Exercises

01

Linearity of curl operation

Show that $\nabla \times (\mathbf{F}+\mathbf{G})=\nabla \times \mathbf{F}+\nabla \times \mathbf{G}$.

02

Conservative vector fields have no curl

Show that if $\mathbf{F}=\nabla f$ for some scalar field $f(x,y,z)$, then $\nabla \times \mathbf{F}=0$ everywhere.

03

Verify Stokes’ Theorem

Verify Stokes’ Theorem by computing the boundary line integral for the hemisphere example above, with $\mathbf{F}=(z,x,1)$.