Calc & Analyt Geom IV (HONORS)
Calc & Analyt Geom IV (HONORS) MATH 2443
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This 8 page Class Notes was uploaded by Mason Larson DDS on Monday October 26, 2015. The Class Notes belongs to MATH 2443 at University of Oklahoma taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/229299/math-2443-university-of-oklahoma in Mathematics (M) at University of Oklahoma.
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Date Created: 10/26/15
Math 2443 Calculus 55 Analytic Geometry 4 Murphy Fall 2004 1 169 Consider the parametric equations z r cos0 y r sin0 z 2 Calculate the determinant Bu 31 ill Bu 31 2 169 Consider the parametric equations z r cos0 sin y r sinl9 sin z z cos Calculate the determinant 81 3a 31 aw 3a 31 aw g g g 3a 31 3w 3 169 Use a transformation to nd the area of the ellipse Hint Use the transformation z au7 y by First verify that this transformation yields the desired ellipse 4 03 5 00 p H 0 171 Sketch the vector eld zf y 2 171 Find the gradient of the function ay W967 WWW x2 ys Sketch the vector eld Parametrize the curve y x2 Represent the curve as a vector function And draw the curve using the vector function representation 171 Now go backwards start with the vector eld ux 7 2963 3W from the example above Find the function ay for which this vector eld is the gradient This means that you need a function f so that 73f 2x and 73f 3y2 Note 31 By how many potential functions are there 171 Consider the vector eld 1396711 42 296 Try to nd a potential function for 172 From the de nition of line integral7 calculate the work done by the force eld Fwy xf y 2jin moving an object along one arch of the cycloid 770 t 7 sinti 17 costjfor 0 S t S 27139 173 Use the FTLl to calculate the work done by the force eld xi y 2 j in moving an object along one arch of the cycloid t t 7 sin t 3 1 7 cos It for 0 S t S 27139 How would the work done by the vector eld in moving an object from 07 0 to 2W70 change if the path was a curve other than the cycloid Why 173 Consider the vector eld 6xy 7 yg 4y 3x2 7 Bxyz a Without nding a potential function7 show that the eld is conservative b Now that we believe a potential function exists7 call it ay so that we are all consistent with our notation for the moment 7 to minimize confusion and nd it c Convince me that your ay is a potential function for d Use your work to help you evaluate the line integral of F along y 2 71 lt x S 2 12 What is the circulation of any conservative vector eld around any closed path Look up circulation77 on a previous handout if you need to 13 Calculus Problems for a New Century Xii19 p 199 14 Calculus Problems for a New Century Xii110 p 200 15 on a previous handout if you need to b Give an example of a connected region which is not simply connected 16 The vector eld a Give an example of a region which is neither open nor closed Look up these terms 15 137y zj issketchedtotheright lil a Use a graphing calculator to 1 X I X X 39 draw the curve C1 given by I i A x X X 77t t2j7 Ogtgl A 4 lt w x 05 A 4 V lt K kl Transfer the graph to the V x x g t k k k K above picture remember to r A k x r k i t f include the orientation of i y i A A A i i l the curvel O 39 V l V 4 l A A A 4 4 i l f b Use the dot product form to f i i A 7 4 4 l l I 4 f f f calculate the line integral of f F over C1 39 39 39 39 4 A A l f f 05 A n v r 1 I I l f f f f 0 5 0 0 5 1 15 c Draw the curve Cg given by Ft tsj 0 lt t lt 1 Transfer the graph to the picture above d Use the dot product form to calculate the line integral of P over Cg e ls F a conservative vector eld Why or why not 17 Let C Cg 7 C1 where C1 and Cg are given as in the problem above Note that C is piecewise smooth7 simple7 closed7 and positively oriented why7 Note that F has continuous partial derivatives at every point in the region enclosed by C why7 Use Green7s Theorem to evaluate the integral 0 P dF 18 What does Green7s Theorem tell us about conservative vector elds 19 Use Green7s Theorem to compute C F d where 2x 7 y 4i 5y 3x 7 6 and C is the triangle with vertices 00 30 and 32 traversed counterclockwise Could you have used the Fundamental Theorem for Line Integrals instead Why 20 Use the Greens Theorem to compute C F d where x yi m 7 y and C is the triangle with vertices 00 30 and 32 traversed counterclockwise Could you have used the Fundamental Theorem for Line lntegrals instead Why 21 Use the Fundamental Theorem for Line lntegrals to compute CF d where x yi x 7 y and C is the parabola y x2 1 traversed from 01 to 12 Could you have used Green7s Theorem instead Why 2 E0 Compute C F d where 2 7 yi y2 s and C is the parabola y 2 1 traversed from 01 to 12 Could you have used the Fundamental Theorem for Line lntegrals Why Could you have used Green7s Theorem Why rr1 xx A 39 pa 3 26A 3 2 5amp3 in mi 2 b e Ea 3 em 29 SEE mania 3 62562 a 6 56 3 385 a E SEE 88 g a E E mg 585 E 32 E a a Ea 22 is E a g 52 33 a 5 2 a a 28 3 be 2 a 2 Ea a 5 a man E E a 2 E a 2 a a a a as E E g 1 13 E a a 28 3 E E a E E wimwizm 13 2 a 3 Hugo 25 m w z 26 2 2 2 5 00 3 176 Find the tangent plane to the surface Wu 1 1125 112i sinuv at the point 12sin the point is given in Cartesian coordinates 177 Use the ux formula to calculate the ux of the given vector eld through a rectangular net 1357 177 Evaluate where Fyz xzyi 7 3xy2j4y313 and Sis z z2y2 7 9 S below the rectangle 0 S x S 27 0 S y S 1 with downward orientation 179 Find the ux of the vector eld Fthrough out of the sphere with radius R centered at the origin in te uxo atroug were az7y72 xiHL x2j yz an is 179F dh th hSh F 3 22 32k dS bounded by z 4 7 x2 7 y2 and the zy plane7 and is oriented outward 178 Calculate ffcurl dg where zyz yz 25 2j x2 y2 I and S is the part S of z 2 y2 that lies inside the cylinder x2 y2 17 oriented upward
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