Find an example of a vector field F(x, y, z) = (F1(x, y, z), F2(x, y, z), F3(x, y, z)) defined and smooth on all of R³ but not conservative, but satisfies ƏF3 dF3 and dz he dz

Q: Consider the vector field F(x, y, z) = (3z + 4y)i + (3z + 4x)j + (3y + 3x)k a) Find a function f…

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Q: Find curl(curlF→(x,y,z)) of the vector field F→(x,y,z)=x^9zi→−9x^8zj→+yzk→. Enter the exact answer.…

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Q: & b. su j+ byz²

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Q: Show that vector field F(x, y, z)=yzi+xzj + (xy + 2z)k, is conservative, then evaluate fF.dr, where…

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Q: Let + y³)%¸ æyz 7 4 F = x² + be a vector field on U = {(x, y, z) E R° | (x, y) + (0, 0)}. Find its…

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Q: Suppose F = (P(x, y, z), Q(x, y, z,), R(x, y, z)), is conservative vector field. Here P, Q, R are…

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Q: Exercise * F (x₁y₁ z) = (ay³ - 2 z)i + (2xy² + 4yz)j + (bx - cy²) K a) find a, b and c which makes…

A: (a) To find: Value of a,b, and c such that the vector filed F=ay3-2zi+2xy2+4yzj+(bx-cy2)k  is…

Q: Let F and G be vector fields with differentiable components. Express curl (F × G) in term of div and…

A: Let F and G be vector fields with differentiable components.Let We will compute .The cross product…

Q: Problem #6: Consider the vector field F(x, y) = (Axy+ 4xy + By)i+ (x* + 2x2 + 8xy)j (a) Find the…

A: A vector field is said to be conservative if it is irrotational. Also, if a vector field F→ is…

Q: Let F(x, y, z) be the vector field (x²(1 + y²z²), Q(x, y, z), exy cos(2z)), where is a…

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Q: Which of these vector fields are irrotational? Identify all of those that satisfy this property. A B…

A: The given vector fields, A) u=-x, y, 0B) u=-y, x, 0C) u=-x, z, 0D) u=x, y, z We have to find the…

Q: Find the divergence of the vector field At the point (2,4,1). F(x, y, z)=x²2√√yi + -j + zk

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Q: HIf vectors x, y, z, u in the real vector space p (the polynomials) are defined by x(t)=1, y (t)=t,…

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Q: Sketch the vector field on given byF(x, y, z )= z K

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Q: What is the curl of the vector field F(z, y, z) = ryi+ zyzj + yzk O a. (-ry + z)i + (yz – 2)k O b.…

A: We have to find

Q: Find the potential function for the vector field given as 2 F(x.y.?) = Oa. $(x, y, z) =x*y + x²2?…

A: Answer:   The correct option is B.

Q: Let F = be a vector field in space Find the potential function f(x,y,z) of F

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Q: (4) Find a function f such that F = Vf. Consider the vector field F(x, y, z) = y cos(xy)i+(x cos(xy)…

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Q: What is the divergence of the vector field F of F(x,y) = x² i + 2y²j?2 div F = 2x + 4y div F = 4x +…

A: Given: F→=x2i+2y2j

Q: Let F₁ (x, y, z) = (y, x-z²+2, -2yz) and F₂(x, y, z) = (x − z², -y +2, 2yz). (a) Find the curl of…

A: Given vectors  and  we need to find the curl of each vectors and then determine which vector is…

Q: Verify Gauss's theorem for the vector field = y²i+xz³j+(2-1)³ and it is the enclosure bounded by the…

A: Divergence Theorem:The divergence theorem relates the flux of a vector field over a closed smooth…

Q: Compute the curl of the vector field F = (xy+ z²,x², xz – 2 curl(F(x, y, z)) = %3D What is the curl…

A: A vector field is irrotational if it is zero at a point.

Q: Use Green’s Theorem to evaluate the work from the vector field F(x,y) = 〈-4x2+y2, 2x2+y〉along the…

A: Step 1:

Q: Bonus Problem: Evaluate | y'dx + xydy where C is the boundary of the region bounded by y = Vx,y =0,…

A: We have to evaluate ∫Cy2dx+xydy, where C is the boundary of the region bounded by y=x, y=0 and x=9…

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = z² sin(xy). F(x, y, z) = (

A: Given: The function .To find the gradient vector field  of the given function. Concepts Used:The…

Q: f(x,y) = x y

A: In this question, we find the volume by using double integral by changing into a polar coordinate.…

Q: Verify Green 's theorem for the vector field  F (x, y) = (x^2y, -xy^2) in the triangle OAB defined…

A: The solution  is given by using the line integral of vector field and verify the greens theorem as…