28. Green's Theorem.— If in Gauss's equa tion we replace a by U V V where U and V are scalar functions we get ffUV' V. dS=fffV. (UV V)dzdyds= f (V U. V V + UV2V)dxdyis This is called Green's theorem and has many applications; e.g. (1) if a continuous scalar function vanishes on the boundary of S, and satisfies Laplace's equation V't V=0 within S, then it vanishes at all points within S. To prove this, let U=. V then by the conditions V)Idxdyds = 0, hence V V= Oat all points within, therefore V is a constant, which is zero by the boundary condition. (2) There is only one solution to the problem of finding a con tinuous function V which shall take assigned values on S, and shall satisfy Poisson's equa tion V = 4irp within S where p is a given continuous function of (x, y, 4. For, suppose that there are two such functions and V,, then the function U.= Vs — V, is zero on S and satisfies V =0 within, hence by (1), U-0, that is, Vim.. V2, at all points within S.
29. Symbolic Vector Product V X a Curl of a Vector.— From (17) we have a 0 kni) ax ay (aas_ Oa, \ 4. 4 Oa] k acts \ay as I \ az ax/ \ax ay/. It is shown in hydrodynamics that if a is the fluid velocity at (x, y, s), the vector just written represents double the molecular velocity of rotation at (x, v, s), or the vortex motion. This vector is called the curl of a even when a is not a velocity. When curl a=0, the three com ponents vanish, and these are the three con ditions that as, a,, as shall be the partial deriva tives of a function V, so that itx=2 V V, where V is the scalar potential of a. Thus the lines of a are normals to the family of level surfaces V='C, which divide the field into lamina which are regarded as infinitesimally thin, and are hence called lamella. The magnitude of a, that is dV/dn, varies inversely as the thick ness do of the lamella at (x, y, z), dV being constant; and any vector whose curl is zero is hence called a lamellar vector. Heaviside has given the name divergent to a vector whose curl is zero but whose divergence is not zero (as the velocity of a compressible fluid at a point where there is no vortex, or the electric force at a point where there is a charge) and the name circuital to 1 vector whose divergence is zero and whose curl is not zero (as the velocity of an incompressible fluid at a point where there is a vortex) ; a vector whose diver gence and curl are both zero is sometimes called a Laplacian vector; it is both lamellar and solenoidal, as in the case of an electric field at a point where there is no charge. Helm
holtz has shown that any vector field can be resolved into a divergent field and a circuital field. The curl of any vector is solenoidal, for v. (v x a) = 0 as in (17), or by differ entiation. The curl of curl a is given by (20), that is V x(V 30. Relation between Curl and Circula tion — Stokes' Theorem.— If AB is any arc in the field of a, the line integralf A a•ds is called the flow of a along arc AB. When AB Is a closed circuit, this integral is called the circulation of a around the circuit. Stokes' theorem asserts that this circulation equals the flux of curl a through any surface which caps the circuit; in symbols = 1 J p x a•dS.
Sketch of Kelvin's proof : If S be ruled up into elementary areas, the circulation around the boundary equals the algebraic sum of the culations about the elementary areas (by cancel-. ing flow on lines described in opposite tions); and similarly the circulation around any element dS at P equals the sum of its tions around the three projections dS,, dS,, dS, on planes drawn through P parallel to the ordtnate axes. Now if we take a parallelogram PARS with sides parallel to dx and dy, then the tangential components of a on the sides PQ, RS are a,, —tai — dy) , and similarly for ay the other two sides, hence the algebraic sum of flow along the four sides is (— dxdy.
ax aa, Summing this over dS,, and taking similar sums over dS, and 4U,, we get the circulation around dS equal to ( v s a)•01S; then integrating over the cap we get the circulation around the boundary.
As an application of Stokes' theorem, in fluid motion, the circulation around a circuit equals the sum of the strengths of all the vortex tubes enclosed by the circuit.
31. Circulation in Electromagnetic Cir cuit—Maxwell's Equations.— Faraday showed experimentally that the total electromotive ,force around a closed circuit (i.e., the line integral of the electric force E) equals the negative of the time-rate of change of the total magnetic induction through the circuit; but this total induction is the surface integral of the mag netic induction vector B (x, y, z) taken over any surface that caps the circuit; hence Faraday's result may be written fE .d s-= — d and comparison with Stokes' formula shows that v s dB — which is Maxwell's equation dt for magnetic current in an electric field. Again Ampere's experiments prove that the work done in carrying a unit magnetic pole around a closed circuit equals 4r times the electric flux through the circuit, that is H f •ds .dS.