Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F across the surface S in the direction away from the origin. F= 4yi + (5 - 5x)j + (z - 2)k S: r(), 0) = (V7 sin o cos 0) i+ (17 sin o sin 0) j+ (V7 cos o) k, 0

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Q: A uniform magnetic field B has constant strength b teslas in the z-direction [1.e., B = (0, 0, b) 1…

A: Stoke's theorem ∫CF→.dr→=∫∫S curlF→.dS→ Where S is a smooth surface bounded by a smooth closed curve…

Q: Use the fact that \bF = (2xy + 3cosy) bi + (r? - 3xsiny) bj, is a gradient vector field to evaluate…

A: Given F = (2xy+3cosy)i+(x2-3x siny)j As F is a gradient vector field, it can be rewritten as F=∇f…

Q: A vector field F = x(x² – y²) + ŷ(2xy + y?) a) f d.č =? c on the closed curve in the figure b)…

A: a) Consider the figure:

Q: QUESTION 1 a) Evaluate the line integral F. dr in terms of a where F = cos yi + rj+ ye*k and C is…

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Q: y² cos(xy?) – ay sin(æ²y) Let it be 7 : R² → R², (2ary cos (ay?) – ža² sin(a²y)/ a vector field.…

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Q: (a) Show that the vector field F (x, y, z) = (x² + y², x, z³) is not conservative.

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Q: a) Calculate the line integral of the vector field F(x, y) = yi − 5xj from the point (0, 3) to the…

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Q: Find the curl of the vector field F curl F = (2y cos(x), 5x sin(y))

A: F = (2ycosx , 5xsiny) find curl F =?   curl F  = dF2/dx - dF1/dy here  F1 = 2ycosx  F2 = 5xsiny now…

Q: ) Verify Stokes' Theorem for the hemisphere S : x²+y²+z² = 9, z > 0, %3D its bounding circle C: 1?…

A: Stokes theorem  If a smooth surface S bounded by closed ,smooth boundary curve C oriented in…

Q: Find the flow rate of F = across the portion of the surface z = over the unit square [0, 1] × [0, 1]…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

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Q: Verify that Stokes' Theorem is true for the vector field F = (-y³, x³) along the circle of radius R…

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Q: Compute the line integral of F along the oriented smooth curve C shown below. (2,0, 2) (1,4, 3)

A: F =(ysinz)i +(xsinz)j+( xycosz)k dr =(dx)i +(dy)j+(dz)k line integralx-21=y-0-4=z-2-1=t (x,y,z)…

Q: Let I be the curve parametrised by 2t 2(4) = ( sin(zt), 41' Vt +1 for 0<t<1 t2 +1' and let F be the…

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Q: Consider the line integral (Image 1) , where C is defined by the vector function: r (t) = - t2i +…

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Q: 3. Find the line integral fF.dr where C' is the circle of radius 5 in the xz-plane oriented…

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Q: 3. Consider the vector field F(x, y) = (3y²+ 4x, 5y sin(x)). (a) Calculate curl F at the point (0,…

A: The curl of a vector field F = (P, Q) in two dimensions is given by the formula:∇×F=∂Q/∂x-∂P/∂yHere,…

Q: Find the gradient vector field of f(x, y) = x° sin(9y). F(x, y) = x°i+ sin(9y)j F(x, y) = 9x°…

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Q: r = 2 on the hemispherical surface defined by 0 ≤ Ɵ ≤ π/2 and 0 ≤ɸ ≤ 2π F = 100cosƟar Verify Stokes'…

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Q: SOLVE STEP BY STEP IN DIGITAL FORMAT 3. Is the series n=1 2n convergent or divergent? Hint: quotient…

A: The given series is .