Section165IntegralsinCylindricalandSphericalCoordinates165节积分在圆柱和球面坐标

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1、Section16.5IntegralsinCylindricalandSphericalCoordinatesInthelastsectionwelookedatintegratingoveraregioninthexy-planegiveninpolarcoordinatesWecanextendpolarcoordinatesinto3spacebyaddinginthez-axisTheresultisCylindricalCoordinatesJustassomedoubleintegralsareeasiertodoinpolar

2、coordinates,sometripleintegralswillbeeasiertocomputeinCylindricalCoordinatesCylindricalCoordinatesWecanrepresentpointsin3spacewith.(r,θ,0).(r,θ,z)θryzxCylindricalCoordinatesWhattypeofsurfacesdowegetifr=cwherecisaconstant?Whattypeofsurfacesdowegetifθ=cwherecisaconstant?Whatty

3、peofsurfacesdowegetifz=cwherecisaconstant?ThesearesometimesreferredtoasthefundamentalsurfacesRegionsthataremosteasilydescribedincylindricalcoordinatesarethosewhoseboundariesarefundamentalsurfacesIntegrationinCylindricalCoordinatesRecallthatinpolarcoordinateswefoundthatdA=rdr

4、dθThiswasbasedonthefactthatΔA≈ΔrΔθNowinrectangularcoordinatesΔV≈ΔxΔyΔzandΔA≈ΔxΔysoΔV≈ΔAΔzPuttingthesetwolinestogetherwegetΔV≈rΔrΔθΔzJustaswithotheriteratedintegrals,ourorderofintegrationwilldependonourproblemLet’stakealookatthefirst2problemsontheworksheetSphericalCoordinates

5、Wecanrepresentpointsin3spaceusing.(r,θ,0)=(x,y,0).(r,θ,z)=(x,y,z)θryzxρSphericalCoordinatesWhattypeofsurfacesdowegetifρ=cwherecisaconstant?Whattypeofsurfacesdowegetifθ=cwherecisaconstant?Whattypeofsurfacesdowegetif=cwherecisaconstant?Thesearesometimesreferredtoasthefundament

6、alsurfacesRegionsthataremosteasilydescribedinsphericalcoordinatesarethosewhoseboundariesarefundamentalsurfacesIntegrationinSphericalCoordinatesWeneedtoexpressthevolumeelement,dV,insphericalcoordinatesLet’stakealookatwhatavolumeelementlookslikeinsphericalcoordinatesWecanseeWh

7、enweintegrateinsphericalcoordinates,wehaveLet’srevisitthesecondproblemontheworksheetNowlet’strysomeoftheotherproblems