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prac4asol

# prac4asol - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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18.02 Practice Exam 4A - Solutions lb) We begin with fz = e5yz fy = exz + 2yz f, = e5y + y2 + 1 Integrating f, we get f = e"yz + g(y, z). Differentiating and comparing with the above equations we get Integrating g, we get g = y2z + h(z). Then g , = y2 + hl(z) so comparing with the second equation above we get hl(z) = 1. Hence h = z+C. Putting everything together we get lc) N, = 0 and P, = 1 hence the field is not conservative 2a) Consider the figure n' = \$(x, y, z) hence z = 2 cos r+6 and dS = 2' sin r+6 d4 dB hence we get 6" 4 sin 4 d4 dB = 1671 cos2 4 sin r\$d4 = - 1 6 ~ 2b) n' = *(x, y,O) hence 2. n' = 0. So the flux is 0. = 1 hence ,(R) = /L 1 d v = /K d i u ~ d v
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