91Ó°ÊÓ

Determine whether the vector field \(\mathbf{F}\) is conservative. If it is, find a potential function for the vector field. \(\mathbf{F}(x, y, z)=\frac{x}{x^{2}+y^{2}} \mathbf{i}+\frac{y}{x^{2}+y^{2}} \mathbf{j}+\mathbf{k}\)

Evaluate the integral \(\int_{C}(2 x-y) d x+(x+3 y) d y\) along the path \(C\). C: line segments from \((0,0)\) to \((0,-3)\) and \((0,-3)\) to \((2,-3)\)

Find the divergence of the vector field \(\mathbf{F}\). \(\mathbf{F}(x, y, z)=6 x^{2} \mathbf{i}-x y^{2} \mathbf{j}\)

Evaluate the integral \(\int_{C}(2 x-y) d x+(x+3 y) d y\) along the path \(C\). \(C:\) arc on \(y=1-x^{2}\) from \((0,1)\) to \((1,0)\)

Find the divergence of the vector field \(\mathbf{F}\). \(\mathbf{F}(x, y, z)=x e^{x} \mathbf{i}+y e^{y} \mathbf{j}\)

Evaluate the integral \(\int_{C}(2 x-y) d x+(x+3 y) d y\) along the path \(C\). \(C: \operatorname{arc}\) on \(y=x^{3 / 2}\) from \((0,0)\) to \((4,8)\)

Find the divergence of the vector field \(\mathbf{F}\). \(\mathbf{F}(x, y, z)=\sin x \mathbf{i}+\cos y \mathbf{j}+z^{2} \mathbf{k}\)

Evaluate the integral \(\int_{C}(2 x-y) d x+(x+3 y) d y\) along the path \(C\). \(C:\) parabolic path \(x=t, y=2 t^{2}\), from \((0,0)\) to \((2,8)\)

Find the divergence of the vector field \(\mathbf{F}\). \(\mathbf{F}(x, y, z)=\ln \left(x^{2}+y^{2}\right) \mathbf{i}+x y \mathbf{j}+\ln \left(y^{2}+z^{2}\right) \mathbf{k}\)

Evaluate the integral \(\int_{C}(2 x-y) d x+(x+3 y) d y\) along the path \(C\). \(C\) : elliptic path \(x=4 \sin t, y=3 \cos t\), from \((0,3)\) to \((4,0)\)