Show that the form under the integral sign is exact in the...

Show that the form under the integral sign is exact in the plane (Probs. $1-4$ ) or in space (Probs. $5-8$ ) and evaluate the integral. (Show the details of your work).
$$\int_{120}^{(2 \pi)}\left(\cos ^{2} y d x-2 x \cos y \sin y d y\right)$$

Key Concepts

Fundamental Theorem for Line Integrals

The fundamental theorem for line integrals states that if a vector field is conservative (i.e., it is the gradient of a scalar potential), then the line integral over any path between two points is equal to the difference in the potential function at those points. This theorem allows one to bypass the complexity of integrating along a specific path, provided the vector field is conservative.

Potential Functions

A potential function is a scalar function whose gradient recovers the original differential form. Finding this function involves integrating one component of the form with respect to its variable and then verifying consistency with the other component. Once the potential function is known, it simplifies the computation of the integral, as the line integral of an exact form is given by the difference in the potential function values at the endpoints.

Exact Differential Forms

An exact differential form is one that can be expressed as the exact derivative of a potential function. In the context of multivariable calculus, when a form is exact, it means there exists a scalar function whose differential equals the given form. This property is particularly useful for evaluating line integrals because the integral will depend only on the values of the potential function at the endpoints of the integration path.

Conditions for Exactness

To determine if a differential form is exact, one typically checks certain partial derivative relationships. In the plane, for a form M(x, y) dx + N(x, y) dy, the necessary condition for exactness is that the mixed partial derivatives should be equal, i.e., ?M/?y must equal ?N/?x, assuming the functions involved are continuous. This criterion confirms that a potential function exists.

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