Thursday March 6 Integrating vector fields 1 Consider the vector field F y 0 on R2 a Draw a sketch of F on the region where 2 x 2 and 2 y 2 Check you answer with the instructor b Consider the following two curves which start at A 2 0 and end at B 2 0 namely the line segment C 1 and upper semicircle C 2 R R Add these curves to your sketch and compute both C 1 F d r and C 2 F d r Check you answers with the instructor c Based on your answer in b could F be f for some f R2 R Explain why or why not 2 Consider the curve C and vector field F shown below a Calculate F T where here T is the unit tangent vector along C Without parameterizing C evaluate R R C F d r by using the fact that it is equal to C F T ds b Find a parameterization of C and a formula for F Use them to check your answer in a by comR puting C F d r explicitly 3 Consider the points A 0 0 and B 2 Suppose an object of mass m moves from A to B and experiences the constant force F mg j where g is the gravitational constant a If the object follows the straight line from A to B calculate the work W done by gravity using the formula from the first week of class b Now suppose the object follows half of an inverted cycloid C as shown below Explicitly parameterize C and use that to calculate the work done via a line integral c Find a function f R2 R so that f F Use the Fundamental Theorem of Line Integrals to check your answers for a and b Have you seen the quantity f anywhere before If so what was its name 4 If you get this far work 52 from Section 16 2
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