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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


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.

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© 1996-9 Eric W. Weisstein