5. Is the vector field F(x, y, z) = (x(z² – y²), y(x² – z²), z(y² – x²)) incompressible? Justify your answer. %3D

Q: The gradient vector field of for,y)= In +2y) is 2. T True A False

A: The gradient vector field of a function, is defined by the formula below. , where are the partial…

Q: Find the divergence of the vector field F <7", 22°, zy* div F -

A:

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: (1) Is the vector field F(x, y) = (2æ*y^ + æ) î + (2a*y³ + y) Î conservative? Support your answer…

A: Note: As per our guidelines we are supposed to answer only one question. Kindly repost other…

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: Q3) Compute the work done by the force field F(x,y,z) = (3x, xz, y) on the curve C, where C is the…

A:

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

A:

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

A:

Q: Show that the vector field F = (x² cos y, e+y) is not a gradient vector field. %3D

A:

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

A:

Q: Calculate the divergence of the vector field F=x²y³z¹i-xyzj+(x+y+z) k

A:

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

A:

Q: 5. Is the following vector field F(r, y) = (2x – y, -x + 3y?) conservative? If so, find a potential…

A:

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

A:

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: A) Consider the vector-field F(x, y) = 2xy³i+ 3y²x²j. 1. Show that F is conservative. 2. Find a…

A: The given vector field is  F⇀(x,y)=2xy3i⇀+3y2x2j→ We have to show that F→ is conservative

Q: 8. Show that the vector field, -y a² + y?' x² + yj²' ) is not conservative in R³, despite the fact…

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: Compute the line integral of the vector field F = (3zy1, 4z, -y) over the path c(t) = (e*, e', t)…

A:

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = In(4x + y + 62) . F(1, y, 2) =

A: we have to find the gradient vector field of f

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

A:

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

A:

Q: (c) Determine the flux of the vector field F = x³i – 2yĵ + 7z²k across a cone z = x2 + y2 + 3 that…

A:

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: 1. Determine whether F(x, y) et sin yi + e" cos syj is a conservative vector field.

A:

Q: a) Determine whether or not the vector field F 8yz, – 6 cos(y) – 8xz, – 8xy + 10z > - is…

A:

Q: Find the divergence of the vector field F. F(x, y, 2) = In(9x+ 8y² )i + 72xyj + In(8y² + 5z? )k #6…

A: Solution is done in step 2 and 3

Q: 4. Show, in detail, that any vector field of the form (*,J, 2) = J (*) +8(y)/+h(z)k is irrotational,…

A: To show that the vector field is irrotational.

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

A:

Q: Show that the following vector field is conservative and find the potential F(r, y, z) = (2ry, x² +…

A: The answer is given as follows.

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: Find the divergence of the vector field F(x, y, z) = In(x² + y? )i+xyj+ln(y² + 2²)k. 2x 2z +x + x2 +…

A:

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

A:

Q: Find the divergence of the vector field F. F(x, y, z) = In 5x²+ + 10xyj + In( 2y2 + 722)k div F(x,…

A:

Q: 1. Is the vector field V (x, y, z) = (2ry, x² – 2², –2yz) conservative?

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: Show that the vector field F(x, Y, z) = (x² + y²)i – 2xyj is not conservative on R³.

A:

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

A:

Q: Show that the vector field F(x, y)= (12x² +3y² +5y)i+(6xy– 3y² +5.x)j is a conservative vector field…

A:

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

A:

Q: 3. Determine if the vector field F(x, y, z) = e" sin yzi + ze² cos yzj + ye² cos yzk is conservative…

A:

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: Compute the flux of the vector field F = 2xi + 3yj through the cylindrical surface shown below, if a…

A: We have to compute the flux of the  given vector field.

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

A:

Q: Compute Sc F.d3 for the vedor field in the Cylinder with out ward facing and the surface which…

A: It is given that F→=xi+yj+ezk and the surface is x2+y2=4,1≤z≤5