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P xy fa. I x i á y i ¥ dot p r o duct amounts t o b eing able to ca rry all geometric constructions fo rmulated in t erms o f a ngles, lengths and distances cos( x , y ) =!. @ A s P s y j ߌ f ⒮ K o A t L b Y X y X ŐV Ë@ ԏ N b ɂȂ Ă ֌W @ ւł B ϐ a @. F á dr = 0foreveryclosedC lying in the planeHintChoose a,b,cso that curl(F) lies in the plane 24 Let F =!.

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Curl(F(P)) á e forP & SrProvethat m(r) ' 1 # r2 %% Sr curl(F) á dS ' M(r) (b) Prove that curl(F(Q)) á e = lim r( 0 1 # r2 % Cr F á dr This proves that curl(F(Q)) á e is the circulation per unit area in the planeS 173 Divergence Theorem We have studied several ÒFundamental TheoremsÓ Each of these is a relation of the type Integral. For any infinite sets x and P(x) (P(x) is the power set of x), if x < y < P(x), then either y=x or y=P(x) In other words, for any set x and its power set (P(x)), there is no set whose cardinality is strictly in between x and P(x) In the following pages we will demonstrate that the Generalized Continuum Hypothesis is consistent with the ZermeloFrankael axiom system That is, we will prove. °Â S a ¦ ¯ À W¶ 15 ° ¯ À¢ À a Y ¦ u § ^ W S ¦ W v ¦ ¾ ^¢ À W¶ 15 À p ¦ W¶ °Â a ¾ ( À n ¦ À ).

ä f ^ á x x x f f _ á \ x x z x y ä F ä X X _. To begin with, if F = á M, N, P ñ, then the divergence of F( x,y,z) is defined to be div F = M x N y P z Notice that the divergence of a vector field is a function and not a vector field Similarly, the curl of F( x,y,z) is defined to be the new vector field curl F = áP yN z, M zP x, N xM y ñ Equivalently, the curl of a vector field can be defined formally by curl F = Ñ×F. I!!" (# i, # o) u s (x, y ) r e ik r d S FresnelKirchhof f diffraction integral Fraunhofer dif fraction in 1D !.

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Flatarheildi sundurleitni vigursviðs F=(P(x,y),Q(x,y)), þe div F, á fleti D er jafnt, ferilheildi af þverþætti vigursins F á jaðri D Vigurinn n=(dy, dx) er þverill jaðars D Aljöfnurnar = og ()= () eru notaðar til að setja fram lögmál Greens með eftirfarandi hætti = () þar sem dA er flatareining og C er jaðar D Lögmál Gauss er samsvarandi lögmál í þremur. That is, the 2form dF corresponds to the curl of the vector field F = á M, N, P ñ, so that Stoke's theorem in terms of differential forms is as follows Stoke's Theorem If F is a 1form with differentiable coefficients and S is an oriented surface, then ¶S F = ó õ ó õ S dF This general version of Stoke's theorem not only includes all the preceding statements of the fundamental. " x y z # $ = 0, (%) for su itab le p aram eters a, b,c, d,e,f O b serve th at th e m atrix in th e m id d le is n ecessarily sy m m etric E x p an d in g th e ab ove p ro d u ct ex p ression y ield s th e followi n g q u ad ratic eq u ation a áx 2 c áy 2 f áz2 2b áx y 2d áx z 2e áy z = 0 W e call su ch an ex p ression of th e form.

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