Question

Help Entering Answers (1 point) Verify that Stokes' Theorem is true for the vector field F = yzi - yj + xk and the surface S the part of the paraboloid z= 4 a2 ythat lies above the plane z = 3, oriented upwards. curl FdS To verify Stokes' Theorem we will compute the expression on each side. First compute S curl F = Σ curl F.dS Σ dy dπ (y-1)-2y)+z where 3 -sqrt(9-x^2) Σ 3 sqrt(9-x^2) curl F dS = Σ F.dr Now compute cos(tsin(t) The boundary curve C of the surface S can be parametrized by: r(t) Σ 4-cos(t)^2-sin(t = 0 t2T (Use the most natural parametrization) MM W MM curl F.ds / F.dr Now compute ( The boundary curve C of the surface S can be parametrized by: r(t) cos(t) Σ 4-cos(t) 2-sin(t sin(t) ). 0t27T (Use the most natural parametrization) 2m F-dr = Σ dt F.dr Σ If you don't get this in 3 tries, you can see a similar example (online). However, try to use this as a last resort or after you have already solved the problem. There are no See Similar Examples on the Exams!

Answer 1

o,-1 i) CurF -2 2 S : 4-x2- Te=く0」-2x) Z 3 Tu XYy =く 2x,2y> dydu. dl S = 2x,21 ) y <oy-リ-z).< 2x2 CuslF.dS= 12y2-2y-2oty di /2y-2y+x2y*-47 32-21-47 x2 2こ.3- 4- intericHon at 2 + Cul F. ds ) Convertng to polan : 2rsino-47rdr do. 2 1 Y C C 3sin20 4- Coso -+ 2sinO 27de 3 4 27 2 sino 4 2 cos(20) 2sino 2ス 7 4

22 Sinl2o) 2cosO 3 37 - 3 co 37 -iv) culFdS rC4) =くostJsintl)4-cos?t-sinzt E costSin t|1]3 2. chr ) <-sin i , costj o) cli : FC) = く3sint,-Sint) cost> F.dr (-3 sint- sint cos t) clt 2 7 3 sin tsintcost clt 2え sin (2t) 3 cos(2t) 33 2 2 7.9 3sin(2t) 3t Cos(2 t) 2 4 37