True-False Determine whether the statement is true or false....

True-False Determine whether the statement is true or false. Explain your answer. (In Exercises $16-18$, assume that $C$ is a simple, smooth, closed curve, oriented counterclockwise.) If $$ \int_{C} f(x, y) d x+g(x, y) d y=0 $$ then $\partial g / \partial x=\partial f / \partial y$ at all points in the region bounded by $C$.

True-False Determine whether the statement is true or false. Explain your answer. (In Exercises $16-18$, assume that $C$ is a simple, smooth, closed curve, oriented counterclockwise.) If
$$
\int_{C} f(x, y) d x+g(x, y) d y=0
$$
then $\partial g / \partial x=\partial f / \partial y$ at all points in the region bounded by $C$.

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Key Concepts

Green's Theorem

Green's Theorem connects a line integral around a simple, closed, and positively oriented curve to a double integral over the region it encloses. It states that the line integral of (f dx + g dy) equals the double integral of (?g/?x - ?f/?y) over the region. This theorem is key in understanding how local properties of a vector field (through partial derivatives) are related to its global circulation around a closed curve.

Conservative Vector Fields and Necessary Conditions

A vector field is conservative if its line integral over any closed curve is zero, which is equivalent to the existence of a potential function for the field. A necessary condition for a two-dimensional vector field to be conservative is that the curl, given by (?g/?x - ?f/?y), vanishes continuously throughout a simply connected region. However, having a zero line integral around a single closed curve does not guarantee that this condition holds pointwise everywhere in the region, because the vanishing of the integral can occur even if the curl is nonzero in some parts, provided that the positive and negative contributions cancel out over the entire region.

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