Use the surface integral in Stokes' Theorem to calculate the rotation of the field F = (y? + z?)i + (x² + y²)j+ (x² +y²)k around curve C which is a square bounded by x = ±1 and y = +1 at xy plane, in a counter clockwise direction. (Hint: n = k)

Q: use the surface integral in Stokes’ Theorem tocalculate the circulation of the field F around the…

A: Use the surface integral in Stokes’ Theorem to calculate the circulation of the field F around the…

Q: - Let the vector field, F = x³ i+y³ j+z³k. Verify Gauss's Theorem for surfaces S to be closed…

A: As per the question we are given a vector field F = x3i + y3j + z3k And a closed cylindrical surface…

Q: Given the following vector field and oriented curve C, evaluate F•Tds. C F = (x,y) on the parabola…

A: Given that, Fx,y on the parabola rt=4t,9t2 for 0≤t≤1. Obtain the r't. r't=ddt4t,9t2=4,18t

Q: 4. Use Stoke's theorem to calculate the flux of the curl of the field F = (x – y)i + (y – z)j+ (z –…

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Q: Let C be the ellipse in which the plane 2x + 3y - z = 0 meets the cylinder x2 + y2 = 12. Show,…

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Q: use the surface integral in Stokes’ Theorem to calculate the circulation of the field F around the…

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Q: Find the flux of the field F(x, y, z) = z2 i + x j - 3 z k outward through the surface cut from the…

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Q: Which of the following is the circulation of the field F in the direction indicated on the curve C,…

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Q: find the circulation and flux of the field F aroundand across the closed semicircular path that…

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Q: 2. Verify both the circulation AND flux versions of Green's Theorem for the field F = 2xi – 3yj…

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Q: Verify Stokes theorem for the vector field F = 2(-+x)i- z? yj-zyk over the upper half x² +y? +z?…

A: To verify Stokes' Theorem for the vector field               F→=2-y2+xi→-z2yj→-zy2k→ over the upper…

Q: Evaluate the line integral of the vector field F(x, y, z) = (x²y³, e™y+z, x+ z²) around the circle…

A: Solution is as below

Q: 3. Consider the vector field F = (y, -x, x² + y?). Find the flux of F through S if: 9– x2 – y? above…

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Q: Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated…

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Q: Let a vector field F(х,у) 3sinx 1 + ху ј be given. What is the integral representing the work done…

A: Workdone by force along the path is integration (f.dl) Where f is force and dl is distance vector…

Q: 3. Verify Stokes' Theorem for the vector field F= 2zi +3xj+5yk, taking S to be a portion of the…

A: Given F=2zi+3xj+5yk

Q: Consider the surface integral of the vector field F = [xy?, 3x², x²z] over a surface S, where S…

A: We will use of gauss divergence theorem

Q: Find the circulation and flux of the field F = - xi – yj around and across the closed semicircular…

A: The circulation of a vector F=M i+ N j along the curve C is defined by ∮CF·T ds=∮CMdx+Ndy. Here…

Q: Let f(x, y, z) = (x² + y² + z²)¬/2. Show that the clockwise circulation of the field F = Vf around…

A: Calculate: F=∇f   =∂∂xx2+y2+z2-1/2,∂∂yx2+y2+z2-1/2,∂∂zx2+y2+z2-1/2   =-x2+y2+z2-3/2x,y,z

Q: 12. Evaluate the surface integral JJ, F ·dS for the vector field F(x,y,z)=(-x,-y,z'), where S is the…

A: This is the problem of multivariable calculus, surface iuntegral.

Q: Verify Stokes Theorem for the vector field F(x, y, z) = (y-z)i + 3xy j + 5zk by evaluating the line…

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

Q: find the circulation and flux of the field F aroundand across the closed semicircular path that…

A: The given vector field is Fx,y=-yi+xj.

Q: Use Stokes' Theorem to calculate the circulation of the field F around the curv F=xi+xyj+yk; C: the…

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Q: Use the Stokes' Theorem to caleulate the circulation of the field F = -6y*7 + 6x*J° + 7z*R° around…

A: Given field, F→=-6y3i^+6x3j^+7z3k^ where C is the curve of the paraboloid x2+y2=z cut by the…

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: use Green’s Theorem to find the counterclockwisecirculation and outward flux for the field F and…

A: Solution:GivenF=(xy+y2)i+(x-y)jFormula:Outward Flux=∫F.n ds=∬ ∂M∂x+∂N∂y dx dyCounter Clockwise…

Q: Calculate the flux of the field F across the closed plane curve C where F=xi+ vj and the curve is…

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Q: Find the flux of the field F(x, y, z) = 4x i + 4y j + 2 k outward (away from the z-axis) through the…

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Q: Find the surface integral of the field F(x,y,z)=9xi +9yj +9zk across the portion of the sphere x² +…

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Q: Let a vector field F (x,у) 3sinx 1 + ху ] be given. What is the integral representing the work done…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F = (4x, y, - 2z) across the…

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Q: Consider the vector field f (x, y, z) = xz i + yz j+ x²y² k . Let S be the surface of the sphere of…

A: Using parametrization of the surface and curve

Q: Consider the field F(x.y.z) = 1 (x +y) i+1 (x5+y)j+1 (x2 +y2) k. (a) Integrate F.ds around the…

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Q: Let ƒ(x, y, z) = (x2 + y2 + z2)^(-1/2). Show that the clockwise circulation of the field F = ∇ƒ…

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Q: Let E be the solid region bounded by the upper half-sphere x² + y? + z2 = 4 and the plane z = 0. Use…

A: Flux of a vector field F→ along surface S =∫∫SF→.dS Divergence theorem  ∫∫SF→.dS→=∫∫∫E divF→dV Where…

Q: 12. Evaluate the surface integral F-dS for the vector field F(x, y,z) =(-x,-y.z), where S is the…

A: The given problem is to evaluate the given surface integral in the part of the cone  between the…

Q: Let a vector field F(x,y)=sinx 1 + xy j be given. What is the integral representing the work done by…

A: Acoording to expert guidelines, I am allowed to answer first question only. Kindly repost the rest…

Q: Find the circulation of the fieldF = (x2 - y)i + 4zj + rk around V? + y?, the curve C in which the…

A: Field F→=x2-y i^+4z j^+ x2 k^Surface S is cone z=x2+y2at plane z=22=x2+y2x2+y2=4So, f=x2+y2-4

Q: find the surface integral of the field F over the portion of the given surface in the specified…

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Q: Use Green's theorem to find the counterclockwise circulation and outward flux for the field F =…

A: F=(y3-ey sinx) i + (y3+ey cos x) jr2=sin(2θ) we have to calculate the counterclockwise circulation…

Q: 18. F = 1 + i + (tan' y)j x² + y? = 1

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Q: 1. Use Green's Theorem to find the counterclockwise circulation of the field F around the curve C as…

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Q: Find the circulation of the field F = -2xi-2yj around the closed semicircular path that consists of…

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Q: find the surface integral of the field F over the portion of the given surface in the specified…

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Q: Use the Divergence Theorem to compute the net outward flux of the field F = (4x, 3y, - 3z) across…

A: Let's find.

Q: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and…

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Q: 1 Find the line integral | Fdrof the vector field F :(xi + yj+ zk) where C is the boundary of the…

A: see below the answer

Q: 1. Use the surface integral in Stokes' Theorem to calculate the circulation of the field F around…

A: We can solve this using Stoke theorem

Q: Calculate the áux of the field F(x, y) = (x - y)i + xj across the circle y2+x2 = 1 in the xy-plane.

A: Given F→x,y=x−yi^+xj^ We have to find the flux of the given field across the circle x2+y2=1 in the…

Q: 03 Verify Stokes theorem for the vector field F= 2(-+x)i-zyi-zy'k over the upper hlf surface of…

A: To find- Verify Stokes theorem for the vector field F→ = 2-y2 + xi→ + z2yj→ - zy2k→ over the upper…