1: Let Let F = (yz², tan-1(xy), 1 + xy³). %3D a) Show that F is not conservative. b) Show that F cannot be the curl of any vector field.

Q: ector product of two vector fields: V × (F x G) = (V · G)F – (V · F)G + (G · V)F – (F · V)G %3D

A: Part iii Given, The identity of the vector product of two vector fields is proved.

Q: The surface H is defined by (x² + y²)² + z = 1; z ≥ 0. The vector field A(r) is given by A = (x, y,…

A: The surface H  is defined by    .The vector field  is given by   .We have to sketch the surface H.

Q: Find the gradient vector field of f(x, y) = ln(5x + y)

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Q: F(x, y, z) = (x²)Î + (2y³2)ĵ + (32)Â is a vector field. (a) div(F) (b) curl(F) (c) C is the…

A:

Q: ) Let C be the path r(t) = (sin(t), te', cos(t)), 0 <t < . Calculate F. dr.

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Q: Consider F = ⟨x, xy, xyz⟩. Does there exist a vector field G such that curl G = F? Why or why not?

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Q: Let F=(9xy,7y,8z)=(9xy,7y,8z). The curl of F=( Is there a function f such that F=∇f=∇?  (y/n)

A: The given function .We have to find the .

Q: Let F = zi + 2yzj + (x +y2)k be a vector field. Find a function f such that F = ▽f

A: To find out the function f.

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: a) Show that F is conservative while G is not. b) Evaluate GdR, where C3 is defined by R(t) = tî +…

A: This is a problem of vector calculus.

Q: Let F (8xy, 3y, 8z). = The curl of F = (000). Is there a function f such that F = V f? ☐ (y/ (y/n)

A: Step 1: Step 2: Step 3: Step 4:

Q: Let f(x, y, z) = 3xyz + y?z. Find the gradient field Vf.

A: Formulae of gradient field is given below

Q: calculate div(F) and curl(F). F = (xy, yz, y² – x³)

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Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

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

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Q: 1. Consider the function f (x, y, z) = y- + y² +z? Determine the gradient vector field of f, F =Vf…

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Q: Ex: Find the work done by the vector field F = (y-x²)i + (z - y²)j + (x-z²2)k Over the curve r(t) =…

A: Given: F=y-x2i+z-y2j+x-z2krt=ti+t2j+t3k

Q: (a) Consider the vector field G = (xyz, yz, xz). Find div(G). (b) (1,2,3). Using the G from part…

A: We are given the vector field  G=xyz,yz,xzwe have to find the divG , line integral ∫cG.dr for line…

Q: Determine the vactor feld of F. F(z, y) = yi - j

A: This is a problem related to vector field calculation. Based on the general concept we will provide…

Q: Compute the curl of F = (3y2, ay, 22).

A:

Q: (d) Is F a conservative vector field? Justify your answer.

A: given: F→=(zexz+yz2, xz2, xexz+2xyz)