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20E-FinalSolutions

# 20E-FinalSolutions - Final Examination Math 20E — Vector...

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Unformatted text preview: Final Examination Math 20E — Vector Calculus Instructor — J. Verstraete Allotted time — 3 hours Answers are to be written clearly and legibly Calculators are allowed State clearly any theorems used without proof Total 50 points Student Name 30 LUTWNS. Student ID Number Question 1. Q (2 (a) Define limxaa ﬁx) 2 L when f : R” —+ R. (b) Prove that the following limit does not exist: E) 2 lim (5v — y) . (140400) x2 + 312 Gr «2 P‘)‘; l) @ (a ] Boorwoﬁt O (bl Pul‘ 11:5 3% L 1 Question 2. Find the maximum value of the function f (as, y) = a: + y subject to the con— 4 straint 11:2 -l— 3/2 = 2. MAJ-vehm . UM Law “‘1 L) X _ x-ew ”f r\(?’—')(L ¢(K,‘1/>\ I X/ E high-60 000-" 015” Vﬂ‘: 9— " \ c \/ bag 7 O ’5 ‘—L’<\ O 5 37‘ *O ‘ DE ‘0 ’A "2“ J” Em t \ l —' gm“ ¥2.,p\I9’.—.L m 6% >4 3 {xeﬂ | H’ ua Wk ” a“ “W‘Qf > L {4‘1“ (II c go (>er L(O/JJ’\— J; L maYlMM , / W M ’th)?2' ‘v Question 3. E (a) State Fubini’s Theorem for a continuous function f : R2 —> R E on a rectangle [(1,1)] x [0, d]. (b) Evaluate the double integral 27r 27r sin :1: + siny dydm. 3 f0 /0 < > El Question 4. l 6 l Consider the transformation from Cartesian coordinates (a:,y) to coordinates (u, 1)) given by the following equations: 1/3 1/3. u:a:y v=y —a: . . . . l (a) Find the Jacobian determinant for this transformation. l3 (b) For what values of a: and y is the transformation invertible? E (M “ALCOHOL; W R ll . Owl AW‘ “‘0. (la, Tacolom» Ml‘ lot W % Question 5. ¥- Use the transformation in Question 4 to evaluate the integral f/D sin(1n(y1/3 — x1/3))dA D={(:z:,y):1§\$y<4,1<y1/3—xl/3Se”,\$20,y20}. where The curves bounding the region D are shown below. 2000 \$y= 1500 1000 500 Question 5 continued... Question 6. E (a) Prove that the vector field f(a:,y,z) = (y,a:,z) is conservative. (b) Evaluate the line integral f“7 f - dr for each of the oriented curves 7 below:@ (0,1,1) (0710) (M V‘J; :6 cm“ l1 , CJ Vr‘c 2/» 967%; 7* 1 a 3'31) a L :3, f; ’J a): a) ah‘ 0 ° Question 7. g (a) State Stokes’ Theorem. 3/L (b) Let f(a:,y,z) : (3:2 + y2 + 752,3: + y + z,z) and let 2 be the surface 2 = 1 — 51:2 — y2 for z 2 0 with upward orientation, shown below. Evaluate the. / ’2, surface integral //V><f.dR 2 27r You may use the fact that / cos2 tdt = 7r. 0 Question 7 continued... Question 8. E (a) State the divergence theorem. ] 3] (b) Let 2 denote the surface of the unit box [0,1] x [0,1] x [0,1], with outward orientation. Determine //2deR E) f(:r, 1/, Z) = (932 + 6Z7 1/2 + 62, 22). where : 3 Blank Page ...
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20E-FinalSolutions - Final Examination Math 20E — Vector...

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