Please show step by step solution Suppose that C is the curve of interesection between the plane z = x and the cylinderx² + y? = 1 (an ellipse), oriented counterclockwise when viewed from above. Let Sbe the interior of this ellipse, oriented with an upward pointing normal vector. IfF(x, y, z) = xi+ zj+ 2y k, verify Stokes' theorem by showing thatF. drcurl F· dSCHint: Show both sides are equal.

Name: Please show step by step solution Suppose that C is the curve of inter
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: Please show step by step solution Suppose that C is the curve of interesection between the plane z = x and the cylinderx² + y? = 1 (an ellipse), oriented counterclockwise when viewed from above. Let Sbe the interior of this ellipse, oriented with an

Consider the given function. z=x and x2+y2=1 As, the function S be the interior of this ellipse then, x2+y2≤1 The function S(x,y) is defined as, Sx,y=x,y,x Find the derivative with respect to x and y. Sx=1,0,1Sy=0,1,0 And the curve C curve of intersection between the plane z=x and x2+y2=1.The parametric form is defined as, rθ=cosθ,sinθ,cosθ Find the derivative with respect to θ. drθ=-sinθ,cosθ,-sinθdθ The function F(x,y,z) is defined as, Fx,y,z=xi+zj+2yk=x,z,2y Convert in the parametric form. Fθ=cosθ,cosθ,2sinθPut all the values in the function ∫CF·dr. ∫CF·dr=∫02πcosθ,cosθ,2sinθ·-sinθ,cosθ,-sinθdθ=∫02πcos2θ-2sin2θ-cosθsinθdθ=∫02π12+cos2θ2-1-cos2θ-sin2θ2dθ=∫02π-12+3cos2θ2-sin2θ2dθ=-θ2+34sin2θ+cos2θ402π=-2π2+34sin4π+cos4π4--02+34sin0+cos04=-π+14-14=-πNow, find the right side. Find the first curl F. F=x,z,2y And the curl F is, curlF=ijk∂∂x∂∂y∂∂zxx2y=2,0,1Further simplify the integral. ∬Scurl Fds=∬S2,0,1⋅−1,0,1dxdy=∬S−2+0+1dxdy=−∬Sdxdy Convert into polar function. −∬Sdxdy=−∫02π∫01rdrdθ=−∫02πr2201dθ=−12θ02π=−2π2=−π Hence, it has been proved that both the sides are equal.