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).
$$\int_{0,3,0}^{(0,1)}\left(z e^{m} d x+d y+x e^{x x} d z\right)$$

Key Concepts

Fundamental Theorem for Line Integrals

The Fundamental Theorem for Line Integrals states that if a vector field (or differential form) is conservative—that is, if it is the gradient of some scalar potential function—then the line integral of the field along a curve depends solely on the values of the potential function at the endpoints of the curve. This theorem justifies the evaluation of the integral by a simple difference, thus greatly simplifying the problem once exactness is proven.

Conservative Vector Fields

A conservative vector field is one that has a potential function, meaning that its curl is zero in simply connected regions (or, in the language of differential forms, the form is closed and hence exact under appropriate conditions). Recognizing that a given field is conservative allows one to use the properties of exact forms and the Fundamental Theorem for Line Integrals to evaluate the integral without the need for detailed path computations.

Exact Differential Forms

An exact differential form is one that can be written as the differential of a scalar function. In other words, if ? is an exact form, there exists a potential function u such that ? = du. This property is crucial because it provides a method to evaluate integrals by simply finding the potential function, hence circumventing the need to perform a potentially more complicated integration along a curve.

Potential Function

The potential function is the scalar function whose differential gives rise to the exact differential form (i.e., u with du = ?). Once the potential function is determined, the value of the integral along any path depends only on the values of this function at the endpoints of the path. In the context of the given problem, finding the potential function directly allows for evaluating the integral by computing the difference u(endpoint) - u(starting point).

Transcript

Please give Ace some feedback