Advanced Calculus fi..

340 Advanced Calculus, Fifth Editionwhen F is a gradient vector. For this reason, force fields that are gradientsare termed conservative force fields.b) Fluid dynamics. If u is the velocity vector and p is the density for a fluidmotion, then, as in Section 5.11, the equation of continuitya~- + div (pu) = 0at(5.116)*e t?\folds. This can also be written, by virtue of an identity of Section 3.4, asfollows:The first two terms are the Stokes total derivative of p:and describe the rate of change of p as one stays with a particular particleof fluid in the motion. For an incompressible fluid, Dpl Dt = 0, so that theequation of continuity becomesdiv u = 0, (5.117)and u is solenoidal.Another interpretation of div u and a new proof of the continuity equationare given in the problems that follow this section.The integral 1, u~ ds about a closed curve C has already been introducedas the circulation of the velocity field. If this is zero for every closed path C,then by Theorem I11 of Section 5.13,r .J.curl u = 0. (5.118)If (5.1 18) holds, the flow is called irrotational. This implies, by Theorem I11of Section 5.13, that the circulation is zero on every closed path, provided thatattention is restricted to a simply connected domain D. If this last assumptionis made, then u = grad 4 for some scalar 4, termed the velocity potential.If the flow is both irrotational and incompressible, then 4 must satisfythe equationthat is,div grad 4 = 0;rand 4 is harmonic in D.c) Electromagnetism. An electromagnetic field is described, in accordance withMaxwell's theory, by two vector fields E and H, where E is the electric forceand H is the magnetic field strength. Both E and H in general vary with time t.