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.