Mathematics of Physics and Engineering

Integration and Differentiation 139

differential operator

and the symbolic vector representations of the divergence and curl:

divF = V-F, curlF = V x F. (3.1.29)

We can also interpret grad/ as a multiplication of the vector V by the
scalar /. Then it is easy to remember the following identities:

div(curl F) = 0 : V • (V x F) = 0; curl(grad/) = 0 : V x (V/) = 0;

V-(/F) = V/-F + /V-F, Vx (/F) = V/xF + /VxF;

V • (F x G) = G • (V x F) - F • (V x G); V • (V/ x Vg) = 0.

(3.1.30)

Not all formulas are nice and easy, though: the expressions for V(F • G)

and curl(F x G) are rather complicated.

EXERCISE 3.1.19.A Recall that, for two vector fields F, G, we define the

vector field (F • V)G; see (3.1.11) on page 126. Show that

{F • V)G = | (curl(F x G) + grad(F • G) - G divF + F divG

-GxcurlF-F x curlGV

Use the result to derive the following formulas:

curl(F x G) = F divG - G divF + (G • V)F - (F • V)G,

grad(F -G) = Fx curlG + G x curlF + (F • V)G + (G • V)F.

For a scalar field /, div(grad /) is called the Laplacian of / and denoted

by V^2 /; an alternative notation A/ is also used.

EXERCISE 3.1.20.c Verify that, in cartesian coordinates,

V^2 f = fxx + fyy + fzz. (3.1.31)

For a vector field F, we define

V^2 F = grad(div F) - curl(curl F). (3.1.32)

EXERCISE 3.1.21. (a)A Verify the equalities in (3.1.30) Hint: choose

cartesian coordinates so that F = F\i. (b) Verify that if F — F\ t+F2 3+F3 k,