Determine whether or not 𝐅 is a conservative vector field. If it is, find a function f such that

step 1 In this problem, we are given the shown information, and we're asked to show if f is a conservat

Determine whether or not 𝐅 is a conservative vector field....

Determine whether or not $\mathbf{F}$ is a conservative vector field. If it is, find a function $f$ such that $\mathbf{F}=\nabla f$.
$\mathbf{F}(x, y)=\left(y e^{x}+\sin y\right) \mathbf{i}+\left(e^{x}+x \cos y\right) \mathbf{j}$

Key Concepts

Integration to Find Potential Functions

Once a vector field is confirmed to be conservative, a potential function can be found by integrating one of its component functions with respect to its corresponding variable, and then including an unknown function of the other variable. This unknown function is subsequently determined by differentiating with respect to the other variable and comparing it with the corresponding component of the vector field.

Mixed Partial Derivative Test

A primary technique for determining if a vector field is conservative is to verify that the mixed partial derivatives of its components are equal. In a two-dimensional field, this requires checking whether the partial derivative of the first component with respect to the second variable equals the partial derivative of the second component with respect to the first variable, assuming the domain is simply connected.

Conservative Vector Field

A conservative vector field is one that can be written as the gradient of a scalar potential function. This means that there exists a function f such that the vector field is equal to ?f. Conservative fields are notable because the line integrals taken over any path between two points depend only on the end points, not the specific path chosen.

Potential Function

A potential function is a scalar function whose gradient yields the given vector field. Finding this function involves integrating the partial derivatives that make up the components of the vector field, ensuring that the resulting function’s mixed partial derivatives are consistent to represent the original field.

Gradient

The gradient is an operator that produces a vector from a scalar function, indicating the direction of the steepest increase of the function and having components composed of the function's partial derivatives. It is central to the concept of conservative fields since a conservative vector field is exactly the gradient of some potential function.

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