Tuesday, October 16 ∗∗ 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 theinstructor.(b) Consider the following two curves which start at A = (−2,0) and end at B = (2,0), namely the linesegment C1and upper semicircle C2.Add these curves to your sketch, and compute bothRC1F· dr andRC2F· dr. Check you answers withthe 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 , evaluateRCF · dr by using the fact that it is equal toRCF · Tds.(b) Find a parameterization of C and a formula for F. Use them to check your answer in (a) by com-putingRCF · dr explicitly.3. Consider the points A = (0, 0) and B = (π , −2). Suppose an object of mass m moves from A to B andexperiences the constant force F = −mgj, 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 theformula from the first week of class.(b) Now suppose the object follows half of an inverted cycloid C as shown below. Explicitly parameter-ize 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 checkyour answers for (a) and (b). Have you seen the quantity − f anywhere before? If so, what was itsname?4. If you get this far, work #52 from Section
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