## A Course of Mathematics for Engineers and Scientists. Volume by Brian H. Chirgwin By Brian H. Chirgwin

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1 0 . If A d e n o t e s t h e integral i φ r χ d r , t a k e n a r o u n d a closed s k e w c u r v e G, show that η ·r = η · A is t h e area b o u n d e d b y t h e p r o j e c t i o n of G o n t h e plane constant. 1 1 . S t a t e Stokes's t h e o r e m , a n d use it t o e v a l u a t e t h e integral J curlF t a k e n o v e r t h a t p a r t of t h e surface 2 x 2 + 4y 2 + z — 2z = 4 · dS §1:5 V E C T O R lying above the plane ζ = 0 , given that 3 F = (x 3 z — y) i — xyzj + */ k. 1 2 . F i n d t h e cartesian c o m p o n e n t s of curl suitable circle of radius r, 55 A N A L Y S T S a b y evaluating ψ a · ds around a centre a t (χ, y, z) a n d t a k i n g t h e l i m i t as r - > 0 of 1 3 .

10). T h e conditions of smoothness, continuity and absence of double points ensure that the transformation is continuous and reversible (see V o l . I I , § 5:10). Therefore W e now apply Green's theorem for a plane to the r. h. side of this equation and obtain 34 A C O U R S E O F M A T H E M A T I C S But etc. 39). 35) for a plane is the t w o dimensional form of b o t h the Divergence theorem and Stokes's Theorem. 11. note that the vector element of arc (dx 1} dx2) can be replaced b y an element of 'area' η as directed normal t o the curve, (Fig.

37a) we write the line- and surface-integrals A 36 C O U R S E O F M A T H E M A T I C S in the invariant forms (where ijk is summed over the cyclic permutations of 123). The element of arc ds represents the change of position when a point Ρ m o v e s an infinitesimal distance along the curve of integration. Therefore the element of arc is sometimes denoted b y dr and the lineintegral b y j a · dr. 3) the volume element dx1 dx2 is replaced b y dx3 d f j d £ 2 d | 3. The Jacobian in this case is I d e t L which is unity.