1. Show that:
(a) \nabla \times (\nabla \times \vec{a}) = \nabla(\nabla \cdot \vec{a}) - \nabla^2 \vec{a}
(b) \nabla \cdot \left[ \hat{n} f(r) \right] = \frac{\hat{n}}{r} f + \frac{df}{dr}, where \hat{n} = \frac{\vec{r}}{r} and f(r) is a function of r = |\vec{r}|
(c) \nabla \cdot (\vec{a} \times \vec{b}) = \vec{b} \cdot (\nabla \times \vec{a}) - \vec{a} \cdot (\nabla \times \vec{b})
(d) \nabla \cdot (f \vec{g}) = \nabla f \cdot \vec{g} + f \nabla \cdot \vec{g}
(e) \nabla \times (f \vec{g}) = \nabla f \times \vec{g} + f \nabla \times \vec{g}
2. Prove that:
(a) For \vec{a} \neq \vec{0}, \delta(a x) = \frac{1}{|a|} \delta(x)
(b) \delta[g(x)] = \sum_{m=1}^{M} \frac{1}{|g'(x_m)|} \delta(x - x_m), where x_m (m = 1, ..., M) are the zeros of the function g(x) and
for which g'(x_m) \neq 0
3. Using the divergence theorem, show that:
(a) \int_V \nabla \phi \, d^3 x = \int_S \phi \hat{n} \, da
(b) \int_V \nabla \times \vec{A} \, d^3 x = \int_S \hat{n} \times \vec{A} \, da
4. Using Stokes's theorem, show that:
(a) \int_S \hat{n} \times \nabla \phi \, da = \oint_C \phi \, d\vec{l}
