Determine whether the statement is true or false. Explain...

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.)
It must be the case that
$$
\int_{C} e^{x^{2}} d x+\sin y^{3} d y=0
$$

Key Concepts

Conservative Vector Fields

A conservative vector field is one where the line integral between any two points does not depend on the specific path taken, and as a consequence, the line integral over any closed curve is zero. This characteristic arises from the existence of a scalar potential function whose gradient yields the components of the vector field.

Potential Functions

A potential function is a scalar function whose gradient is equal to the given vector field. If such a function exists and the domain of interest is appropriately connected, the line integral over any closed path will vanish because the initial and ending points have the same potential value.

Simply Connected Domains

A simply connected domain is a region without holes, ensuring that any closed curve within the region can be continuously shrunk to a point. This property is crucial in guaranteeing that a conservative vector field (with continuous partial derivatives) has a globally defined potential function, thereby making the evaluation of closed line integrals straightforward.

Equality of Mixed Partial Derivatives (Curl Test)

For a two-dimensional vector field, demonstrating that the partial derivative of the first component with respect to y equals the partial derivative of the second component with respect to x is a necessary condition for the field to be conservative in a simply connected domain. This condition, often linked to the curl of the field being zero, ensures that the field has no rotational component and that closed line integrals will equal zero.

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