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Unformatted text preview: Mathematics 20E - Spring 2009 - Homework 2 April 26, 2009 Please complete the following problems and turn them in to the homework drop box in the sixth floor of AP&M labelled “Math 20E / Ryan Szypowski” by Friday, April 24 at 5:30pm. Only a subset of the problems will be graded; leave some blank at your own risk. 1. Let f : Ê 2 → Ê be given by f ( x, y ) = x + y. Sketch the vector field given by F = ∇ f and show that c ( t ) = parenleftbigg x y parenrightbigg + t parenleftbigg 1 1 parenrightbigg is a description of the flow line of this field through the point ( x , y ). Solution : Here, we have F ( x, y ) = parenleftbigg 1 1 parenrightbigg . To see that the given curve is a description of the flow line through ( x , y ), first notice that c (0) = ( x , y ), so that it does indeed go through ( x , y ). Next, we must show that c ′ ( t ) = F ( c ( t )). In this case, we have c ′ ( t ) = parenleftbigg 1 1 parenrightbigg which matches up with F since it, too is constant. 1 2. Let g : Ê 2 → Ê be given by g ( x, y ) = xy. Sketch the vector field given by G = ∇ g and show that d ( t ) = 1 2 parenleftbigg ( x + y ) e t + ( x − y ) e − t ( x + y ) e t − ( x − y ) e − t parenrightbigg is a description of the flow line of this field through the point ( x , y ). Solution : Here, we have G ( x, y ) = parenleftbigg y x parenrightbigg ....
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- Vector Space, dy dx, Vector field, Gradient