Advanced Calculus fi..

2&Advanced Calculus, Fifth EditionThe combinations of (i), (j), (k) with (b) and (d) are also discussed in Sections 3.3 to3.5. The results include the important case of scalar constant times scalar or vector.Here we have the general rule:operator on scalar constant factor = scalar factor times operator; (3.30)that is, a scalar constant can be factored out. Thus V(cf) = cVf, V . (cu) = cV . u,and so on. The rules (3.29) and (3.30) characterize what are called linear operators;thus grad, div, and curl are linear operators.If one considers the other possible combinations, one obtains a long list ofidentities, some of which will be considered here. The proofs are left to the problems.All derivatives occurring are to be assumed continuous.Curl of a gradient.Here one has the rule:curl grad f = 0. (3.31)This relation is suggested by the fact that curl grad f = V x (Vf ), that is, has theappearance of the vector product of collinear vectors. There is an important converse:if curl v = 0, then v = grad f for some f; (3.32)further assumptions are needed here, and the rule (3.32) must be used with caution.A proof and full discussion are given in Chapter 5. A vector field v such that curl v =0 is often termed irrotational.Divergence of a curl. Here one concludes thatr u..1div curl v = 0; (3.33)this relation is again suggested by a vector identity, for div curl v = V . (V x v),so that one has an expression resembling a scalar triple product of coplanar vectors(Section 1.4). Again there is a converse:. .if div u = 0, then u = curl v for some v; (3.34)as with (3.32) there are restrictions on the use of (3.34), and one is again referred toChapter 5. A vector field u such that div u = 0 is often termed solenoidal.Divergence of a vector product.Here one hasdiv (U x V) = v - curl u - u . curl v. (3.35)Divergence of a gradient.If one expands in terms of components, one finds thata2f a2f a2fdivgrad f = -+-+-. (3.36)ax2 ay* a22The expression on the right-hand side is known as the Laplacian of f and is alsodenoted by A f or by v2 f, since div grad f = V .(Vf ). A function f (having continuoussecond derivatives) such that div grad f = 0 in a domain is called harmonic3