Example:Find the curl for the function 2ˆ-=ρkρFr, where kis a constant and ρis the magnitude of the position vector in xy-plane. 1.5.2. Second order spatial derivatives 126.96.36.199. The Laplacian The Laplacian of a scalar fis ∂∂∂∂+∂∂∂∂+∂∂∂∂=∇⋅∇≡∇33213221321132132121qfhhhqqfhhhqqfhhhqhhhff(1.42) In cylindrical coordinates it becomes 2222221111111zfffzfzfff∂∂+∂∂+∂∂∂∂=∂∂∂∂+∂∂∂∂+∂∂∂∂=∇φρρρρρρφρφρρρρ(1.43) In these two equations, replacing scalar fby vector Frresults in the Laplacian of vector Fr. Exercise:Express equation (1.42) in (i) rectangular and (ii) spherical coordinates. 188.8.131.52. The divergence of the curl of a vectorFrom equations (1.38) and (1.40), one gets
12()()()()()()()∂∂-∂∂∂∂+∂∂-∂∂∂∂+∂∂-∂∂∂∂=×∇⋅∇211122332113331123213222331321111qhFqhFqhhhqhFqhFqhhhqhFqhFqhhhFr(1.44) Exercise:Express equation (1.44) in (i) Cartesian, (ii) cylindrical and (ii) spherical coordinates. 1.6. Integral calculus 1.6.1. Line, surface and volume integrals The most important integrals in electromagnetism are the line (path) integral, surface integral (or flux) and volume integral. 184.108.40.206. Line integral The line integral of a vector function along a prescribed path between points aand bis ∫⋅balrrdFwhere we take the dot product of the vector Frand the elemental length lrdat each point on the path, with lrdbeing tangential to the path. If the path is closed then the integral is written as ∫⋅cldlrrFIn general these integrals depend on the path taken. There is a special class of vectors for which the line integral does not depend on the path connecting the two points aand b. A force having this property is known as a conservativequantity, and its line integral, which is the work done by it, is independent of he path taken. Example 1: Find the line integral of a vector 2ˆ-=crrvr, with ca constant, from r= r1to r=r2. Soln.: In spherical coordinates the displacement is φθdˆdˆdˆdφθr++=rrlr. Then -==⋅+⋅+⋅=⋅∫∫∫∫∫----21222211ddsinˆˆdˆˆdˆˆd212121212rrcrcrrcrrcrrcrrrrrφφθθφθθφrθrrrvll1lrrwhere the rules for dot products of unit vectors have been applied.