prac4a

# prac4a - value of I . Problem 4. (15) Use the divergence...

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18.02 Practice Exam 4A Problem 1. (15) a) Show that the vector ﬁeld x x x F = e yz, e z + 2 yz, e y + y 2 + 1 ± is conservative. b) By a systematic method, ﬁnd a potential for F . c) Show that the vector ﬁeld G = y, x, y ± is not conservative. Problem 2. (20) Let S be the part of the spherical surface x 2 + y 2 + z 2 = 4, lying in x 2 + y 2 > 1, which is to say outside the cylinder of radius one with axis the z -axis. a) Compute the ﬂux outward through S of the vector ﬁeld F = y i x j + z k . b) Show that the ﬂux of this vector ﬁeld 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 x 2 + 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 ﬁeld F = xz i + y j + y k . c) Write down a ﬂux integral through S which can be computed using the
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Unformatted text preview: value of I . Problem 4. (15) Use the divergence theorem to compute the ﬂux of F = i + j + k outwards across the closed surface x 4 + y 4 + z 4 = 1. Problem 5. (15) Consider the surface S given by the equation z = ( x 2 + y 2 + z 2 ) 2 . a) Show that S lies in the upper half space ( z ² 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 x 2 + y 2 < 1. Compute the ﬂux of F = y i + x j + z k upward across S by reducing the surface integral to a double integral over the disk x 2 + y 2 < 1. 1...
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## This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.

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prac4a - value of I . Problem 4. (15) Use the divergence...

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