Please do like.
Consider: S (x + 2 x + y)z ds, C: r(t)= (2+, 1-37, 57 +5), te[0,1]...
Consider: S x2-yds, C: r(t) = (e"? 2, 1+e'), te[0,2] Which one of the following "regular" integrals represents the above line integral. dt O a. Ob. V 4 dt 0 S'Vertel dat O d.o Question 8 10 point Consider: | <x?,v/dr, C: r(t) = (sint, cost), te[0,1] Which one of the following "regular" integrals represents the above line integral. S". cost sint - cost sint dt O a. o П 1 sin2tdt 0 s "cost sin’t + cost sint dt...
1. Evaluate the line integral S3x2yz ds, C: x = t, y = t?, z = t3,0 st 51. 2. Evaluate the line integral Scyz dx - xz dy + xy dz , C: x = e', y = e3t, z = e-4,0 st 51. 3. Evaluate SF. dr if F(x,y) = x?i + xyj and r(t) = 2 costi + 2 sin tj, 0 st St. 4. Determine whether F(x,y) = xi + yj is a conservative vector field....
Consider: 5x2-y’ds, C:r(t)=<2t, – 1), te[0,11 Which one of the following "regular" integrals represents the above line integral. 312dt 1 oors ! *52-2 + 14t V5S 3+2+2t - 1dt Ос. 1 d. f 31² + 28-1dt 1 -15% 312 +2t - 1dt
Evaluate the surface integral f(x,y,z) dS using a parametric description of the surface. 2 f(x,y,z) x 2 where S is the hemisphere x + y +z2 = 25, for z 2 0 The value of the surface integral is (Type an exact answers, using t as needed.) Evaluate the surface integral f(x,y,z) dS using a parametric description of the surface. 2 f(x,y,z) x 2 where S is the hemisphere x + y +z2 = 25, for z 2 0 The...
(bonus) Prove that operator A: L²(0,1) - L’(0,1), Ar(t) = 5 ts(1 – st)x(s)ds is compact and find its spectrum.
Evaluate the surface integral. 1 (x + y + z) ds, S is the parallelogram with parametric equations x = u + v, y = u - v, z = 1 + 2 + v, osus 6, Osvs 2.
11. (20 pts) Consider the surface integral JJs F dS with F(x, y, 2) - 2xyǐ + zeij + z3k where s is the surface of the cylinder y2 + 2 = 4 with 0-x < 2. (a) Parametrize this surface and write down (but do not evaluate) the iterated integrals for the surface integral. (b) Let S' be the closed surface with outward-facing normals obtained by taking the union of the surface S with the planes x = 0...
R is a closed and bounded region in the polar coordinate and it's given by {(x,y): x 0,1 S$2 + y's 49). R 0, y a. Determine the area of R by using double integral in the polar coordinate. Given the surface z - 8xy + 1, determine the volume between the b. surface z and region R by using double integral in the polar coordinate. R is a closed and bounded region in the polar coordinate and it's given...
2. Consider the conical surface S={(x,y,z)∈R3 : x2 + y2 = z2, 0 ≤ z ≤ 1}, and the vector field (a) Carefully sketch S, and identify its boundary ∂S. (b) By parametrising S appropriately, directly compute the flux integral S (∇ × f) · dS. (c) By computing whatever other integral is necessary (and please be careful about explaining any orien- tation/direction choices you make), verify Stokes’ theorem for this case.
(2) Let F-1 + rj + yk and consider the integral- , ▽ × F. т. dS where s is the surface of the paraboloid z = 1-12-y2 corresponding to z 0, and n is a unit normal vector to S in the positive z-direction (a) Apply Stokes' theorem to evaluate the integral. (b) Evaluate the integral directly over the surface S rectlv over the new surface (2) Let F-1 + rj + yk and consider the integral- , ▽...