Vector Derivative

The basic types of derivatives operating on a Vector Field are the Curl $\nabla\times$, Divergence $\nabla\cdot$, and Gradient $\nabla$.

Vector derivative identities involving the Curl include

$\quad\nabla\times (k{\bf A}) = k\nabla\times{\bf A}$	(1)
$\quad\nabla\times (f{\bf A}) = f(\nabla\times{\bf A})+(\nabla f)\times{\bf A}$	(2)
$\quad\nabla\times ({\bf A}\times{\bf B}) = ({\bf B}\cdot\nabla){\bf A}-({\bf A}\cdot\nabla){\bf B}+{\bf A}(\nabla\cdot{\bf B})-{\bf B}(\nabla\cdot{\bf A})$	(3)
$\quad\nabla\times\left({{\bf A}\over f}\right)= {f(\nabla\times{\bf A})+{\bf A}\times (\nabla f)\over f^2}$	(4)
$\quad\nabla\times ({\bf A}+{\bf B}) =\nabla\times{\bf A}+\nabla\times{\bf B}.$	(5)

In Spherical Coordinates,

$\quad\nabla\times{\bf r}= {\bf0}$	(6)
$\quad\nabla\times\hat{{\bf r}} = {\bf0}$	(7)
$\quad\nabla\times [rf(r)] = f(r)(\nabla\times{\bf r})+[\nabla f(r)]\times{\bf r}= f(r)({\bf0}) + {df\over dr}\hat {\bf r}\times{\bf r}= {\bf0}+{\bf0} = {\bf0}.$	(8)

Vector derivative identities involving the Divergence include

$\quad\nabla\cdot(k{\bf A}) = k\nabla\cdot{\bf A}$	(9)
$\quad\nabla\cdot(f{\bf A}) = f(\nabla\cdot{\bf A})+(\nabla f)\cdot{\bf A}$	(10)
$\quad\nabla\cdot({\bf A}\times{\bf B}) = {\bf B}\cdot (\nabla\times{\bf A})-{\bf A}\cdot(\nabla\times{\bf B})$	(11)
$\quad\nabla\cdot\left({{\bf A}\over f}\right)= {f(\nabla\cdot{\bf A})-(\nabla f)\cdot{\bf A}\over f^2}$	(12)
$\quad\nabla\cdot({\bf A}+{\bf B}) =\nabla\cdot{\bf A}+\nabla\cdot{\bf B}$	(13)
$\quad\nabla({\bf u}{\bf v}) = {\bf u}\nabla\cdot {\bf v}+(\nabla{\bf u})\cdot{\bf v}.$	(14)

In Spherical Coordinates,

$\quad\nabla\cdot{\bf r}= 3$	(15)
$\quad\nabla\cdot\hat {\bf r} = {2\over r}$	(16)
$\quad\nabla\cdot [{\bf r}f(r)]= {\partial\over\partial x} [xf(r)]+{\partial\over\partial y} [yf(r)]+ {\partial\over\partial z} [zf(r)]$	(17)
$\quad {\partial\over\partial x} [xf(r)] = x {\partial f\over\partial x} + f = x {\partial f\over\partial r} {\partial r\over\partial x} + f$	(18)
$\quad {\partial r\over\partial x} = {\partial\over\partial x} (x^2+y^2+z^2)^{1/2} = x(x^2+y^2+z^2)^{-1/2} = {x\over r}$	(19)
$\quad {\partial\over\partial x} [xf(r)] = {x^2\over r} {df\over dr} + f.$	(20)

By symmetry,

$\quad \nabla\cdot[{\bf r}f(r)] = 3f(r) + {1\over r} (x^2+y^2+z^2){df\over dr} = 3f(r)+r {df\over dr}$	(21)
$\quad \nabla\cdot(\hat{\bf r}f(r)) = {3\over r} f(r) + {df\over dr}$	(22)
$\quad \nabla\cdot(\hat{\bf r}r^n) = 3r^{n-1}+(n-1)r^{n-1}= (n+2)r^{n-1}.$	(23)

Vector derivative identities involving the Gradient include

$\quad \nabla (kf) = k\nabla f$	(24)
$\quad \nabla (fg) = f\nabla g + g\nabla f$	(25)
$\quad \nabla ({\bf A}\cdot{\bf B}) = {\bf A}\times (\nabla\times{\bf B})+{\bf B... ...s (\nabla\times{\bf A})+({\bf A}\cdot\nabla){\bf B}+({\bf B}\cdot\nabla){\bf A}$	(26)
$\quad \nabla ({\bf A}\cdot\nabla f) = {\bf A}\times(\nabla\times\nabla f) + \na... ...s(\nabla\times{\bf A})+{\bf A}\cdot\nabla(\nabla f)+\nabla f\cdot \nabla{\bf A}$
$\qquad = \nabla f\times(\nabla\times{\bf A})+{\bf A}\cdot\nabla(\nabla f)+\nabla f\cdot \nabla{\bf A}$	(27)
$\quad \nabla \left({f\over g}\right)= {g\nabla f-f\nabla g\over g^2}$	(28)
$\quad \nabla (f+g) = \nabla f+\nabla g$	(29)
$\quad \nabla({\bf A}\cdot {\bf A})=2{\bf A}\times(\nabla\times{\bf A})+2({\bf A}\cdot\nabla){\bf A}$	(30)
$\quad ({\bf A}\cdot\nabla){\bf A}=\nabla({\textstyle{1\over 2}}{\bf A}^2)-{\bf A}\times(\nabla\times{\bf A}).$	(31)

Vector second derivative identities include

$\quad\nabla^2t\equiv \nabla\cdot(\nabla t) = {\partial^2t\over\partial x^2}+{\partial^2t\over\partial y^2}+{\partial^2t\over\partial z^2}$	(32)
$\quad\nabla^2{\bf A}=\nabla(\nabla\cdot {\bf A})-\nabla\times(\nabla\times{\bf A}).$	(33)

This very important second derivative is known as the Laplacian.

$\quad \nabla\times(\nabla t) = {\bf0}$	(34)
$\quad \nabla(\nabla\cdot{\bf A}) = \nabla^2{\bf A}+\nabla\times(\nabla\times{\bf A})$	(35)
$\quad \nabla\cdot(\nabla\times{\bf A}) = 0$	(36)
$\quad \nabla\times(\nabla\times{\bf A}) = \nabla (\nabla\cdot{\bf A})-\nabla^2{\bf A}$
$\quad \nabla\times(\nabla^2{\bf A})=\nabla\times[\nabla(\nabla\cdot{\bf A})]-\nabla\times[\nabla\times(\nabla\times{\bf A})]$
$\qquad = -\nabla\times[\nabla\times(\nabla\times{\bf A})]$
$\qquad = -\{\nabla[\nabla\cdot(\nabla\times{\bf A})]-\nabla^2(\nabla\times{\bf A})]\}$
$\qquad = \nabla^2(\nabla\times{\bf A})$	(37)
$\quad \nabla^2(\nabla\cdot{\bf A}) = \nabla\cdot[\nabla(\nabla\cdot{\bf A})]$
$\qquad = \nabla\cdot[\nabla^2{\bf A}+\nabla\times(\nabla\times{\bf A})] = \nabla\cdot(\nabla^2{\bf A})$	(38)
$\quad \nabla^2[\nabla\times(\nabla\times{\bf A})] = \nabla^2[\nabla(\nabla\cdot{\bf A})-\nabla^2{\bf A}]$
$\qquad = \nabla^2[\nabla(\nabla\cdot{\bf A})]-\nabla^4{\bf A}$	(39)
$\quad \nabla\times[\nabla^2(\nabla\times{\bf A})] = \nabla^2[\nabla(\nabla\cdot{\bf A})]-\nabla^4{\bf A}$	(40)
$\quad \nabla^4{\bf A}= -\nabla^2[\nabla\times(\nabla\times{\bf A})]+\nabla^2[\nabla(\nabla\cdot{\bf A})]$
$\qquad = \nabla\times[\nabla^2(\nabla\times{\bf A})]-\nabla^2[\nabla\times(\nabla\times{\bf A})].\quad$	(41)

Combination identities include

$\quad{\bf A}\times (\nabla{\bf A}) ={\textstyle{1\over 2}}\nabla ({\bf A}\cdot{\bf A})-({\bf A}\cdot\nabla){\bf A}$	(42)
$\quad\nabla\times (\phi\nabla\phi) =\phi\nabla\times(\nabla\phi)+(\nabla\phi)\times (\nabla\phi) ={\bf0}$	(43)
$\quad ({\bf A}\cdot\nabla)\hat {\bf r} = {{\bf A}-\hat {\bf r}({\bf A}\cdot\hat {\bf r})\over r}$	(44)
$\quad\nabla f\cdot{\bf A}=\nabla\cdot(f{\bf A})-f(\nabla\cdot{\bf A})$	(45)
$\quad f(\nabla\cdot{\bf A})=\nabla\cdot(f{\bf A})-{\bf A}\nabla f,$	(46)

where (45) and (46) follow from divergence rule (2).

See also Curl, Divergence, Gradient, Laplacian, Vector Integral, Vector Quadruple Product, Vector Triple Product

References

Gradshteyn, I. S. and Ryzhik, I. M. ``Vector Field Theorem.'' Ch. 10 in Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, pp. 1081-1092, 1980.

Morse, P. M. and Feshbach, H. ``Table of Useful Vector and Dyadic Equations.'' Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 50-54 and 114-115, 1953.