91Ó°ÊÓ

For the following exercises, find the curl of \(\mathbf{F}\) at the given point. $$\mathbf{F}(x, y, z)=\mathbf{i}+\mathbf{j}+\mathbf{k} \text { at }(2,-1,3)$$

For the following exercises, find the curl of \(\mathbf{F}\) at the given point. $$\mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+x \mathbf{k} \text { at }(1,2,3)$$

For the following exercises, find the curl of \(\mathbf{F}\) at the given point. $$\mathbf{F}(x, y, z)=e^{-x y} \mathbf{i}+e^{x z} \mathbf{j}+e^{y z} \mathbf{k}\( at \)(3,2,0)$$

For the following exercises, find the curl of \(\mathbf{F}\) at the given point. $$\mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k}\( at \)(1,2,1)$$

For the following exercises, find the curl of \(\mathbf{F}\) at the given point. $$\mathbf{F}(x, y, z)=e^{x} \sin y \mathbf{i}-e^{x} \cos y \mathbf{j}\( at \)(0,0,3)$$

Let \(\mathbf{F}(x, y, z)=\left(3 x^{2} y+a z\right) \mathbf{i}+x^{3} \mathbf{j}+\left(3 x+3 z^{2}\right) \mathbf{k}\) For what value of \(a\) is \(\mathbf{F}\) conservative?

Given vector field \(\mathbf{F}(x, y)=\frac{1}{x^{2}+y^{2}}(-y, x)\) on domain \(D=\frac{\mathbb{R}^{2}}{\\{(0,0)\\}}=\left\\{(x, y) \in \mathbb{R}^{2} |(x, y) \neq(0,0)\right\\}\) is \(\mathbf{F}\) conservative?

Given vector field \(\quad \mathbf{F}(x, y)=\frac{1}{x^{2}+y^{2}}(x, y)\) on domain \(D=\frac{\mathbb{R}^{2}}{\\{(0,0)\\}},\) is \(\mathbf{F}\) conservative?

Compute divergence \(\mathbf{F}=(\sinh x) \mathbf{i}+(\cosh y) \mathbf{j}-x y z \mathbf{k}\)

Compute curl \(\mathbf{F}=(\sinh x) \mathbf{i}+(\cosh y) \mathbf{j}-x y z \mathbf{k}\)