EE2011 Lecture 1 - Review of Vector Calculus

System 1 1a az a rar rr r z spherical coordinate

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Unformatted text preview: operator in Cylindrical and Spherical Coordinate Systems – Cylindrical Coordinate System 1 1A Az A rAr rr r z – Spherical Coordinate System A 1 R 2 AR R2 R Prof Joshua Le-Wei Li, EM Research Group Prof 1 R sin 24 A sin 1 R sin A EE2011: Engineering Electromagnetics Electromagnetics Divergence of a Vector Field • Example 2 Prof Joshua Le-Wei Li, EM Research Group Prof 25 EE2011: Engineering Electromagnetics Electromagnetics Divergence of a Vector Field • Divergence Theorem Edv V E ds S • Solenoidal, if ·E = 0 • Distributive, because of ·(E1+E2) = ·E1+ · E2 • For a constant E, the entering and leaving fluxes are the same and the divergence is zero, the field is thus divergenceless. divergenceless Prof Joshua Le-Wei Li, EM Research Group Prof 26 EE2011: Engineering Electromagnetics Electromagnetics Divergence of a Vector Field • Example 3a Prof Joshua Le-Wei Li, EM Research Group Prof 27 EE2011: Engineering Electromagnetics Electromagnetics Divergence of a Vector Field • Example 3b Prof Joshua Le-Wei Li, EM Research Group Prof 28 EE2011: Engineering Electromagnetics Electromagnetics Curl of a Vector Field • Basic Concept of the Curl B dl Circulation C – Circulation of a uniform field is zero, for instance, in the case of (a) – For case (b), we have 0I Bˆ 2r – So we have B dl Circulation C I ˆ 2r 0 C Prof Joshua Le-Wei Li, EM Research Group Prof 29 ˆ rd 0 I EE2011: Engineering Electromagnetics Electromagnetics Curl of a Vector Field • Basic Concept of the Curl B B curl B 1 ˆ lim n B dl s0 s C ˆ yB y ˆ zB z ˆ x B ˆ xBx ˆ y max ˆ z x Bx ˆ Bz x y By y z Bz By z Prof Josh...
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