Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.5 SURFACE INTEGRALS x z y R dA α S k dS Figure 11.6 A surfaceS(or part thereof) projected onto a regionRin the xy-plane;dSis ...

LINE, SURFACE AND VOLUME INTEGRALS where|∇f|and∂f/∂zare evaluated on the surfaceS. We can therefore express any surface integral ...

11.5 SURFACE INTEGRALS dS S z C a a a x y dA=dx dy Figure 11.7 The surface of the hemispherex^2 +y^2 +z^2 =a^2 ,z≥0. 11.5.2 Vect ...

LINE, SURFACE AND VOLUME INTEGRALS dr r O C Figure 11.8 The conical surface spanning the perimeterCand having its vertex at the ...

11.5 SURFACE INTEGRALS
Find the vector area of the surface of the hemispherex^2 +y^2 +z^2 =a^2 ,z≥ 0 ,by evaluating the line in ...

LINE, SURFACE AND VOLUME INTEGRALS In particular, when the surface is closed Ω = 0 ifOis outsideSand Ω = 4πifO is an interior po ...

11.6 VOLUME INTEGRALS V O S r dS Figure 11.9 A general volumeVcontaining the origin and bounded by the closed surfaceS. cannot b ...

LINE, SURFACE AND VOLUME INTEGRALS 11.7 Integral forms forgrad,divandcurl In the previous chapter we defined the vector operator ...

11.7 INTEGRAL FORMS FOR grad, div AND curl to the surface integral from these two faces is then [(φ+∆φ)−φ]∆y∆zi= ( φ+ ∂φ ∂x ∆x−φ ...

LINE, SURFACE AND VOLUME INTEGRALS R Q P S x y z T h 1 ∆u 1 eˆ 1 h 2 ∆u 2 eˆ 2 h 3 ∆u 3 eˆ 3 Figure 11.10 A general volume ∆Vin ...

11.8 DIVERGENCE THEOREM AND RELATED THEOREMS 11.8 Divergence theorem and related theorems The divergence theorem relates the tot ...

LINE, SURFACE AND VOLUME INTEGRALS R y C x dx dy dr nˆds Figure 11.11 A closed curveCin thexy-plane bounding a regionR.Vectors t ...

11.8 DIVERGENCE THEOREM AND RELATED THEOREMS vector fieldφ∇ψwe obtain ∮ S φ∇ψ·dS= ∫ V ∇·(φ∇ψ)dV = ∫ V [ φ∇^2 ψ+(∇φ)·(∇ψ) ] dV. ( ...

LINE, SURFACE AND VOLUME INTEGRALS 11.8.3 Physical applications of the divergence theorem The divergence theorem is useful in de ...

11.8 DIVERGENCE THEOREM AND RELATED THEOREMS Sincevhas a singularity at the origin it is not differentiable there, i.e.∇·vis not ...

LINE, SURFACE AND VOLUME INTEGRALS 11.9 Stokes’ theorem and related theorems Stokes’ theorem is the ‘curl analogue’ of the diver ...

11.9 STOKES’ THEOREM AND RELATED THEOREMS is the circlex^2 +y^2 =a^2 in thexy-plane. This is given by ∮ C a·dr= ∮ C (yi−xj+zk)·( ...

LINE, SURFACE AND VOLUME INTEGRALS Substituting this into (11.26) and takingcout of both integrals because it is constant, we fi ...

11.10 EXERCISES everywhere except on the axisρ=0,wherevhas a singularity. Therefore ∮ Cv·dr equals zero for any pathCthat does n ...

LINE, SURFACE AND VOLUME INTEGRALS in section 11.3, evaluate the two remaining line integrals and hence find the total area comm ...