ABSTRACT
Let f and u = u e1 + v e2 + w e3 be arbitrary sufficiently smooth scalar and vector functions, and let e1, e2, and e3 be the unit coordinate vectors corresponding to the Cartesian coordinates x, y, and z. Then, by definition, we have
∇f = fxe1 + fye2 + fze3, div u = ∇ · u = ux + vy + wz, curl u = ∇× u = (wy − vz)e1 + (uz − wx)e2 + (vx − uy)e3, ∆f = fxx + fyy + fzz,
where the subscripts x, y, and z stand for the derivatives. The following differential relations hold:
curl∇f = 0, div curl u = 0, div∇f = ∆f, curl curl u = ∇ div u−∆u, ∆(xf) = x∆f + 2∇f, x = (x, y, z), curl[∆(xf)] = curl(x∆f), ∆(x · u) = x · (∆u) + 2 div u,
which are often used in what follows in Sections 12.5-12.19.
