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.

symbol	vector derivative
	gradient
	Laplacian or vector Laplacian
or	directional derivative
	divergence
	curl
	convective derivative

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

			(1)
			(2)
			(3)
			(4)
			(5)

In Cartesian coordinates

			(6)
			(7)

In spherical coordinates,

			(8)
			(9)
			(10)

Vector derivative identities involving the divergence include

			(11)
			(12)
			(13)
			(14)
			(15)

In Cartesian coordinates,

			(16)
			(17)
			(18)
			(19)
			(20)
			(21)

In spherical coordinates,

			(22)
			(23)
			(24)
			(25)
			(26)
			(27)

By symmetry,

			(28)
			(29)
			(30)
			(31)
			(32)

Vector derivative identities involving the gradient include

			(33)
			(34)
			(35)
			(36)
			(37)
			(38)
			(39)
			(40)
			(41)

Vector second derivative identities include

			(42)
			(43)
			(44)

This very important second derivative is known as the Laplacian.

			(45)
			(46)
			(47)
			(48)
			(49)
			(50)
			(51)
			(52)
			(53)
			(54)
			(55)
			(56)
			(57)
			(58)
			(59)
			(60)

Identities involving combinations of vector derivatives include

			(61)
			(62)
			(63)
			(64)
			(65)

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

REFERENCES:

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 " 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.