,dx}=0$.

This implies that $f’(x)^*(h(x))=0$ for all $x \in \mathbb{D}^n$.

In particular, for every smooth vector field $V$ on $\mathbb{D}^n$, $f’(x)(V(x)) \cdot h(x)=f’(x)^*(h(x)) \cdot V(x)=0$.