Vector Fields

Theory of Vector Fields Exercise 1 : Basic Operations on Vector Fields Given the vector fields: $\mathbf{F}_1 = x^2 \hat{\mathbf{z}}$ $\mathbf{F}_2 = x \hat{\mathbf{x}} + y \hat{\mathbf{y}} + z \hat{\mathbf{z}}$ $\mathbf{F}_3 = yz\hat{x} + zx\hat{y} + xy\hat{z}$ Calculate $\nabla \cdot \mathbf{F}_i$ and $\nabla \times \mathbf{F}_i$ for $i=1,2,3$. Which fields are conservative? Find scalar potentials where possible. Which fields are solenoidal? Find vector potentials where possible. Show that $\mathbf{F}_3$ can be expressed both as a gradient and a curl. Exercise 2 : Helmholtz Theorem Implications For a vector field $\mathbf{F}$ in 3D, prove the following implications: ...