Is the gradient theorem equivalent to the curl theorem if F is conservative?

[Meow12]

The gradient theorem states: [imath]\displaystyle\int_C \nabla f\cdot d\mathbf{r}=f(\mathbf{r}(b))-f(\mathbf{r}(a))[/imath]
where C is a smooth curve parametrized by the vector-valued function [imath]\textbf{r}(t)[/imath] for [imath]a\le t\le b[/imath].

So, if we consider a closed loop beginning and ending at [imath]t=a[/imath],

[imath]\displaystyle\oint_C \nabla f\cdot d\textbf{r}=f(\textbf{r}(a))-f(\textbf{r}(a))=0[/imath] -------- (1)

The curl theorem states: [imath]\displaystyle\oint_C\textbf{F}\cdot d\textbf{l}=\iint_S \text{curl }\textbf{F}\cdot d\textbf{S}[/imath]

If [imath]\mathbf{F}[/imath] is conservative, [imath]\text{curl }\textbf{F}=\mathbf{0}[/imath].

So, the curl theorem implies that [imath]\displaystyle\oint_C\textbf{F}\cdot d\mathbf{l}=\iint_S\mathbf{0}\cdot d\textbf{S}=0[/imath] -------- (2)

Equations (1) and (2) are the same.

 

[blamocur]

Not sure about equivalency of the theorems, but, to me, you've shown that [imath]\nabla f[/imath] is conservative.