Advanced Calculus fi..

Chapter 3 Vector Differential Calculus 183can be interpreted as measuring angular motion of a fluid (see Problem 16 followingSection 3.6) and the conditioncurl v = 0 (3.25)for a velocity field v characterizes what are termed irrotationalJows. The analogousequationcurl E = 0 (3.26)for the electric force vector E holds when only electrostatic forces are present.The curl satisfies the basic laws:The proofs are left to the problems.curl (u + v) = curl u + curl v, (3.27)curl (f u) = f curl u + grad f x u. (3.28)As a result of the new definitions, we now have at our disposal the operations listedin Table 3.1.The theory of vector algebra, discussed in Section 1.2, concerns the properties -of the algebraic operations and their combinations. The theory of vector differentialcalculus concerns the theory of the differential operations (g) to (k) and theircombinations with each other and with the algebraic operations (a) to (f).The combinations of (g) and (h) with the algebraic operations are discussed inProblem 3 following Section 2.13.The combinations of (i), (j), and (k) with (a) and (c) are discussed in Sections 3.3,3.4, and 3.5. The results can be summarized in the one rule:operator on sum = sum of operators on terms. (3.29)AlgebraicOperationsDifferentialOperationsTABLE 3.1Where DiscussedOperation Symbols (Section No.)I(a) sum of scalars(b) product of scalars(c) sum of vectors(d) scalar times vector(e) scalar product(f) vector product((g) derivative of scalarI (h) derivative of vectorI(i)gradient of scalar(j) divergence of vector(k) curl of vectorVf = grad fV.v=divvV x v 5 curl v