19. Is there a vector field \( \mathbf{G} \) on \( \mathbb{R}^{3} \) such that \( \operatorname{curl} \mathbf{G}=\langle x \sin y, \cos y, z-x y\rangle \) ? Explain.
20. Is there a vector field \( \mathbf{G} \) on \( \mathbb{R}^{3} \) such that \( \operatorname{curl} \mathbf{G}=\left\langle x y z,-y^{2} z, y z^{2}\right\rangle \) ? Explain.
21. Show that any vector field of the form
\[
\mathbf{F}(x, y, z)=f(x) \mathbf{i}+g(y) \mathbf{j}+h(z) \mathbf{k}
\]
where \( f, g, h \) are differentiable functions, is irrotational.
22. Show that any vector field of the form
\[
\mathbf{F}(x, y, z)=f(y, z) \mathbf{i}+g(x, z) \mathbf{j}+h(x, y) \mathbf{k}
\]
