Verify that each of the following force fields is...

Verify that each of the following force fields is conservative. Then find, for each, a scalar potential $\phi$ such that $\mathbf{F}=-\nabla \phi$.
$$\mathbf{F}=2 x \cos ^{2} y \mathbf{i}-\left(x^{2}+1\right) \sin 2 y \mathbf{j}$$

Key Concepts

Conservative Vector Fields

A conservative vector field is one that can be expressed as the gradient of a scalar potential function. In such fields, the work done along a path between any two points depends only on the endpoints, implying path independence. A necessary condition for a vector field to be conservative in a simply connected region is that its curl is zero.

Scalar Potential Function

A scalar potential function is a single-valued function whose gradient (or negative gradient, depending on the context) equals the given vector field. Finding a scalar potential involves integrating the components of the vector field with respect to their respective variables and ensuring that these partial integrations yield a consistent function.

Gradient Operator

The gradient operator transforms a scalar function into a vector field where each component is the partial derivative of the function with respect to one of the coordinates. It indicates the direction of steepest ascent of the function and is fundamental in relating scalar potentials to vector fields.

Curl of a Vector Field

The curl measures the tendency of a vector field to circulate about a point. For a field to be conservative, its curl must vanish, indicating that there is no local rotation. This condition is critical in confirming whether a potential function exists.

Integration in Multiple Variables

When finding a potential function, integration is performed with respect to each variable independently. Since the constant of integration in one variable may actually be a function of the other variables, it is essential to differentiate the integrated result and compare it with the given vector field component to determine any missing functions of the other variables.

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