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

Q: Find the curl of the vector field at the given point. F(x, y, z) = x²zi – 2xzj + yzk; (7, –9, 5)

A: Given vector field F(x,y,z) =x2 z i -2xz j +yz k   ; (7 , -9 , 5)

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

A:

Q: What is the flux of the vector field F(x, y, z) = [9y, 2xy, 4yz + 234xy] through the unit cube [0,1]…

A: We find the flux of the vector field F(x, y, z) = [9y, 2xy, 4yz + 234xy] through the unit cube [0,1]…

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

A:

Q: Find the work done by vector field F(x, y, z) = xi + yj + (x + z)k on a particle moving along a line…

A:

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

A:

Q: : Find the divergence of the vector field At the point (2,4,1). F(x, y, z) = x² √yi+j+zk

A: We have to find the divergence of the vector field F(x, y, z)=x2y i^ +yxj^+zk^ at the point (2, 4,…

Q: Find the curl of the vector field at the given point. F(x, y, z) = x2zi – 2xzj + yzk; (9, -9, 5)

A:

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

A:

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

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

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: Find the gradient vector field of f(x,y) = x2 + y2 and sketch it.

A: Solution

Q: How do you determine whether a vector field in ℝ2 is conservative(has a potential function w such…

A:

Q: Determine the line integral if the vector field is F(x, y) = x²yî – 2xyf On a rectilinear path that…

A:

Q: (a) Show that the vector field F (x, y, z) = (x² + y², x, z³) is not conservative.

A:

Q: Find the conservative vector field for the potential function by finding its F(x, y) = x² + 5y2

A:

Q: 2. Show that V(V × F) = 0, here vector field is twice differentiable. %3D

A: This question is from vectors.

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: Determine if the vector field F=(yz,xz,xy) is conservative?

A:

Q: 13

A: We have to find out conservative or not

Q: Is there a vector field Ğ on R³ such that curl Ġ = (xyz,–y*z³,y*z*)?

A: the given function is, G→=xyzi^-y4z3j^+y3z4k^ If there exists a vector field G then the divergence…

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: Determine if the vector field F is conservative. F= cos x cos yi + sin x sin yj - sec |- sec zk O…

A:

Q: Find the divergence of the vector field F = < yx², xzª, zy³ div F =

A: f = y x2  ∂f/ ∂x = 2xy   g = x z4  ∂g/ ∂y = 0   h = z y3  ∂h/ ∂z = y3

Q: Apply fundamental theorem for conservative vector field to find the line integral fF dr where F =…

A:

Q: the gradient vector field of f(z, y) = In(x + 4y) %3D Work

A:

Q: Find curl F for the vector field at the point (9, -9, 5). F(x,y,z) = x²zi - 2xzj + yzk curl F =

A:

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

A:

Q: Find the curl of the vector field F. F(x, Y, 2) = arcsin yi+ V1 - xzj + yêu y²k

A:

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: Find the curl of the vector field at the given point. F(x, y, z) = x²zi – 2xzj + yzk; (9, -9, 9)

A:

Q: Compute the work done by the vector field F(x, y) = (ry, x² + y³) when moving a particle…

A:

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

A:

Q: Let be the vector field defined on the plane by F(x, y) = (x¹+3y, 6x - siny).

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: Determine if the vector field F(x,y,z)= (yz,xz+2y,xy+1) is conservative. If it is, find its…

A:

Q: xi + yj F(x, y) = k- x² + y2'

A:

Q: Find the curl of the vector field at the given point. F(x, y, z) = x²zi – 2xzj + yzk; (9, -9, 7)

A: The curl of a vector field F is defined as  ∇×F=i^j^k^∂∂x∂∂y∂∂zf1f2f3

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

A:

Q: The divergence of a vector field F = 3x² yz? î + 2yz ĵ – xz² k at he point (1,0, 1) is -2 O 2 O 1

A:

Q: Find the curl of the vector field F(x,y,z)

A: The given vector field F(x,y,z)=yex, 2x+3y, ycosx we have to find curl F(x,y,z)=?

Q: Suppose a vector field F(r, y) = construct a potential function o(r, y). i+ j is conservative. Then

A:

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

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: 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: Find the curl of the vector field F. F(x, y, z) = arcsin yi + V1 - x2j + y2k

A: To find the curl of the given vector field F

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…

Q: Find the curl of the vector field at the given point. F(x, y, z) = x²zi - 2xzj+yzk; (9,-9, 9) curl F…

A: