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Unformatted text preview: Math 32B, winter 2006 Practice Midterm II Solutions Prepared by Jeffrey Hellrung 1. Find out which of the following two vector fields on R 2 has a potential and calculate the potential of that vector field. (a) F 1 ( x,y ) = ( x 2 + y 2 , 0) (b) F 2 ( x,y ) = ( x 2 + y 2 , 2 xy ) Solution Recall that a necessary condition for F ( x,y ) = ( M ( x,y ) ,N ( x,y )) to be conservative (i.e., have a potential function) is that N/x = M/y on the domain of definition of F . F 1 fails to satisfy this condition, while F 2 does in fact satisfy it, and since the partials of the components of F 2 exist on all of R 2 (which is simply connected), F 2 must have a potential function. Inspection or integration gives that parenleftbigg 1 3 x 3 + xy 2 + C parenrightbigg = F 2 ( x,y ) . 2. Consider the following vector field defined on R 2 : F ( x,y ) = parenleftBig 2 y + e x 2 , 2 x + sin y parenrightBig ....
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This note was uploaded on 06/03/2011 for the course PHYSICS 1B 1b taught by Professor Corbin during the Winter '10 term at UCLA.
- Winter '10