a) For the given a vector field A = 3x*yz a, + x*z a,+ (x*y – 2z) a, check whether given vector 'A' is Irrotational or Solenoidal? b) Show that the vector A= xza, + z?a, + yzn, satisfies the vector identity VX (V XA) = v (V.A) – v²A. %3D

Q: If the vector field v = [mz+x, -y, x]is irrotational, the mis equal to O A. 2 O B. 0 O C. 1 O D.…

A: Curl of v is zero.

Q: Find the flux of the vector field V(x, y, z) = 3xy²i + 5x²yj + 2z³k out of the unit sphere. Flux =

A:

Q: For the vector field F(x,y,z) =  2yi +xyj + (2x-xz)k. (i) Show that  (ii) Determine whether if the…

A:

Q: 4. Consider the vector field given in Cartesian coordinates by A(r) = (x² + y³, 3xy², z²). (a)…

A: Given : Ar=x2+y2,3xy2,z2

Q: If F (x, y) conservative = axi + byj is a vector field, then a = = b

A: It is a vector calculus problem and let us determine the curl of the given vector field  and also…

Q: Given a vector field F = 5(x² + y²)i+ 2xyj - 3zk, evaluate SSS, V· FdV for which the solid D is…

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: 7. Determine if the given vector field F is conservative on some domain and, if it is, find the…

A:

Q: (b) Verify Green's Theorem for a vector field F(r, y) = ry²i – yx²j on a disc with center (0,0) and…

A:

Q: Evaluate , (V. A) dV for the vector field A = x²yi +xz³j- y²z²kwithin the volume of a cube defined…

A:

Q: Given the 2-D vector field : G(x, y) = (2y)î + (x)j and the triangular curve T : line segments from…

A: Given: G→x,y=2yi^+xj^ Triangular curve T, line segments from point A0,0 to point B4,0 to point D4,2…

Q: Is the vector field F = y? – 2x², 2xy + 3| conserva Yes No yz Is the vector field F y? – x2 con Vy?…

A: According to the guidelines issued by the company we are supposed to answer only first question.…

Q: (b) Given a vector field F = 5(x² + y²)i+ 2xyj – 3zk, evaluate JI, V· FdV for which the solid D is…

A: Here we use polar form of circle .

Q: 2. Show that any vector field of the form F(r, y, z) = f(y, z)i + g(x, z)j+ h(x, y)k is…

A:

Q: (2) For a differentiable vector field f, which of the following is (are) incorrect? (a) ▼ × (▼ × ƒ)…

A: Since we know Formula a→×(b→×c→)=(a→·c→)b→-(a→·b→)c→

Q: A vector field is given as G = (1/(√x² + xy))(e¯²ax +xzay), find the unit vector which is normal to…

A: The given vector fields are: G=1x2+xye-zax+xzayP=3x2ax-4yay We have to find the unit vector which is…

Q: For the vector field F = ((y/z2+1)i + (-2x/z2+1)j + (3x2+3y2/(z2+1)2)k , calculate curlF

A: We have given vector field F=yz2+1i+1-2xz2j+3x2+3y2z2+12k=yz2+1,1-2xz2,3x2+3y2z2+12=F1,F2,F3…

Q: a) Find the curl and the divergence of the vector field: F = (3x + 2z?)i + (*)j- (2- 7x)R.

A:

Q: Determine whether the vector field F(x, y, z) = (2x³ — 2x) i + (2y³ – 2y) j + (2z³ – 2z) k - is free…

A:

Q: If a vector field F (x, y, z) = (y² + z², xe cos z, - xe³ cos z), which of the following values is…

A: Given vector field is Fx, y, z=<y2+z2, xey cosz, -xey cosz>.   To Find: div F.

Q: b) Evaluate both sides of the divergence theorem for the vector field D = 2xyx + x²9 [C/m²] and a…

A:

Q: Given the vector field F (x, y, z) = x²i + 2xy²j¬ +x²zk> a. Find the divergence of the vector field…

A: F→ = x2i→ + 2xy2j→ + x2zK→divergence :∇.F = ddxx2 + ddy(2xy2) + ddz(x2z) ∇.F = 2x + 4xy + x2

Q: Determine the divergence of these vector fields: (a) P = xʻyz a, + xz a, (b) Q3D p sin фа, + p'za,…

A: The given problem is to find the divergence of the given vector fields, We have to use Cartesian ,…

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

A:

Q: Compute w(v), where w =(x-y)dx+(x+y)dy is a covector field and v = (x/y)d/dx-(y/x)d/dy is a vector…

A: Given: w =(x-y)dx+(x+y)dy  v = (x/y)d/dx-(y/x)d/dy

Q: Is the vector field F (x, y) = (2xy – y³) î + (x² – 3xy²) ĵ solenoidal, conservative, both or…

A:

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

A:

Q: Let Σ be the sphere x2`y2`z2 “ 25 with outward orientation and let F~ be a vector field with divF~ “…

A:

Q: 1 Find the divergence of the vector field F -(xi + yj+ zk) Vx? +y? +Z 2 x² + y² + z? 1 x² + y? + z?…

A: If the vector field is given by F=F1i+F2j+F3k then the divergence of F is ∇·F=∂F1∂x+∂F2∂y+∂F3∂z.

Q: 2x 2y Consider the vector field F = |x² + y2’ x² + y² (a) If C1 is the unit circle x² + y² = 1,…

A: Compute the circulation and flux using parameterization:  ∮C1Fdr=∮C12xx2+y2,2yx2+y2·dx,dy…

Q: A constant vector field F(r, y, z) = ai + bj + ck is a gradient vector field. True

A:

Q: Compute the flux of the vector field i = -2i+ j- 2 k through the rectangular region shown below,…

A:

Q: 3 Given the vector field F(x, y, z) = 2xy²i+3xz° j+ y²zk A) Find and simplify the divergence of the…

A:

Q: Determine whether or not the vector field F(x, y, z) = is conservative, and if so, find a function…

A: Given the given vector field is Fx,y,z=-xyex, -xex+ex+2y, 2z The vector field is conservative if…

Q: Is it true or false that the vector field below is conservative? -y -X -Z F(x, y, z) = (x² + y? +…

A: Given,                              =-(xi+yj+zk)(x2+y2+z2)32

Q: Determine if the vector field F = y ln zi – x ln zj + xyk is a conservative vector field.

A:

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…

Q: Find the divergence of the vector field F(x, y, z) = In(x² + y?)i+ cyj+ ln(y² + z² )k 2x + x + 2z x2…

A:

Q: What is the value of a so that the vector field F(xy)=(In y+2xy³) i+(ax3y2+ x/y) j is conservative?…

A:

Q: Compute the flux of the vector field F⃗ =xi⃗ +3yj⃗ F→=xi→+3yj→ through the cylindrical surface shown…

A: Given that the vector field is F(x,y)=xi+3yj Let us parametrize the cylinder as x=cosθ, y=sinθ, z=r…

Q: 1. An illustration of the vector field F(r, y) = j is given, (a) Find paths C, Ca, and C, from P to…

A: The general form of a vector-valued function in three dimensions is given by F→=F1i→+F2j→+F3k→,…

Q: For the vector field F(x,y,z) =  2yi +xyj + (2x-xz)k. (i) Show that  (ii) Determine whether if the…

A: Given that vector field is  Fx, y, z=2yi+xyj+2x-xzk We have to prove that (1) 12∇·F=div∇×F Consider…

Q: Find the divergence of the vector field V(x, y, z) = -2e*'i – e³xyj+ 7e%yzk.

A:

Q: Show that the vector field A (3x?z + y?)a,+ 2xya, + x³a, is conservative. %3D

A: Given problem:- Show that the vector field A = (3x²z+ y²)ax + 2xyay + x³az is conservative. =

Q: 5. If F = A × OP, where A is a constant vector field, show that V x F = 2A 6. If R = OP,r = ||R||,…

A: The solution are next step

Q: Given the vector field F(x,y) = [x/((x^+y^2)^(3/2)), y/((x^+y^2)^(3/2))], Find a fucntion f(x,y)…

A:

Q: If a vector field A(x,y,z)=xz3ex - 2x2yzey+2yz4ez , calculate the divergence ∇·A at point…

A: A vector field is given as  and

Q: Find the divergence of the vector field F. F(x, y, 2) = In(7x2+ 6y? )i + 42xyj + In(6y2. + 72² )«…

A: