91Ó°ÊÓ

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

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

Find div 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 \(\mathbf{F}(x, y, z)=\left(y^{2}+z^{2}\right)(x+y) \mathbf{i}+\left(z^{2}+x^{2}\right)(y+z) \mathbf{j}+\left(x^{2}+y^{2}\right)(z+x) \mathbf{k}\).

Find the divergence of \(\mathbf{F}\) for vector field \(\mathbf{F}(x, y, z)=f_{1}(y, z) \mathbf{i}+f_{2}(x, z) \mathbf{j}+f_{3}(x, y) \mathbf{k}\).

Use \(r=|\mathbf{r}|\) and \(\mathbf{r}=(x, y, z)\). Find the \(\operatorname{curl} \frac{\mathbf{r}}{r}\)

Use \(r=|\mathbf{r}|\) and \(\mathbf{r}=(x, y, z)\). Find the \(\operatorname{curl} \frac{\mathbf{r}}{r^{3}}\).

Use \(r=|\mathbf{r}|\) and \(\mathbf{r}=(x, y, z)\). Let \(\mathbf{F}(x, y)=\frac{-y \mathbf{i}+x \mathbf{j}}{x^{2}+y^{2}}, \quad\) where \(\mathbf{F}\) is defined on \(\\{(x, y) \in \mathbb{R} \mid(x, y) \neq(0,0)\\} .\) Find \(\operatorname{curl} \mathbf{F}\)

Use a computer algebra system to find the curl of the given vector fields. $$ \mathbf{F}(x, y, z)=\arctan \left(\frac{x}{y}\right) \mathbf{i}+\ln \sqrt{x^{2}+y^{2}} \mathbf{j}+\mathbf{k} $$