91Ó°ÊÓ

For the following exercises, find the divergence of \(\mathbf{F}\) $$\mathbf{F}(x, y)=\frac{x}{\sqrt{x^{2}+y^{2}}} \mathbf{i}+\frac{y}{\sqrt{x^{2}+y^{2}}} \mathbf{j}$$

For the following exercises, find the divergence of \(\mathbf{F}\) $$\mathbf{F}(x, y)=x \mathbf{i}-y \mathbf{j}$$

For the following exercises, find the divergence of \(\mathbf{F}\) $$\mathbf{F}(x, y, z)=a x \mathbf{i}+b y \mathbf{j}+c \mathbf{k}$$ for constants a, b, c

For the following exercises, find the divergence of \(\mathbf{F}\) $$\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+x z \mathbf{k}$$

For the following exercises, determine whether each of the given scalar functions is harmonic. $$u(x, y, z)=e^{-x}(\cos y-\sin y)$$

For the following exercises, determine whether each of the given scalar functions is harmonic. $$w(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2}$$

If \(\mathbf{F}(x, y, z)=2 \mathbf{i}+2 x \mathbf{j}+3 y \mathbf{k}\) \(\mathbf{G}(x, y, z)=x \mathbf{i}-y \mathbf{j}+z \mathbf{k}\) find \(\operatorname{curl}(\mathbf{F} \times \mathbf{G})\)

If \(\mathbf{F}(x, y, z)=2 \mathbf{i}+2 x \mathbf{j}+3 y \mathbf{k}\) \(\mathbf{G}(x, y, z)=x \mathbf{i}-y \mathbf{j}+z \mathbf{k},\) find \(\operatorname{div}(\mathbf{F} \times \mathbf{G})\)

Find \(\operatorname{div} \mathbf{F}\) given that \(\mathbf{F}=\nabla f\) where \(f(x, y, z)=x y^{3} z^{2}\)

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