10. For a vector field F = <2xy + y³, x² + 3xy²+ 2y7 find a function f such that F = Vr.

Q: 4. Is this vector field F(x,y)=(ycos x)ĩ +(sin x+ y)j being the path-independent field? Please show.

A: To check if the given vector field is path-independent field or not. Given:

Q: When is the vector field continuous?

A: given,       Continuous vector field .

Q: Show that work performed by a constant force field F over any path C from P to Q is equal to F - PÓ.

A: A vector has 2 components, magnitude and direction which is represented by cos and sin angles. A…

Q: 4. Is the vector field F(r, y, :) = (yz,rz, ry) conservative? Justify your answer.

A:

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

A:

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: 41.) Use given grid to sketch the vector field F = (y, -x). Include vectors at all integers points…

A: Given vector field is F=12y,-x

Q: (b) Suppose C is a curve of length N, and ||F|| < M, where M is some positive number. Prove that ·…

A:

Q: What is the value of a so that the vector field F(x.y)=(In y+2xy³) i+(ax²y2+ x/y) j is conservative?…

A: To find: the value of a for which given vector field is conservative.

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

A:

Q: Find the gradient vector field of the function f. f(x, у, 2) 3 3x соs Vf(x, у, 2) %3D

A:

Q: 3) Sketch the vector field F. F(1, y, 2) = yi

A: F(x,y,z)=yi

Q: 13 Is this vector field F(x, y)=(ycosx)i +(sin x+ y)j being conservative? Please show.

A:

Q: 5. Compute div F and curl F if F = (2x – cos y)i – x²e32j + (z² – 7x)k. 6. Determine if the vector…

A:

Q: 9. Sketch the vector field F(x, y) : :(yi — х). Vx? + y?

A: Consider the given filed, Fx,y=1x2+y2yi−xj Further simplify. Fx,y=yx2+y2i-xx2+y2i=〈yx2+y2,-xx2+y2〉

Q: 3. Use the fact that F = is a conservative vector field to evaluate SF. dr when C is the path r(t)…

A:

Q: Find the divergence of the vector field F = div F =

A:

Q: 4. Is the vector field F(x, y, z) = (yz, xz, xy) conservative? Justify your answer.

A: Calculate divergence of given vector field. If divergence is zero then field is conservative.

Q: Find the divergence of the vector field F = div F

A:

Q: 5. Suppose that the vector field F = is conservative. Find a potential function of the vector field…

A:

Q: 14. Find the curl of the vector field F = x²i – xevj+ 2xyzk

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Q: 3. Let F be the vector field (ze²z + yz², xz², xe*z + 2xyz)

A:

Q: Q5) If the vector field T = (axy + Bz³)a, + (3x² – yz)a, + (3xz² – y)a, is irrotational, determine…

A:

Q: compute dr for the vector field where Cis the line segment from Co.o) to Cli3)

A: Given vector field is, F=1+x2,xy2  The objective is to compute ∫CF·dr, where C is the line segment…

Q: Determine whether the vector field F(x, y) = conservative. yi + xj 1 + xy is

A:

Q: (d) For any vector field F, show that div(curl F) = 0.

A: Note: As per our company guidelines we are supposed to answer only one question at a time. Kindly…

Q: 6. What is the corresponding vector fieldV for the scalar potential p(x, y)= In(x+ ¥)?

A:

Q: Evaluate both sides for the field- rectangular parallelepiped X = 0 and 1 Y=0 and 2 and Z=0 and 3.…

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Q: 4. Show the vector field Vu x Vv, where u and v are scalar fields, is solenoidal.

A:

Q: Find potential 9, for the conservati ve field the vector F (x, y,a)- (2xe? - y*sinx) î * sinx) î +…

A:

Q: Find the divergence of the vector field F div F =

A:

Q: Consider the vector field F(x, y) = (x², −y). If (x, y) is in the first quadrant then the vector…

A: Part 1. (x, y) is in the first quadrant.  =&gt; x &gt; 0 and y &gt; 0 =&gt; x2 &gt; 0 and - y &lt; 0…

Q: 2. Sketch the vector field F(x, y, z) = xi + yj + zk.

A:

Q: Select the irrotational vector field. OAyz 1-2xz1 + xyz k OB. yz 7 + xz 7 + xz R OCZ7 + z? Î + yz R…

A: We will find out the required value.

Q: 2. Find the divergence of the vector field F div F = %3D

A:

Q: 5. Is the vector field F(x, y, z) = (x(z² – y²), y(x² – z²), z(y² – x²)) incompressible? Justify…

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Q: i) Show that F is a conservative vector field.

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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: Which graph represents the gradient vector field of f(z,9) = Sz + y

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Q: Find a vector potential field for Ġ = [6xz+x³, –3r²y – y², 4.x + 2zy – 32²], i.e. find F such that V…

A: The solution is below.

Q: 3. Show that the vector field F = -C with C> 0, is conservative and find its potential.

A:

Q: the vector field F is shown in the xy plane. y 21 2 Is div F positive, negative, or zero? Positive…

A: Given : The vector field F is shown in the xy plane :   We have to determine that : Is div F…

Q: Let F(x, y, z) =⟨3x2y+az, x3,3x+ 3z2⟩ be a vector field. For what values of a is F conservative?

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Q: Find the divergence of the vector field F %3D div F =

A:

Q: B) Find the work done by the conservative vector field F (x, y, z) = yzi+xzj+xyk along a smooth…

A: Since, it is given that the field is conservative.

Q: Choose the option in which the vector field is conservative. Select one: F =(xy,x+y> F = (2xy² -…

A:

Q: ermine whether the vector field F(x, y, z) = (x³, 3zy², y³) is conservative on R³. If so, find a…

A: