Find Values of a,b for F(x,y,z) = 0 for All (x,y,z)eR^3

[savageheart]

F(x,y,z)=y^2z(x hat) + axyz (y hat) + (bxy^2 +4z) (zhat)
(a) Find the values of the constants a and b such that \nabla x F(x,y,z) = 0 for all (x,y,z)eR^3

 

[jvicens]

To find the values of constants a and b, we need to take the partial derivatives of F(x,y,z) with respect to x, y, and z and set them equal to 0.

∂F/∂x = y^2z + aby^2 = 0
∂F/∂y = 2yz(x hat) + axz + bxy^2 = 0
∂F/∂z = y^2(x hat) + axy + 4(zhat) = 0

From the first equation, we can solve for a: a = -1/z
Substituting this into the second equation, we get:
2yz(x hat) - (x/z)xy^2 = 0
Simplifying, we get:
2y(x hat) - xy = 0
Solving for y, we get y = 0 or 1/2.
If y = 0, then the third equation becomes:
axy + 4(zhat) = 0
Solving for x, we get a = 0.
If y = 1/2, then the third equation becomes:
(x hat) + (1/2)ax + 4(zhat) = 0
Solving for x, we get a = -8.

Therefore, the values of constants a and b that satisfy ∇xF(x,y,z) = 0 for all (x,y,z) in R^3 are a = 0 and b = -1/z or a = -8 and b = -1/z.