By Robert Steel Edgar (auth.)

"Electromagnetic thought is a unusual topic. The peculiarity is living no longer loads within the stratification - superposed layers of electrostatics. magnetostatics. regular currents and time-varying fields - as within the failure that has attended all makes an attempt to weld those layers right into a logical entire. the bottom layer. electrostatics. defines yes strategies. akin to E. D, ~, in a manner that's commonly passable just for the static case. but the try is made to strength those specialized definitions into the better strata, with advert hoc alterations while worthy. the coed, in searching through his textual content books on electromagnetics, can locate normal definitions in simple terms with trouble. if in any respect; or even the main complex treatises fail to provide a carefully logical improvement of the subject". 1 So wrote Moon and Spencer a few 30 years in the past; and their feedback is still pertinent at the present time. 2 extra lately. a senior physicist of the nationwide Bureau of criteria has expressed his hindrance in comparable phrases: "A logically constant set of definitions of the electromagnetic box amounts is intensely tricky to discover within the literature. so much textual content books both sidestep the matter or current definitions which are appropriate in basic terms to big cases".

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**Sample text**

S)p" Since this holds independently of the magnitude of As we may let As+O. s 0 1\ But this remains true for all orientations of PQ (ie of s) so that which is the required result. THE DIFFERENTIAL AND INTEGRAL CALCULUS OF VECTORS 45 EXERCISES 1-30. 10-1) for a surface with a single internal boundary by means of an analysis similar to that of (a) Sec. 8 (b) Sec. 9. 1-31. By proceeding from first principles as in Sec. 8 derive the form of Stokes's theorem appropriate to a vector point function which is defined within a plane region and has no component normal to the plane (planar field).

Show from first principles that, so long as U is a function of V, W is single-valued in R and grad W - U grad V. 8 Stokes's Theorem Curl of a Vector Field An open two-sided surface defines a boundary curve around which a closed line integral may be taken. The direction of integration is conventionally chosen as anti-clockwise in relation to an observer who sees the positive side of the surface; in other words, the positive sense of the normal at the surface and positive motion around the closed curve bear a right-handed screw relationship.

This will be designated ~S. The projection of ~S on the xy (or k) coordinate plane. viz is ~Sz' positive for the particular orientation of S adopted in the figure. because the positive normal at makes an angle of less than 90 0 with k. ~S ~Sz is numerically equal to the product of ~x and the projection of as on the xy plane. If ~y is the increment of the y coordinate on passing from S to a then it is clear that ~y~x - -~SZ. By projecting onto the xz plane we may show in a similar manner that ~z~x - +~S, y where ~z is the increment of z on passing from S to a.