finalReview - MA1C ANALYTIC FINAL REVIEW NOTES ALDEN...

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MA1C ANALYTIC FINAL REVIEW NOTES ALDEN WALKER ([email protected]) Contents 1. Notice About Citing These Notes 1 2. Info About the Final 1 3. Tips 1 4. A Whole Bunch of Examples 2 5. Line Integrals 7 6. Line Integrals With Respect to Arc Length 8 7. Gradient Fields 8 8. Green’s Theorem 9 9. Green’s Theorem for “Multiply Connected” Regions 10 10. Curl 11 11. Divergence 12 12. Change of Variables 12 13. Surface Area 15 14. Surface Integrals 15 15. Stokes’ Theorem 17 1. Notice About Citing These Notes I have tried to make these notes as accurate as possible. However, they haven’t been edited by anybody else, so I would encourage you to rely on a source which has been, namely your book, or at least the class notes. Therefore, please don’t cite these notes on the final . If you find something in here which is helpful, cite (and check the hypotheses of) the book’s version. This way, you don’t make mistakes that are my fault. 2. Info About the Final It will be available Friday, June 5 and it will be due Wednesday, June 10 at noon . Make sure to read the rules on the front of the exam. It is basically open book. You are allowed to use notes prepared by a TA (such as these). The final covers everything after (but not including) formal (delta-epsilon style) integration. You can find notes at the following places: http://www.its.caltech.edu/~awalker/09Ma1c/09Ma1c.html http://www.its.caltech.edu/~bsimanek/09Ma1c/09Ma1c.html http://www.its.caltech.edu/~dongping/Notes.html 3. Tips If you are asked to do an multiple integral, always check: (In two dimensions) Is the integrand of the form ∂Q ∂x - ∂P ∂y with a nice boundary? Use Green’s theorem (In any dimension) Is there a change of basis that will parameterize my region nicely? For a line integral, check: Is the integrand a gradient? If so, you are done. 1
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2 ALDEN WALKER ([email protected]) Is the curve the boundary of a nice surface or region? If so, apply Green’s theorem or Stokes’ theorem. (Sort of the same) Take the curl of the integrand. If it’s zero, you just have to exhibit a surface with that curve as a boundary to show that the integral is zero. To calculate area, always consider Green’s theorem! 4. A Whole Bunch of Examples 4.1. Line Integrals. Let F ( x, y ) = ( - y, x ). Let’s do the line integral of this vector field counterclockwise around the circle C of radius 1 in the plane. That is, c ( t ) = (cos t, sin t ). We find c 0 ( t ) = ( - sin t, cos t ) and thus that Z C F · = Z 2 π 0 F ( c ( t )) · c 0 ( t ) dt = Z 2 π 0 ( - sin t, cos t ) · ( - sin t, cos t ) dt = Z 2 π 0 dt = 2 π Recall the common other notation for line integrals, which doesn’t record the parameterization but only the function. For example this integral would have been written Z C - ydx + xdy 4.2. Line Integrals with Respect to Arc Length. Let’s integrate the function f ( x, y ) = x 2 around the circle C of radius 2 going counterclockwise. Here we must essentially do the same thing as with surface integrals, except here it’s easier; we let c ( t ) = (2 cos t, 2 sin t ) and compute: Z C fds = Z 2 π 0 f ( c ( t )) k c 0 ( t ) k dt = Z 2 π 0 4 cos 2 t k 2( - sin t, cos t ) k
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