Let 𝐅=𝐅(x, y)=P(x, y) 𝐢+Q(x, y) 𝐣 be a vector field, where the functions P and Q are continuous

step 1 In this problem of vector field we have given that let F say here F is the function of x and y w

Let 𝐅=𝐅(x, y)=P(x, y) 𝐢+Q(x, y) 𝐣 be a vector field, where...

Let $\mathbf{F}=\mathbf{F}(x, y)=P(x, y) \mathbf{i}+Q(x, y) \mathbf{j}$ be a vector field, where the functions $P$ and $Q$ are continuous on some simply connected, open region $R .$ Suppose $\frac{\partial P}{\partial y}$ and $\frac{\partial Q}{\partial x}$ are also continuous on $R$. Then $\mathbf{F}(x, y)=P(x, y) \mathbf{i}+Q(x, y) \mathbf{j}$ is a conservative vector field on $R$ if and only if _________ .

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Key Concepts

Simply Connected Region

A simply connected region in a plane is one without holes, meaning that any closed curve within the region can be continuously contracted to a point without leaving the region. This property is crucial because it ensures that certain integral tests (like checking the equality of mixed partials for a conservative field) are both necessary and sufficient for the existence of a potential function.

Equality of Mixed Partial Derivatives

The equality of mixed partial derivatives, typically expressed as ?P/?y = ?Q/?x, is central in determining whether a vector field is conservative. Under the conditions of continuous partial derivatives in a simply connected region, this equality guarantees that a potential function exists for the vector field. This condition is a direct consequence of Clairaut’s theorem on the equality of second mixed partials.

Conservative Vector Field

A conservative vector field is one in which the line integral between any two points is independent of the chosen path. This means that the field can be expressed as the gradient of a scalar potential function. Such fields exhibit the property that the circulation around any closed path is zero, which is a key aspect in evaluating work done by forces in physics and other applications.

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