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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!

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

1 Answer 1

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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 $\vec{F}$.

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

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