Assume that Ax, y, z) is a vector field whose i component is only in terms of y and z, whose j component is only in terms of x and z, and whose k component is only in terms of x and y. That is, Ax, y, z) = f(y, z)i + g(x, z)j+ h(x, y)k. Explain why div F= 0.

Q: Show that the vector field F(x, y, z) = xz i + xyz j - y 2 k can’t be written as the curl of another…

A:

Q: If ϕ=3xy+2x2yz−12xz3ϕ=3xy+2x2yz−12xz3, Find ∇×(∇ϕ)

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

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: A vector field in space is conservative. Is the vector field irrotational? Explain.

A: Query-  A vector field in space is conservative. Is the vector field irrotational? Explain. Answer-…

Q: Use the Divergence Theorem to compute the flux across the sphere x2+y2+z2 = 1 within the vector…

A: The answer is as follows:

Q: Give an example of a vector field F(z, y) in 2-space with the properties that F is perpendicular to…

A:

Q: Find the curl for the vector field at the given point. F(x, y, z)= 6xyzi+6yj+6zk (6, 7, 6) O…

A: To find the curl for the vector field  at the given point.Fx, y, z=6xyzi+6yj+6zkPoint=6, 7, 6

Q: Vector fields V and W are defined by V- (2х — Зу + z, -3х — у + 4z,4y + z) W- (2x - 4y — 5z, - 4x +…

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: Verify that the work performed along the segment P Q by the constant ~ vector field F = (2, -1, 4)…

A: We have given; vector field F = (2, -1, 4) (a) P = (0, 0, 0), Q = (4 , 3, 5) (b) P = (3, 2, 3), Q =…

Q: Let F(x,y,z)=(−6xz2,6xyz,6xy3z)F(x,y,z)=(−6xz2,6xyz,6xy3z) be a vector field and…

A: Given, Fx, y, z=-6xz2, 6xyz, 6xy3zfx, y, z=x3y2z

Q: Find the curl of the vector field F→(x,y,z)=e^xsinyi→−e^xcosyj→+xyk→at the point (10,9,24).…

A:

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: Sketch the vector field F(x, y) = xj, the line segment from (1, 4) to (7,4), and the line egment…

A:

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

A: This question is from vectors.

Q: Compute the flux of the vector field (x, - xy*), out of the rectangle with vertices (0,0), (5,0),…

A:

Q: Write a formula for a two-dimensional vector field which has all vectors of length 1 and is…

A:

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

A:

Q: 1. Give an example of a vector field F(x, y) in 2-space such that F has constant direction but || F…

A: The solution are next step

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: There is a vector field F(x, y, z) = (M (x, y, z), N (x, y, z), R (x, y, z)) , such that divF (x, y,…

A: We need to determine the validity of the statement: divF=div(curlF) for a vector field of the form:…

Q: Compute the flux of the vector field F = 3xzi + 5yk through the surface S, which is the hemisphere…

A: The flux is given byflux =∫S  F→·n^  dShere,n^=x, y, zx2+y2+z2 in spherical coordinatesF→·n^ =x, y,…

Q: Show that the vector field F = 2xyi + (x² – 2 y)j is conservative and evaluate the line integral…

A:

Q: Compute the flux of the vector field (x2, - xy*), out of the rectangle with vertices (0,0), (4,0),…

A: Given,

Q: Compute the flux of the vector field (x, (æ°, – ay*), out of the rectangle with vertices (0,0),…

A: To compute the flux of the vector field x3,-xy4, out of the rectanglewith vertices 0,0,2,0,2,4…

Q: If Vis a vector field, prove that: V × (V × V) = (V · V) – V²V.

A:

Q: Find work done by a force field F= in moving object along the line segment from the point (2,1,3) to…

A: Given, F=1y-x,xyz,ex2+y2+z2

Q: Given the vector field F(x,y,z) = [x, y, -2z] satisfies div(F) = 0. Find a vector field G(x,y,z)…

A:

Q: Show that F = (3, 1, 2) is conservative. Then prove more generally that any constant vector field F…

A: Given: F = (3, 1, 2) To show: F is conservative. Also to show any constant vector field F = (a, b,…

Q: Determine whether or not the vector field is conservative F(x,y) = 18x²y²i + 12x³yj conservative O…

A: NOTE: Refresh your page if you can't see any equations. . compare this vector field with  so here…

Q: Find the level surfaces of the scalar field U = ear, where the vectors are: a = (2, 2, 1) and r = =…

A:

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

A:

Q: let c is a differentiable vector field then div(curl)=

A: This question is related to vectors.

Q: Compute the flux of the vector field (x3, – xy³), - out of the rectangle with vertices (0,0), (3,0),…

A: Given: F=x3 ,-xy5with rectangle with vertices (0,0) , (3,0) , (3 ,2) , (0,2) To find flux of vector…

Q: The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In…

A:

Q: Let F(x,y,z)=(2xz^2,−2xyz,9xy^3z) be a vector field and f(x,y,z)=x^3y^2z (∇f=(  ,  , )). (∇×F=(  ,…

A: Given:F(x,y,z)=2xz2,-2xyz,9xy3zf(x,y,z)=x3y2z

Q: Write a formula for a two-dimensional vector field which has all vectors parallel to the y-axis and…

A:

Q: Prove that F = (y' cosx+z')î +(2ysin x-4)ĵ+(3xz² +2)k is a conservative force field, find its scalar…

A:

Q: There is a vector field F (, y, z) such that divF (x, y, z) = curlF (x,y, z). (M (x, y, z) , N (x,…

A:

Q: Find a vector field with twice-differentiable components whose curl is x i + yj + z k or prove that…

A:

Q: Sketch the vector field F(x, y) = xj, the line segment from (1, 9) to (6,9), and the line segment…

A:

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

A:

Q: If the vector field A = âx3x² + âyy – â̟5z³ is given, express à in cylindrical and spherical…

A:

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: Give an example of a vector field F (x, y, z) that has value 0 on precisely one line and such that…

A:

Q: There is a vector field F (x, y, z) such that curlF (x, y, z) = (divF (x, y, z) , (divF (x, y, z))²,…

A:

Q: Find the flux of the vector field F(x, y, z) = [x4, −4yx3, 7z + sin(xey))] through the outwards…

A: