finw99 - Math 20E Final Winter 99 Lindblad 1 Find the...

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Unformatted text preview: Math 20E Final Winter 99, Lindblad. 1. Find the equation for the line of intersection of the two planes 256— 3y+z— 1 = 0 and$+2y—z+3=0. 2. The temperature of space is given by ¢(a:,y, z) = 1 + 303/2. A fly is flying in space and at each point (1:, y, z) of its journey it flies in the direction F(a:, y, z) in which the rate of increase of temperature is maximum. (a). Calculate F (:17, y, z). (b). Find the curve along which the fly travels if it starts at the point (1, —1, 1). 3. (a). Let F be the vector field zew_yi — mew—yj + ew_yk. Find (b with V¢ = F. (b). Calculate f0 F - R, where C' is the curve $(1_$) Z=1_—$ 0<flc<17 oriented so that :23 = 0 at the initial point. (c). Decide whether the vector field 33 i—I— y . —|— Z k $2+y2+22 $2+y2+22J $2+y2+Z2 is conservative and justify your answer. 4. (a) State the change of variable theorem for the double integral over a region in the plane. Let a: : $(u, v) and y : y(u, v) be the change of variables given by {x2vcoshu {coshu=(e“+e_”)/2 where yzvsinhu sinhu= (cu—e_u)/2 Let D: {(33,31); 1 < x 2—y2 < 2, 2y_ < x < 3y} D is the image under the mapping (u, 11) (00,31) of a rectangle 1n the uv- -plane R: {(u, v); a g u g b, c s v S d}. (b) Find the rectangle R. (Hint: a useful identity is cosh2 u — sinh2 u = 1. ) 30189) 301,11). )Find // dmdy —)(1+ac2—y22 (c) Find the Jacobian 5. (a) State Greens theorem for a domain D in the plane bounded by a curve C'. (b) Explain how one can use Greens theorem to find the area of the domain D. a: = t2 (c) Consider the closed curve C given by { t3 t , —\/§ 3 t 3 \/§. 3/ = 3 — Compute the area D bounded by the curve C. (d) Find the length of the curve C. 6. (a) State the Divergence Theorem. Find the flux // F - ndS of the vector field F : i+j + 23k out of the sphere S S : {(3331, z); 11:2 + y2 + Z2 = a2}, where n is the outward unit normal. (b) By direct calculation of the surface integral. (c) By using the Divergence Theorem to convert it to a volume integral and evaluate. 7. (a) State Stokes theorem. Let S be the surface S = {(az,y, z); z = 2 — x2 — y2, (any) 6 D}, where D = {(33, y); x2 + 3/2 S 1}, and let n be the unit normal to S oriented up- wards so n - k > 0. Let C be the boundary of S positively oriented relative to the orientation of 5'. Let F : yi — mj —|— zk. Find the line integral / F - dR. C’ (a) By direct calculation using the usual parameterization of the circle. (b) By using Stokes theorem to convert it to an integral over S and evaluate. 8. A parameterization of the torus (donut ) 2 T = {($,y,z); (V332 +y2 — 4) + z2 3 4} is given by x=4c0s0+2c0s6 cos¢ yz4sin9+23in0cos¢ where { z: 2sin¢ 0 S 0 S 271' 0 S <25 S 27? (a) Find a normal to T at a point with coordinates (9, (b). (b) Calculate the surface area element dA in terms of dfldqi and use it to calculate the area of the surface of the torus T. ...
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