Vector Derivative

A vector derivative is a derivative taken with respect to a vector field. Vector derivatives are extremely important in physics, where they arise throughout fluid mechanics, electricity and magnetism, elasticity, and many other areas of theoretical and applied physics.

The following table summarizes the names and notations for various vector derivatives.

Vector derivatives can be combined in different ways, producing sets of identities that are also very important in physics.

Vector derivative identities involving the curl include

del x(kA)=kdel xA
del x(fA)=f(del xA)+(del f)xA
del x(AxB)=(B·del )A-(A·del )B+A(del ·B)-B(del ·A)
del x((A)/f)=(f(del xA)+Ax(del f))/(f^2)
del x(A+B)=del xA+del xB.

In Cartesian coordinates

del xx=del xy=del xz=0
del xx^^=del xy^^=del xz^^=0.

In spherical coordinates,

del xr=0
del xr^^=0
del x[rf(r)]=0.

Vector derivative identities involving the divergence include

del ·(kA)=kdel ·A
del ·(fA)=f(del ·A)+(del f)·A
del ·(AxB)=B·(del xA)-A·(del xB)
del ·((A)/f)=(f(del ·A)-(del f)·A)/(f^2)
del ·(A+B)=del ·A+del ·B.

In Cartesian coordinates,

del ·x=1
del ·y=1
del ·z=1
del ·x^^=0
del ·y^^=0
del ·z^^=0.

In spherical coordinates,

del ·r=3
del ·r^^=2/r
del ·[rf(r)]=partial/(partialx)[xf(r)]+partial/(partialy)[yf(r)]+partial/(partialz)[zf(r)]

By symmetry,

del ·[rf(r)]=3f(r)+1/r(x^2+y^2+z^2)(df)/(dr)
del ·(r^^f(r))=2/rf(r)+(df)/(dr)
del ·(r^^r^n)=3r^(n-1)+(n-1)r^(n-1)

Vector derivative identities involving the gradient include

del (kf)=kdel f
del (fg)=fdel g+gdel f
del (A·B)=Ax(del xB)+Bx(del xA)+(A·del )B+(B·del )A
del (A·del f)=Ax(del xdel f)+del fx(del xA)+A·del (del f)+del f·del A
=del fx(del xA)+A·del (del f)+del f·del A
del (f/g)=(gdel f-fdel g)/(g^2)
del (f+g)=del f+del g
del (A·A)=2Ax(del xA)+2(A·del )A
(A·del )A=del (1/2A^2)-Ax(del xA).

Vector second derivative identities include

del ^2t=del ·(del t)
del ^2A=del (del ·A)-del x(del xA).

This very important second derivative is known as the Laplacian.

del x(del t)=0
del (del ·A)=del ^2A+del x(del xA)
del ·(del xA)=0
del x(del xA)=del (del ·A)-del ^2A
del x(del ^2A)=del x[del (del ·A)]-del x[del x(del xA)]
=-del x[del x(del xA)]
=-{del [del ·(del xA)]-del ^2(del xA)}
=del ^2(del xA)
del ^2(del ·A)=del ·[del (del ·A)]
=del ·[del ^2A+del x(del xA)]
=del ·(del ^2A)
del ^2[del x(del xA)]=del ^2[del (del ·A)-del ^2A]
=del ^2[del (del ·A)]-del ^4A
del x[del ^2(del xA)]=del ^2[del (del ·A)]-del ^4A
del ^4A=-del ^2[del x(del xA)]+del ^2[del (del ·A)]
=del x[del ^2(del xA)]-del ^2[del x(del xA)].

Identities involving combinations of vector derivatives include

Ax(del xA)=1/2del (A·A)-(A·del )A
del x(phidel phi)=0
(A·del )r^^=(A-r^^(A·r^^))/r
del f·A=del ·(fA)-f(del ·A)
f(del ·A)=del ·(fA)-A·(del f),

where (64) and (65) follow from divergence rule (2).

See also

Curl, Directional Derivative, Divergence, Gradient, Laplacian, Vector Integral, Vector Quadruple Product, Vector Triple Product

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Gradshteyn, I. S. and Ryzhik, I. M. "Vector Field Theorem." Ch. 10 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1081-1092, 2000.Morse, P. M. and Feshbach, H. "The Differential Operator del " and "Table of Useful Vector and Dyadic Equations." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 31-44, 50-54, and 114-115, 1953.

Referenced on Wolfram|Alpha

Vector Derivative

Cite this as:

Weisstein, Eric W. "Vector Derivative." From MathWorld--A Wolfram Web Resource.

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