Verify Stokes' theorem on a region which is a portion of the sphere r+y+2= 16 which lies above the plane z= 2 and with F(r, y, 2) = r²i – 2²j+ yɛk. Suppose that o is differentiable. Prove the identity |vo nds=

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: Use Stokes' Theorem to evaluate curl F. ds. F(x, y, z) = xyzi + sin(xyz)) + xyzk, S is the part of…

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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: Use Stokes’ Theorem to evaluate the flux integral ∫∫(∇×F)·ndS, where F =〈2y, −z, x+y −z〉, S is the…

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Q: Evaluate the surface integral. xz ds, S is the part of the plane 2x + 2y + z = 4 that lies in the…

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Q: Evaluate the surface integral F.n dA, where F(x, y, z) = i + j + z(x² + y²)<k and S is the surface…

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Q: Use Stockes' theorem to calculate the line integral f. F dr where x, y, z 3xy + e?va-1 i + ( 2x2 –…

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Q: 2. Evaluate the surface integral f,yz ds ; o is the part of the plane x + y + z = 2 that lies in the…

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Q: Use Stokes' Theorem to evaluate F.dr for F=(xz,-y,x²y), where C is the boundary of the open surface,…

A: Given : vector field F=xz,-y,x2y and C is the boundary of the open surface, S that is bounded by…

Q: Use Stokes's Theorem to evaluate f.7 X F.dŠ, where F (x, y, z) = (y +29)î – xj + x³yk and S is the…

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Q: Let S be the part of the cone z = Vx2 + y? under the plane z = 4, outside the cylinder +y 1, and in…

A: The given cone is where  The widest section of the surface is obtained at z=4, whereas the thinnest…

Q: (b) Find the volume under the paraboloid z-x+y above the triangle enclosed by the lines y=x,x-0 and…

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Q: Consider the surface S which is the part of the cylinder x^2 + z^2 = 9 , whit   0 ≤ y ≤ 2; x ≥ 0; z…

A: a  x2+y2=9    0≤y≤2,  x⩾0, z⩾0   S is parametrised by     rθ,y=3 cos ϖ, y, 3 sin θ    θ∈0 π2,   y∈0…

Q: Compute the surface integral over the given oriented surface: F=y³i+7j − xk, portion of the plane x…

A: We need to find surface integral.

Q: 5. Evaluate the surface integral S_(x+ y) ds ; o is the part of the plane 2x + y +z = 3 that lies in…

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Q: Verify Stokes theorem for F = (xy -x2)î + yx?j in the region in the xy-plane bounded by y = 0,x = 1…

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Q: Evaluate surface integral F.N dS where F(x, y, z) = xy²i + yz?j + x²zk, S is the boundary of the…

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Q: Compute the triple integral f(x,v.z)dxdydz where D is the sphere with center O and radius R and…

A: Given that ∭Df(x,y,z)dxdydz where f(x,y,z)=R2-(x2+y2+z2). By the spherical coordinates,…

Q: The volume under the plane z – 1² – 2y? = 0 and above the region enclosed by the line y = I- 128 and…

A: To find the volume under the plane  z-x2-2y2=0⇒z=x2+2y2 above the region enclosed by: y=x-126,i.e.…

Q: Compute the surface integral over the given oriented surface: F = y²i+6jxk, portion of the plane x +…

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Q: Evaluate the surface integral. /| xz ds, S is the part of the plane 2x + 2y + z= 4 that lies in the…

A: We need to find surface integral.

Q: Evaluate the surface integral. 2,2 ds S is the part of the cone + y that lies between the planes z =…

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Q: Evaluate the surface integral. J[x²³x² x²yz ds, S is the part of the plane z = 1 + 2x + 3y that lies…

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Q: Evaluate the surface integral fs F. dS where F = (-2x, –5z, 5y) and S is the part of the sphere x? +…

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Q: (b) Verify Stokes' theorem on a region which is a portion of the sphere x? +y? +z² = 16 which lies…

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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: Evaluate the surface integral || x2 z2 dS, where S is the part of the cone z2 = x² + y?, S that lies…

A: Here we solve the given problem of calculus.

Q: Use Stokes's Theorem to calculate the surface integral S Ss F - dS where F(x, Y, z) = xi + (2ze² –…

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Q: Let R be the solid region inside the cylinder x^2 + y^2 = 1 and inside the paraboloid z = 5-x^2-y^2…

A: Given,x2+y2=1Using cylindrical coordinates,x2+y2=r2 0≤r≤10≤θ≤2πNow, z=0z=5-x2-y2z=5-r2 0≤z≤5-r2dv=r…

Q: 2. Use Stokes' Theorem to evaluate , Vx F ·n dS where F(x, y, z) = and C is the part of the sphere…

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Q: 3. Let S be the surface of the solid region above the ry-plane and bounded between the planes r + z…

A: Here,  F⇀=z,ex,1+yz

Q: 1. Find the surface area of the cone z = Jx2 + y2 that lies between the plane y = x and the cylinder…

A: Surface area in xy-plane

Q: 10. Let S be the surface of the solid region above the ry-plane and bounded between the planes x + z…

A: The given curves are x+z=2....(1), -x+z=2...(2),  y=0 and y=3. Also given  F→=z i^+ex j^ +1+yz…

Q: Use Stokes' Theorem to evaluate the line integral (y + 2z)dr + (x + 2z)dy + (x+ 2y)dz where C is the…

A: Given intgeral. ∫Cy+2zdx+x+2zdy+x+2ydz The Curve is formed by the intersection of the sphere…

Q: Use Stokes's Theorem to evaluate f. F dř where F(x, y, z) = (x- y) i+ (y-z)j+ (z - x) k and S is the…

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Q: Let F (x, y, z): = (x + 2y2, y + z2, z + 2x2), and let C be the curve that results from the…

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Q: Evaluate the line integral in Stokes' Theorem to find the surface integral [[ (V × F) · ndS, where…

A: Given: The vector is Fx,y,z=4y,-xz,exyz. To find: The value of∬S∇×F·ndS by using the line integral…

Q: Use Gauss's Theorem to find the surface integral OF •dS across the positively-oriented S sphere of…

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Q: a) Evaluate ||| x²yz dV, where G is solid in the first octant that is bounded by parabolic cylinder…

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Q: (b) Evaluate the surface integral ƒ (x, y, z)dS where f(x,y, z) = z² and S is a portion of the cone…

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Q: (b) Find the volume under the paraboloid z=x+y² above the triangle enclosed by the lines y=x, x=0…

A: To find the volume of the given solid.

Q: Let S be a square plate with boundary C, having each side of length 2 units, lying in the plane x =…

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Q: 1. Find the surface area of the cone z = √x² + y² that lies between the plane y = x and the cylinder…

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Q: Use Stoke's theorem to calculate the surface integral || F. dS, where S F(x,y, 2) = (e*siny i+ (e*…

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Q: 5. Use the Divergence Theorem to calculate the surface integral Ss F dS, where F(r, y, 2) = r'i+…

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Q: Evaluate the surface integral fSs x dS, where S is the cylinder y²+z² = 4 between (2) the planes x =…

A: We have to evaluate the surface integral

Q: Jevaluate the surface integral ff, (y²z2 i+ z?x² j+ z²y²k).n ds, where S is the part of the sphere…

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Q: Evaluate the surface integral a dS where S is the entire surface of the solid bounded by the…

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Q: Evaluate the surface integral s F. dS where F (-x, -4z, 4y) and S is the part of the sphere x? + y?…

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Q: 4. Use Stokes' Theorem to evaluate |F• dr when F(x, y, z)= z' i+y° j+xy k. Let S is the part of the…

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