Use Stokes' theorem to write a line integral that is equal to the surface integral cu aff curt S F(x, y, z) = (4y – 2z, -z,-3y), S is the portion of the paraboloid z = 4x² - y² above the xy-plane, and its boundary C is directed counterclockwise when viewed from the first octant. (Hint: parameterize C by the angle, when 0 ≥ 0.) Provide your answer below: curl F. ds, where de

Q: Evaluate the surface integral. JS (x+y+z) dS, S is the parallelogram with parametric equations x = u…

A: We find the surface integral.

Q: Use Stokes' theorem to evaluate the surface integral I of the two-form w = dŋ, with n = (xz+ yz²) dæ…

A: η=(xz+yz2)dx+(sin(y)+z+5x)dy+xyz dzz=13-x2-y2,   z≥9 F→=xz+yz2i+(sin(y)+z+5x)j+xyzk z=9⇒9=13-x2-y2…

Q: Let F(x, y, z) = (-y, xy, 0) and C be the curve of the intersection of z = x² + y² and z = 2x such…

A: Given clear answer below

Q: An RC circuit with a 100 Q resister and a 10^(-2) F capacitor is driven by a voltage…

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Q: /F-nds, where b) Use Gauss theorem to evaluate F(x, y, z) = (2a²y² + In 2) i – xy° j – tanh(2² + y²)…

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Q: Evaluate the surface integral G(x,y,z) do using a parametric description of the surface. 2 2…

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Q: S[y] given by y = cosh æ + B sinhx, 0 0,

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Q: (5) Use Stokes' Theorem to find the line integral |(* + 2y°)i + (y + z°) +(= + 2r*)k) • dř where C…

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Q: Evaluate the surface integral 1 d.S where o is the portion of the plane z = 4 - 6x – 1y in the first…

A: To find  where  is the portion of the plane  in the first octant

Q: Find the value of the integral fs (V x A). da if the vector A = yi + zj + xk and Sis the surface…

A: Given the vector A=yi+zj+xk and S is the surface defined by the paraboloid z=1-x2-y2

Q: Use Stokes' Theorem to find the line integral [((x² + x + 2y²)i + (y + z³)j + (z + 2x²)k) · dr where…

A: To Evaluate: The integral ∫C(x+2y2)i+(y+z2)j+(z+2x2)k·dr. where C is the boundary of the triangle T…

Q: Verify the formula o F T ds = || (curl F) n dS in Stoke's Theorem. by evaluating the line integral…

A: Given  Fx,y,z=5x-yi+5y-zj+5z-xk We have to verify the Stoke's theorem. x+y+z=3 is the plane.

Q: Let F = yi+zj + xk and C is the boundary of the surface S, which is the paraboloid z = 1 - (x² +…

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Q: Compute the surface integral of the function f(x, y, z) = 2xy over the portion of the plane 2x + 3y…

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Q: Use Stokes' Theorem to evaluate F. dr, where F(x, y, z) = xzi+ xyj+2xz k,•. and C is the boundary of…

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Q: be the surfa enned by the vector function R(u, v) (-/- 2u,v + 5, ve¹) all (u, v) € R². Set - up an…

A: Here, in the question given a surface S defined by the vector function: R ⇀(u, v) =(v5-2u, v+5, veu)…

Q: Calculate the surface integral // y ds, where S is the part of the helicoid defined by the S vector…

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Q: ify Stokes' theor d S is the surface

A: INTRODUCTION: Stoke's Theorem:-…

Q: 311. Evaluate surface integral where S is hemisphere x² + y² + z² = x² + y² + z² = a², z ≥ 0.…

A: Parametrize the surface and calculate corresponding ds.

Q: Evaluate the line integral PF • dr by evaluating the surface с integral in Stokes' Theorem with an…

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Q: Use Stokes' theorem frotF. ds to evaluate F(x, y, z) = (x + tan¯¹yz)i + y²zj + zk, S is the part of…

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Q: (x + y + z) ds, S is the parallelogram with parametric equations x = u + v, y = U-v, z = 1+2u+v, 0 ≤…

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Q: (b) Use Stoke's Theorem to evaluate the line integral $ (3? – 22) dr + (z² – x²) dy + (y² – x²) dz,…

A: Stokes theorem  ∮CF.dr=∫∫SCurl F .dS Where S be a smooth surface that is bounded by a  closed,…

Q: . Use Stokes' Theorem (and only Stokes' Theorem) to evaluate F dr, where F(r,y,z) = (3y, -2x, 3y)…

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Q: Use Stokes' Theorem (in reverse) to evaluate (curl F) ndS where F = 6yzi + 9xj +6yzeªk,S is the…

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Q: Evaluate ff F.n ds where F(x, y, z) = 12xi +24j+24x²k and is that portion of the paraboloid z = x² +…

A: We will find normal vector n and region of integral on xy plane to find value of integral .

Q: Verify the formula F.T ds = [[(curl F). ndS in Stoke's Theorem C by evaluating the line integral and…

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