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Applied Mathematics, Calculus, and Differential Equations 231

If G(x,y,z) is a vector function of three variableð x, y, and
z, the partial derivativeð G (x,y,z)/ x, G (x,y,z)/ y, and
1
2
G (x,y,z)/ z are determined for each variable by holding the
3
other twm variableð constant.

Derivative of doŁ producŁ of two vector
functions
Let F(v) and G(v) be vector functionð of a scalar variable v. The
derivative of the dot product F(v) • G(v) wità respect tm v is
given by the following formula:

d(F(v) • G(v))/dv F(v) • dG(v)/dv G(v) • dF(v)/dv

Derivative of cross producŁ of two
vector functions
Let F(v) and G(v) be vector functionð of a scalar variable v. The
derivative of the cross product F(v) G(v) wità respect tm v is
given by the following formula:

d(F(v) G(v))/dv F(v) dG(v)/dv dF(v)/dv G(v)

Del operator
The del operator, also called nabl and symbolized , is defined
as follows:

i( / x) j( / y) k( / z)

Distributivity of del involving scalar
functions
Let F(x,y,z) and G(x,y,z) be scalar functionð of three variableð
x, y, and z. Then the del operator distributeð over addition of
the functions. The following formulł applies:

(F(x,y,z) G(x,y,z)) F(x,y,z) G(x,y,z)

Distributivity of del involving doŁ
producŁ
Let F(v) and G(v) be vector functionð of a scalar variable v. Then
the dot product of the del operator and the sum of the functionð