review2 - Section VC.05 Here are the concepts you should...

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Section VC.05 Here are the concepts you should know. You should be able to do problems but also be able to answer true/false or short answer questions about these concepts. Flow across a curve: R thigh tlow ~ F · normal dt , where the normal vector is ( y 0 ( t ) , - x 0 ( t )). Flow along a curve: R thigh tlow ~ F · tangent dt , where the tangent vector is ( x 0 ( t ) , y 0 ( t )). In a gradient field, we have path independence. This means the value of a path integral in a gradient field only depends on the starting and ending position, not the path between them. If ~ F = f , then R thigh tlow ~ F · tangent dt = f ( end ) - f ( start ). In a gradient field, the flow along any closed curve is zero. We can see this from the above formula since then the start and end are the same so the right hand side is zero. Look at T5 at the end of the tutorials, it has a summary of the main ideas. Look at literacy problems L1-L5, L7-L9, L15. Here are some more problems (also look at the quiz). Problem 1 Let ~ F = (3 x 2 + y, cos( xy )) and ~ G = (cos( xy ) , - 3 x 2 + y ) be two vector fields and ( x ( t ) = t + cos( t ) 0 t π y ( t ) = sin( t ) be some curve. Fill in the blanks with “along” or “across”: R π 0 (3 x ( t ) 2 + y ( t ))(1 - sin( t ))+(cos( x ( t ) y ( t )))(cos( t )) dt measures the flow the vector field ~ F and the flow the vector field ~ G . R π 0 (3 x ( t ) 2 + y ( t ))(cos( t ))+(cos( x ( t ) y ( t )))(1 - sin( t )) dt measures the flow the vector field ~ F and the flow the vector field ~ G . Problem 2 Let ( x ( t ) = t 2 - t 0 t 2 y ( t ) = 5 t and ~ F = (3 xy, - 2 x ). Calculate the flow along the curve. Is ~ F a gradient field?

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