Step 1: Take a curve that stays on the level surface.
Let \( \mathbf{r}(t) \) be any curve lying entirely on the surface \( \varphi = \text{constant} \), so \( \varphi(\mathbf{r}(t)) = c \) for all \( t \).
Step 2: Differentiate along the curve.
By the chain rule, \( \dfrac{d}{dt}\varphi(\mathbf{r}(t)) = \nabla \varphi \cdot \mathbf{r}'(t) = 0 \), since \( \varphi \) does not change along the curve.
Here \( \mathbf{r}'(t) \) is a tangent vector to the surface at that point, and it can point along any direction within the surface, since the curve can be chosen freely.
Step 3: Read off the geometric meaning.
Since \( \nabla \varphi \) has a zero dot product with every possible tangent vector of the surface, it cannot have any component lying inside the surface.
Final Answer:
\( \nabla \varphi \) has to be entirely normal to the surface, which rules out the parallel, zero, and minimum rate options directly.
\[ \nabla \varphi \perp \{\varphi = \text{constant}\} \]