real analysis – Do these integral orthogonality relationships involve point-to-point relationships?

I have the following problem:

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

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

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

I ask if integral orthogonality relationships involve point-to-point relationships.

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

* Just to clarify, the hypothesis is
$$int _ { mathbb {D} ^ n} langle h, df (V) rangle_ {e} = 0$$
or $$langle, rangle_ {e}$$ is the Euclidean domestic product.