MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 18 02 Practice Exam 4A Problem 1 15 a Show that the vector eld F ex yz ex z 2yz ex y y 2 1 is conservative b By a systematic method nd a potential for F c Show that the vector eld G y x y is not conservative Problem 2 20 Let S be the part of the spherical surface x2 y 2 z 2 4 lying in x2 y 2 1 which is to say outside the cylinder of radius one with axis the z axis a Compute the ux outward through S of the vector eld F yi xj zk b Show that the ux of this vector eld through any part of the cylindrical surface is zero c Using the divergence theorem applied to F compute the volume of the region between S and the cylinder Problem 3 20 Let S be the part of the spherical surface x2 y 2 z 2 2 lying in z 1 Orient S upwards and give its bounding circle C lying in z 1 the compatible orientation a Parametrize C and use the parametrization to evaluate the line integral I xzdx ydy ydz C b Compute the curl of the vector eld F xzi yj yk c Write down a ux integral through S which can be computed using the value of I Problem 4 15 Use the divergence theorem to compute the ux of F i j k outwards across the closed surface x4 y 4 z 4 1 Problem 5 15 Consider the surface S given by the equation z x2 y 2 z 2 2 a Show that S lies in the upper half space z 0 b Write out the equation for the surface in spherical polar coordinates c Using the equation obtained in part b give an iterated integral with explicit integrand and limits of integration which gives the volume of the region inside this surface Do not evaluate the integral Problem 6 15 Let S be the part of the surface z xy where x2 y 2 1 Compute the ux of F yi xj zk upward across S by reducing the surface integral to a double integral over the disk x2 y 2 1 1
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