Use Stokes' Theorem to find the value of the line integral of the vector field Faround the curve Cwhere C is in the direction of positive orientation (that is, counterclockwise when looking down from the positive z-axis): f. v - dr - [[ (cart 1) F. = (curl F). n ds where F(x, y, z) = xyi + x²j + z²k and Cis the intersection of the paraboloid z = x² + y² and the plane z = y.

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Q: Verify the validity of the Stokes's theorem for the vector A = e-bz sin ø a, + pe-bz , over the open…

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Q: Use Stokes' theorem to evaluate Ĝ - dĩ, vhere Ĝ(1, Y, 2) = (2r, y,0) and D is any closed curve.

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Q: Show that this curve y represents the vector (1,0) as v = 1- +0- in the sense that, for an arbitrary…

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Q: a) By first sketching its projection onto the yz-plane, sketch the curve r(t) = (t, cos(t), 2…

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Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

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Q: Calculate the curl(F) and then apply Stokes' Theorem to compute the flux of curl(F) through the…

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Q: r(t) = sin(t)i + cos(t)j + ek

A: We have to solve given problem:

Q: 6. Use the Normal Vector to a Smooth Parametric Surface to find the equation of the tangent plane to…

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Q: Evaluate the line integral a) | 3x'yds, where Cis the curve r(t)=5ti + (3t-7)j. Ostsl. b) (xy +…

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Q: Let C be a curve defined by some vector function R with unit binormal B such that R(0) = (-2, 5, 4)…

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Q: H=pz coup ap te'sing a4 tp'ar At point (1, ,0)

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Q: 2. Consider the curve traced out by the vector valued function r(t) = (sin t cos t, cos? t, sin t)…

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Q: (b) Sketch a condition to prove that the curl of a gradient is zero (0). Briefly explain your…

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Q: For the vector function: r(1)=(Sf,21,5). find %3D (a) r'(t) and r'(4) vileuponi od yd bolbni PA pe…

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Q: 7. Let F(x, y,z)=-2yi +2xj. Use Stokes’ Theorem to find F dr where C is a circle (a) Parallel to the…

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Q: Q3 – A// If P(1,2,-1),Q(2,-1,1), R(-1,-1,2) are three point in space. 1- Find the vector…

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Q: True or False? Vector functions r 1 = ti + t 2 j, 0 ≤ t ≤ 1 , and r 2 =(1 — t )i+(1 — t ) 2 j, 0 ≤ t…

A: Given: Vector functions: r1=ti+t2j, 0≤t≤1r2=(1-t)i+(1-t)2j, 0≤t≤1 We are required to find: If the…