Divergence, Curl, Line Integrals ((c) of M55 Upd)
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Mathematics Second Long Exam 55ExercisesI. Divergence and Curl1. Compute for the curl and divergence of the following vector fields.(a) F(x,y,z) = (b) G(x,y,z)=(x – y)ˆ i + (x + z)ˆ j + (y – z)k ˆ(c) H(x,y,z) = (d) F(x,y,z) = (siny + 3z)k ˆ(e) G(x,y,z) = 2. Using the fact that we can write F(x, y) = as F(x,y,z) = , compute for thedivergence and curl of the following:(a) F(x, y) =ˆ i + (ex – y + 2z)ˆ j + (3x + 2y + z)〈-〉(b) G(x, y)=2xx y,x 1ˆ i + 3yˆ j (c) H(x, y) = <3x2y,-2xy3>II. Conservative Vector Fields1. Determine if the following vector fields are conservative. If yes, find a potential function forthem.(a) F(x, y) = <2x + y,x + 2y - 2> (b) F(x, y) = (c) F(x,y,z) = <6xy sinz + 2x,3x2 sinz + 5,3x2y cosz> (d) F(x,y,z) = (e) F(x,y,z) = III. Triple Integrals1. Evaluate∫10∫ √1−x20∫1−x2−y20(1 + x2 + y2) dx dy dz using cylindrical coordinates.2. Let ∫∫∫SS ((x2 be + the y2) solid dV .bounded by the paraboloid z = 1 – x2 – y2 and the xy-plane. Evaluate3. (a) Write the equation of the cone z =√3(x2 + y2) spherical coordinates. (b) Set-up the integral in spherical coordinates that gives the volume of the region inside thesphere x2 + y2 + z2 = 1 and above the cone in the previous item.4. Find the volume of the solid bounded by the following surfaces: S1: z2 – x2 – y2 = 0, S2: z2 – x2 – y2 = 1, and S3: z = 55. Set-up a triple integral in spherical coordinates xy-plane that lies within the sphere x2 + y2 + z2 that = 4 and gives below the volume the cone of z the =√solid 3×2 + above 3y2.the6. Evaluate using cylindrical coordinates∫2−2∫ √4−z2∫5−y0x2+y2dz dy dx7. Evaluate∫2∫/4−y200∫//x2+y28−x2−y2x2 + y2 1+ z2dz dx dy using spherical coordinates.8. Set-up the iterated triple integral in cylindrical coordinates that gives the volume of the solidin the first octant bounded by the cylinder x2 + y2 = 1 and the plane x = z.
9. Let coordinates S be the to region evaluatebounded ∫∫∫S√by x2 the + y2 spheres + z2 ρ = dV .2, ρ = 4, and the cone φ = π 4. Use spherical10. Set-up paraboloid the x2 integral +y2 +z in = cartesian 6 and the form cone that z =will √x2 give + y2. the The mass density of the function solid S of bounded S at any by point theis ρ(x,y,z) = kz.11. Let Q be the solid bounded by the planes y = 0, x = y, 2x+3z = 6, and the xy-plane. Set-up the iterated triple integral that will calculate the mass of Q if the density at any point (x,y,z) on Q is directly proportional to its distance in the xy-plane.IV. Vector Fields, Line Integrals, and Green’s Theorem1. Given F(x, y) =(3x2y +)ˆ i +(x3 -)(a) Show that the vector field F is conservative. (b) Evaluate1 xeylnx ey+ 2y∫CF · dr where C is any path from (e,0) to (1,1).2. Let F(x, C y) be = ythe ˆi + part xˆ j. Evaluateof the parabola ∫CF y = · T dr.x2 from the point (0,0) to the point (1,1), and let3. Evaluate the line integral given below where C is the circle centered at (2,3) of radius 2.∮(6y + x) dx + (y + 2x) dy4. Given F(x, y)=(ex lny + cosxcosy)(ex y)ˆ j.(a) Show that F is a conservative vector field. (b) Evaluateˆ i +- sinxsiny∫CF · dr, where C is a path from (π 2,1) to (0,e).5. Use a line integral to find the area of the region enclosed by the ellipsex2 9+y2 4= 16. Given F(x, y) =(lny -cosx y)ˆ i +(x y+sinx y2) + 2yˆ j.(a) Show that F is conservative.∫ (b) EvaluateC F · 7. Use the Green’s Theorem dr along any path from (0,2) to (π,1).to evaluate∮(ey -3y2) dx+(xey +6xy) dy, where C consists of the line segment from (-1,-1) to (2,2), and the portion of y2 – 2 from (2,2) to (-1,-1).