Give an example of a vector field F(z, y) in 2-space with the properties that F is perpendicular to Ğ = (z + y) i– (1+z*) jat every point. F(z,y)

Q: Assume that Ax, y, z) is a vector field whose i component is only in terms of y and z, whose j…

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Q: Show that if a vector field has the form F = ⟨ƒ(y, z), g(x, z), h(x, y)⟩, thendiv F = 0.

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Q: Compute the flux of the vector field F 3.ri + 3yj through the cylindrice surface shown below, if a=3…

A: To calculate the flux of the vector field.

Q: A vector field is described as F = (-z + 7x, x³y²z=1,3x + 2z²). (i) Calculate curl F. (ii) Determine…

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Q: Determine whether the vector field F(x, y) = 5y² (yi+2xj) is conservative. F is not conservative F…

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Q: If a vector field F (x, y, z) = (evz-, In (x² z), sin z), which of the following values is equal to…

A: We have to find div F(-1,0,1).

Q: Let F be the vector field (ze"z + yz², xz², xeaz + 2xyz) (a) Find the divergence of F.

A: We will find out the required value.

Q: Determine whether the vector field F(x,y,z)=(y,x,1) is conservative on R° . If so, determine a…

A: Given the vector field Fx,y,z=y,x,1=yi→+xj→+k→.

Q: Compute the curl of the vector field F = - (x + 3y) i + (y – 3z) } – (z + 3x) k curl

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Q: . Consider the vector field F(r, y, 2) = (8r sin y, 4x² cos y – e"z, -e" – sec² z). %3D | F. dR…

A: For a vector field F→, the line integral about the path C is ∫CF→·dr. Let rt be the parametric…

Q: Show that the vector field F = ( -z, 0, x) is orthogonal to the position ----+ vector OP at each…

A: Given data: The vector field in the Cartesian co-ordinates is given by, F=-z i+x k.  Here, I and k…

Q: Show that Gauss's Theorem holds for the vector field F(x, y)=(xy2, -yx2) in a side square 3.

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Q: Consider the vector field F (x, y, z) = (y - zsinx, x, 2z + cosx). The work that performs the F…

A: Given that, F(x,y,z)=y-zsinx, x, 2z+cosxA body is displaced from point A(3π,-1,1) to point Bπ,2,0

Q: (b) Verify Green's Theorem for a vector field F(r, y) = ry²i – yx²j on a disc with center (0,0) and…

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Q: Use the divergence theorem to calculate the flux of the vector field F = x3i - 3x2yj + 6zk through…

A: The divergence theorem is defined by the formula, ϕ=∮∂VF·ndS=∮V∇·FdV Consider the given vector…

Q: 1. Give an example of a vector field F(x, y) in 2-space such that F has constant direction but || F…

A: The solution are next step

Q: Is there a vector field G on R³ such that curl G = (x sin y, cos y, z - xy)? Explain

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Q: Give an example of a vector field F (x, y, z) that has value 0 on a surface and such that curl F is…

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Q: Compute the curl of the vector field F = (xy+ z² ,x², xz – 2). curl(F(x, y, z)) =

A: Please refer the attached image for complete solution.

Q: What is the flux of the vector field (x2,y2,z1 ) through a cuboid with x ranging from 0 to 8.3, y…

A: use divergence theorem

Q: Given the vector field F(x,y,z) = [x, y, -2z] satisfies div(F) = 0. Find a vector field G(x,y,z)…

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Q: Compute the flux of the vector field F 3yi + 4j – 3xzk through the surface S, which is the surface y…

A: We apply formula for finding this

Q: Show that the vector field F(x, y, z) = (2y cos(-3x), –3x sin(2y), 0) is not a gradient vector field…

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

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Q: Find the line integral F.dr of the vector field F=ax] over a curve C defined by a C rectangle in the…

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Q: Find the level surfaces of the scalar field U = ear, where the vectors are: a = (2, 2, 1) and r = =…

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Q: Explain how to compute the divergence of the vector field F = ⟨ƒ, g, h⟩ .

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Q: Compute the flux of the vector field F(x,y, z) = yi– 2xyk over the triangle T with vertices (1,0,0),…

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Q: 3. Find a field F(r,y) (M(r, y), N(r, y)) in the xy-plane with the property that at each point (r.…

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Q: Verify Stoke's theorem for the vector field F = (2x - y) I- yz*J-yzK over the upper half surface of…

A: Stokes theorem is used

Q: . Verify Stoke's theorem for the vector field F = (2x - y) I– yz²J -y z K over the upper half…

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

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Q: The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In…

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Q: Prove that F = (y' cosx+z')î +(2ysin x-4)ĵ+(3xz² +2)k is a conservative force field, find its scalar…

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Q: Let F be the vector field (x², 2r, 2²). Let C be the boundary of Lv **---- the surface S given by…

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Q: Calculate the flux of the vector field F 6i + 2x2j- 2k, through the square of side 4 in the plane y…

A: Given vector field is F→=6i→+2x2j→−2k→ We have to find the flux of the given vector field. The flux…

Q: Explain why Green’s Theorem proves that if gx = ƒy,then the vector field B = ⟨ƒ, g⟩ is conservative.

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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: The vector field F is shown in the xy plane and looks the same in all other horizontal planes. 3- Y…

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Q: Show that Green's Theorem holds for the vector field F(x, y)=(x +y, 3x) at a square of side 2.

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Q: Let C be the curve of intersection of z = 1 – 2x2 and the cylinder x2 + y2 = 1 oriented…

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Q: Calculate F=, around the triangle with vertices: (0,0,2), (2,0,0) and (0,2,0). Can you use Stoke's…

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Q: Sketch the vector field F(x, y) = (-x − y, x - y) at the points (x, y) where x, y = {−1,0,1}.

A: Consider the given vector field, F→x,y=-x-y,x-y Here we need to sketch the vector field at the point…

Q: Give an example of a vector field F (x, y, z) that has value 0 on precisely one line and such that…

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Q: Determine whether the vector field F(x, y) = 5y2(yi + 2xj) conservative. F is conservative Fis not…

A: Just check if curl of F Is zero or not.

Q: : Verify the divergence theorem for A=4xi +(2Y+z Vector field ) | +(52-2x)k in the regim beunded by…

A: According to the given information, it is required to verify the divergence theorem for a vector…

Q: Give an example of a vector field F (x, y, z) that has value 0 at only one point and such that curl…

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Q: a) What is the flux of the constant vector field F = (a, b, c) through any closed surface? b) Is the…

A: Given That : A constant  vector field F=a,b,c and F=x3,y3,z3 To Find : i) The Flux through of…