# Differentiable scalar fields question.

Let $$f,g:\mathbb{R}^3\to\mathbb{R}$$ be twice continuously differentiable scalar fields. Which of the following statements is false?

A: If $$S$$ is any simple piecewise-smooth surface with unit normal vector $$\mathbf{n}$$, then $$\iint_S (\nabla\times\nabla f)\cdot\mathbf{n} \,dS=0$$
B: If $$S$$ is the unit sphere with the outward pointing normal vector $$\mathbf{n}$$, then $$\iint_S \nabla f\cdot\mathbf{n} \,dS=4\pi$$
C: If the directional derivative of $$f$$ along the unit tangent vector of a simple piecewise-smooth curve $$C$$ is zero at every point on $$\space\space\space\space\space C$$, then $$\int_C \nabla f\cdot d\mathbf{r}=0$$
D: The vector field $$\nabla f \times \nabla g$$ is solenoidal

What I know so far...
So, I believe statement B to be false since I know that $$\iint_S dS=4\pi$$ where $$S$$ is the unit sphere. Also, I know D to be true if $$f$$ and $$g$$ are twice continuously differentiable scalar fields - so that rules that out. My gut tells me C to be true since $$C$$ is zero at every point. It's from here I'm not sure which is false... any help would be great!

• Did you compute $rot(\nabla f)$?
– PQH
Nov 18, 2020 at 17:53

For A: compute $$\nabla\times \nabla f=rot(\nabla f)$$ using the definitions.

For B: take a constant field and compute.

For C: use that $$\frac{d f}{d \vec{v}}=\nabla f\cdot \vec{v}$$ and that $$\int_C \vec{F} \cdot d\vec{r}=\int_C(\vec{F}\cdot \vec{T}) dr$$, for vector fields $$\vec{F}$$.

• You have to know the definitions in order to proceed. If it is not the case, you have to go look for them.
– PQH
Nov 18, 2020 at 18:37