So I have a vector field (see below) where I need to draw 3 oriented closed curves on it, where the line integral of the vector field over the curve is to be positive, negative, and zero. I know how to draw oriented curves on it but am not sure how to draw an oriented closed curve.
Here is the vector field with the oriented curves:
The blue arrow means it's negative, the green means positive and the Red means zero since its perpendicular to the vector's direction.
So my question is, how do I draw a oriented closed curve ? I think I know how to do it, but am unsure if it is the correct way to do.
Here is how I think is the way to do it, but I am dobutful.
When you traverse a path in the direction of the field, it contributes positively and when you move against the field, it contributes negatively. And when you move along the path perpendicular to the field, it's neutral (contributes nothing to the integral).
For a positive line integral, consider a clockwise rectangle sitting above the $x$-axis. You can see that along the top (positive contribution) will be greater than along the bottom (negative contribution) because the vectors have greater magnitude along the top. Along the sides, the contribution is zero. A counter-clockwise rectangle below the $x$-axis would also work.
For a negative line integral, use a clockwise rectangle down below or a counter-clockwise rectangle up above.
For a zero integral, consider a rectangular path that's balanced across the $x$-axis.
Does this make sense?