ELEC3002-Vector%20Calculus%204

ELEC3002-Vector%20Calculus%204 - ELEC3002/ELEC7005 Vector...

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ELEC3002/ELEC7005 Vector Calculus 4 Topics in Integration (Glyn James, Ch 7.4) The relationship between integrals: Gauss’s divergence theorem (7.4.11) Stokes’ theorem (7.4.12) Applications solution of operator equations.

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2 Gauss’s divergence theorem Gauss’s theorem relates surface and volume integrals. Consider closed volume V with surface area S ( V = int S ) d S = n dS, where n is the outward normal of the surface S enclosing the volume V the proof of this important theorem is in the book
3 Gauss’s divergence theorem (an example) ∫∫∫ = ∫∫ = S V S dv S d D int ρ G G ∫∫∫ = ∫∫ = S V S dv D S d D int div G G G We use Gauss’s theorem to convert surface to volume integral and… We obtained ME III in differential form – a PDE = ∫∫∫ = ∫∫∫ D dv dv D V V G G div div Gauss’s law or Maxwell’s third equation in integral form

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4 Stokes’ Theorem Stokes’s theorem is the generalisation of Green’s theorem and relates surface integrals to line integrals in three dimensional space Flux of vector field curl F ( r ) through any surface S spanned over the same contour C , is the same. the proof of this important theorem is in the book
5 Stokes’ Theorem (an example) ∫∫ = S C S d t B d E G G A G G We use Stokes’ theorem to convert line to surface integral and… We obtained ME I in differential form – a PDE Faraday’s law or Maxwell’s first equation in integral form ∫∫ = S C S d E d E G G A G G curl ∫∫ = ∫∫ S S S d t B S d E G G G G curl t B E = G G curl

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6 Examples – Integration and Theorems See problems Q1 – Q7
7

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ELEC3002-Vector%20Calculus%204 - ELEC3002/ELEC7005 Vector...

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