7. Let F = F1 i + F2 j + F3 k be a smooth vector field and let f(x, y, z) be a smooth scalar function. Match the follwing (a) ду dz (b) dy dz (c) dx dy dz F1 F2 F3 (d) Vf (e) V.F (f) V × F with one of (i) divF (ii) curl F (iii) grad f

Q: Show that if E is a finite extension of a field F and [E : F]is a prime number, then E is a simple…

A: Let, α∈E be such that α∉F. As we know that, If E is the finite extension field F and K is finite…

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A: Given:The vector field  is given by .To Find:We have to sketch the given vector field.

Q: Find the curl of the vector fleld F(z, y) = 9ye" i+2ejat the point (-2, 0). Enter an exact answer.…

A: F(x, y)=9y2 ex3 i +2x2 ey jcurl F→=∇×F→curl F→=ijk∂∂x∂∂y∂∂z9y2 ex32x2 ey0         =∂∂x(2x2…

Q: 4. Let ƒ,g be C² scalar fields on R³. Verify that V x (ƒVg) = Vƒ × Vg. V × (ƒVg+gVf). V· (SVg) =…

A: As give by the question There are given that the scalar field is f, g be C2 on ℝ3. And, In the above…

Q: (c) For the vector fields defined in part (b), evaluate separately each term in the iden- tity given…

A:

Q: Let F(z, y) be the vector field plotted below, and let C be the piecewise-linear path from (8, –5)…

A: Draw the linear path from 8,-5 to 8,5 to 10,5, which is given below.

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…

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A:

Q: Sketch the vector field: F(x, y) = (1, 2y) Carefully draw at least four vectors in each of the four…

A:

Q: Determine whether or not F(x,y) = (xy + y²,x² + 2xy) is a conservative vector field.

A: Given vector field is  Fx,y=xy+y2,x2+2xy Fx,y=F1,F2 A vector field is said to be conservative if…

Q: FTTT 10 5 10- -5- -10- X TTTTI 1C d.

A: We draw the sketch of vector field with the help of direction and length of vectors .

Q: Show that the vector field F(x, y, z) = (-ycos(8x), 8x sin(-y), 0) is not a gradient vector field by…

A:

Q: Determine the vector field of F. F(x, y) = yi - xj

A: Given vector field: Fx,y=yi-xj To sketch: The given vector field and determine which is correct…

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…

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Q: Determine the vector field of F. F(x, y) = 81-4)

A: The given vector is F=8i-4j

Q: Sketch the vector field of F = (-y,0) in the xy-plane. %3D

A:

Q: Find the curl of the vector field F =< yx5, xz², zyб curl F = i + 1:0 12

A: