Is a normal vector to the boundary of a surface well-defined?

Question

I am undergraduate studying physics.

I am applying the 2D divergence theorem:

$$\int_A{\nabla \cdot \textbf{F}}*da=\oint_L{\textbf{F}\cdot\hat{n}*dl}.$$

Is there a general way to find $\hat{n}$?

Thank you!

Answer 1

No, it depends on what data you have specfying $L$.

One particular case:

If you have a function $f(x,y) : \mathbb R^2 \to \mathbb R$ such that $L = \{(x,y) \;:\; f(x,y) = 0\}$, then $L$ is a level set of $F$, and the gradient $\nabla f$ is perpendicular to level sets, so $\hat n = \pm\frac{\nabla f}{|\nabla f|}$. The gradient points in a direction where $f$ increases, so that's how you determine which sign to use.