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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Gradient and Divergence / Parallel Table Gradient v = grad u = �u Potential u(x, y ): v1 = Test on v : �u �u , v2 = �x �y Divergence div w = � · w = 0 Stream function s(x, y ): w1 = � w1 � w2 + =0 �x �y Solenoidal: zero source Zero ﬂux through loops: � � Test on w: w · n ds = �s �s , w2 = − �y �x � v2 � v1 − =0 �x �y Irrotational: zero vorticity Zero circulation around loops: � � v · t ds = v1 dx + v2 dy = 0 w1 dy − w2 dx = 0 Kirchhoﬀ’s Voltage Law Equip otentials u(x, y ) = constant v is p erpendicular to equip otentials Kirchhoﬀ’s Current Law Streamlines s(x, y ) = constant w is tangent to streamlines Green-Gauss Formula �� w · grad u dx dy = �� u(− div w) dx dy + � u w · n ds (grad)T = − div from integration by parts: (Au)T w = uT (AT w ) Connections when (v1 , v2 ) = (w1 , w2 ) 1. Equip otentials are p erpendicular to streamlines �� �� � �u � �u 2. Laplace’s equation div(grad u) = + = � · �u = 0 �x �x �y �y 3. Cauchy-Riemann equations 4. Laplace’s equation for s �u �s �u �s = and =− connecting u to s �x �y �y �x �2s �2s �2u �2u + 2 =− + =0 � x2 � y � x�y � y �x 5. Zero vorticity and zero source: Ideal p otential ﬂow 6. In two dimensions: u(x, y ) + is(x, y ) is a function f (x + iy ) ...
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