Verify that Strokes' Theorem is true for the given vector field F and surface S. F(x, y, z) = yi + zj + xk, S is the hemisphere  x2 + y2 + z2 = 1, y ≥ 0,  oriented in the direction of the positive y-axis.

Q: Please compute the work done by the vector field F(r, y) = (sin r+ry?)i+(ev+ a)j in R?, where C is…

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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: Find (a) the divergence of the vector field and (b) the curl of the vector field.

A: Given :   To find : a) The divergence of the vector field. b) The curl of the vector field.

Q: Give an example of a vector field F(z, y) in 2-space with the properties that F is perpendicular to…

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Q: Find the curl of the vector field F. F(x, y, z) = (2y - z)i + e²j + xyzk

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Q: Let F = (0, x), a 2D vector field. Compute fF . T ds, where C is the top half of the circle of…

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Q: Evaluate the surface integral F. dS for the given vector field F and the oriented surface S. In…

A: We have F(x,y,z)=yj-zk, S consists of paraboloid y=x2+z2, 0≤y≤1, and the disk x2+z2≤1, y=1. We have…

Q: ompute the flux of the vector field F = ri + yj+ zk through the surface S, which is a closed…

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Q: (b) Verify Green's Theorem for a vector field F(r, y) = cy'i – ya'j on a disc with center (0, 0) and…

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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: Find the flux of the vector field F(x,y,z) = xi + yj+ 2zk across o, where o is the portion of the…

A: Flux of  a vector field F on the surface S   =∫∫SF ds We will use divergence theorem  If surface S…

Q: 3) Given the vector field F = circulation around the triangle formed by the intersection of the…

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Q: Evaluate the surface integral || F. dS for the given vector field F and the oriented surface S. In…

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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: 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: Show that the vector field F (x, y, z) = xz i + xyz j y 2 k is not conservative.

A: Consider the following vector: Fx,y,z=xzi^+xyzj^+y2k^ Find the Curl of F→:…

Q: Find the flow of the vector field F(x, y, z) = zi + yj + xk through the unit sphere x² + y² + z² = 1…

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Q: Use Stoke's Theorem to determine the circulation of the vector field F (x, y, z) =(3y²) î – (cos z)j…

A: Solution:  Application of Stokes Theorem.

Q: Find the curl and divergence of the vector field. F(x,y,z)=x^2yz i -x x^3y k

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Q: Consider the vector field F(x, Y, z) = 3x?yi+ (x³ + 1) j+9z²k. Let C be the circle cut from the…

A: Given vector field is  F(x,y,z)=3x2yi+(x3+1)j+9z2k C be circle cut from sphere x2+y2+z2=8 by the…

Q: Compute the flux of the vector field F = 3xzi + 5yk through the surface S, which is the hemisphere…

A: The flux is given byflux =∫S  F→·n^  dShere,n^=x, y, zx2+y2+z2 in spherical coordinatesF→·n^ =x, y,…

Q: 4. Verify that Stokes' Theorem is true for the given vector field F and surface S. F(x, y, z) = xyi…

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Q: Please compute the work done by the vector field F(x, y) = (sin x+xy²)i+(e" + a2)j in R?, where C is…

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Q: |Let F = (x³ – e" )i+(y³+sin(2))j+(2³ – xy)k be a vector field. Use the Divergence Theorem to…

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Q: Find the flux of the vector field F(x, y, z) =(x)i+(y)j+(z)k through the surface S of the…

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Q: . Verify that Stokes' theorem is true for the vector field F(r, y, z) = (x, y, 2), where S is the…

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Q: Verify Stoke's Theorem for the vector field F = (2y + 2) i + (x –2)ĵ+(y – x) . over the portion of…

A: The formula for stokes theorem is, ∫Pdx+Qdy+Rdz=∬∇×F.ndS Given that, F=(2y+z)i+(x-z)j+(y-x)k Then,…

Q: Consider the vector field F = 2xy i + (x2 + 2y) j. Verify the Green's Theorem in the closed region…

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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: Verify Stokes’ Theorem for the vector field F = 2xyi + xj + ( y + z)k and surface z = 4 - x2 - y2,…

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Q: Let F = x²yi – xy'j+ (z+2)k be a vector field. Use the Diver- gence Theorem to evaluate ſſ, F ·ndS…

A: In this question, we need to find the flux of vector field by using the Gausses divergence theorem.

Q: Let F = x'yi – xy²j+ (z + 2)k be a vector field. Use the Diver- gence Theorem to evaluate ff,F·ndS…

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Q: Verify that Stokes' Theorem is true for the given vector field F and surface S. F(x, y, z) = -yi +…

A: Concepts introduction: Let S be an oriented smooth surface that is bounded by a simple, closed,…

Q: Evaluate the surface integral F. ds for the given vector field F and the oriented surface S. In…

A: The S is composed of the surface in the paraboloid in the form y=g(x,z) f(x,y,z)=y-g(x,z)=y-x2-y2…

Q: Evaluate the surface integral || F·dS for the vector field F(x, y,z)=3yi– 3xj+6zk where S S consists…

A: ∇F=∂ ∂xx2+y2+z2-4i+∂ ∂yx2+y2+z2-4j+∂ ∂zx2+y2+z2-4k=2xi+2yj+2zk

Q: Let S be the surface in R' that is the part of the ellipsoid 2x2 + y² + z? = 2 satisfying y > 1.…

A: In this question, we use the div.  theorem to evaluate the surface integral. given that…

Q: Find the surface integral to evaluate the flux of the vector field F=, where S is the surface of the…

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Q: Consider the vector field F = ⟨ax + by, cx + dy⟩. Show that F has zero circulation on any oriented…

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Q: 3. Find the flux of the vector field F(x,y,z) = xi+ yj + 2zk across o, where o is the portion of the…

A: Find your answer below

Q: Use the Divergence Theorem to find the outward flux of the vector field F(x,y, z) = x³i + y³j+z³k…

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Q: The divergence of a curl of the vector field v (x, y, z) = xyz i+xz j+ z k [i.e. V· (V xv)] is

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Q: Evaluate the surface integral F F. ds for the given vector field F and the oriented surface S. In…

A: To find- Evaluate the surface integral ∬SF·dS for the given vector field F and the oriented surface…

Q: Verify Stoke's theorem for the vector field F = (2x - y) I– yz"J - y z K over the upper half surface…

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Q: Consider the vector field F = 2xy i + (x² + 2y) j. Verify the Green's Theorem in the closed region…

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Q: Evaluate the integral line of vector field F(x,y) -2y + 2x, 2²+ (y²/4) 2.x F(x, y) = + xy + 1 2² +…

A: Given Fx,y=-2yx2+y24+2x,2xx2+y24+xy+1We need to evaluate the line integral over C:x2+y2=1 oriented…

Q: Evaluate the surface integral F. ds for the given vector field F and the oriented surface S. In…

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Q: Use Gauss's Theorem to evaluate fF.ds for the vector field, F = x³ į+ y³ į+z³k, taking S S to be a…

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Q: Calculate the flux of the vector field through the given surface. F = zi + 2xj + 3yk, and S is the…

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Q: Calculate the vector field flow f➝ (x,y,z) = (xy, y2, z) through the outer surface of solid T in the…

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Q: Verify Stokes Theorem for the vector field F(x,y)=(-2yz) i + y j+3x k on the part of the paraboloid…

A: Given F→x,y =-2yzi→+yj→+3x k→  We have to Verify the Stokes theorem for the given vector field on…