MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 18 02 Lecture 29 Tue Nov 20 2007 Recall statement of divergence theorem S F dS D div F dV Del operator x y z symbolic notation f f x f y f z gradient F x y z P Q R Px Qy Rz divergence Physical interpretation div F source rate ux generated per unit volume Imagine an incompressible uid ow i e a given mass occupies a xed volume with velocity F then D div F dV S F n dS ux through S is the net amount leaving D per unit time total amount of sources minus sinks in D Proof of divergence theorem To show S P Q R dS P Qy Rz dV we can D x separate into sum over components and just show S Rk dS R D z dV same for P and Q If the region D is vertically simple i e top and bottom surfaces are graphs z1 x y z z2 x y with x y in some region U of xy plane r h s is z2 x y Rz dV Rz dz dx dy R x y z2 x y R x y z1 x y dx dy D U z1 x y U z2 x z2 y 1 dx dy so Flux through top dS R x y z2 x y dx dy top Rk dS z1 x z1 y 1 dx dy so R x y z1 x y dx dy Bottom dS bottom Rk dS Sides sides are vertical n is horizontal F is vertical so ux 0 it into vertically simple pieces illustrated for a donut Then Given any region D decompose sum of pieces clear and D S sum of pieces since the internal boundaries cancel each other Di usion equation governs motion of smoke in immobile air dye in solution 2 u u 2u 2u 2 The equation is k u k 2 2 t x2 y z where u x y z t concentration of smoke we ll also introduce F ow of the smoke It s also the heat equation u temperature Equation uses two inputs 1 Physics F k u ow goes from highest to lowest concentration faster if concentration changes more abruptly 2 Flux and quantity of smoke are related if D bounded by closed surface S then S F n dS d dt D u dV ux out of D variation of total amount of smoke inside D This can be explained in terms of By di erentiation under integral sign the r h s is D t u dV integral as a sum of u x i yi zi t Vi and derivative of sum is sum of derivatives and by divergence theorem the l h s is D div F dV Dividing by volume of D we get 1 u 1 dV div F dV vol D vol D D t D Same average values over any region taking limit as D shrinks to a point get u t div F Combining we get the di usion equation u t div F kdiv u k 2 u 1
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