Show that the force field F(x, y, z) = (81xz)i + (1 −6yz³)j + (42²-9y²22) k is conservative and find its potential function. Hence find the work done by F(x, y, z) in moving an object from (1,4,0) to (3, 4, 1) and then to (2,0,3). If C is the boundary of the region bounded by the curves y 1² and y = 1+2, use Green's theorem to evaluate (2y + e√²) dr + (3x + cos (y²)) dy.

Q: Verify that f (x, y, z) = xy + yz 2 is a potential function for the vector field F = (y, x + z2,…

A: Given: Vector field F→=y , x+z2, 2yz To verify that fx , y, z=xy+zy2 is the potential function.

Q: Given two points A(3, 0) and B(), find the work done by the following force field in moving a…

A:

Q: Find the work done by the force field F(x, y) =(2y'+2xy, 6xy+x +1) in moving a particle over the…

A:

Q: Find the work done by the force field F on a particle moving along the given path. F(x, y, z) = yzi…

A:

Q: y' = y*. (a) Find an implicit expression for all solutions y of the differential equation above, in…

A:

Q: Consider a force in R³ defined by F(x, y, z) = (3x²y³,3 x³y² + 2y, 0). Find the potential energy…

A:

Q: Consider a force in R³ defined by F(x, y, z) = (6x²y³,6x³y2 + 2y,0). Find the potential energy U(x,…

A:

Q: 2. Given vector field F= (-2y, z-2x, y + z), find the work done by ♬ on a particle that moves along…

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: Find the work done by the force field F in moving an object from A to B. F(x, y) = 6y³/2 i + 9x√yj…

A: