Question

Consider the vector field F(x,y,z)=(−4y,−4x,5z). Show that F is a gradient vector field F=∇V by determining the function V which satisfies V(0,0,0)=0 V(x,y,z)=

Answer 1

To show the vector field F(x,y,z)=(-4y,-4x,5z) is a gradient field, we find the scalar potential function V(x,y,z) by integrating the components, resulting in V(x,y,z) = -4xy + (5/2)z^2, which satisfies the requirements.

Step-by-step explanation:

To show that the vector field F(x,y,z)=(−4y,−4x,5z) is a gradient vector field F=∇V, we need to find a scalar potential function V(x,y,z) such that F is the gradient of V and V(0,0,0)=0.

Recall that the gradient of a scalar field V in three dimensions is given by ∇V = (∂V/∂x, ∂V/∂y, ∂V/∂z). So, we need to find V such that:

• ∂V/∂x = -4y
• ∂V/∂y = -4x
• ∂V/∂z = 5z

To find V, we can integrate each component separately.

1. Integrate -4y with respect to x, we get V = -4xy + f(y,z)
2. Integrate -4x with respect to y, we notice that it agrees with the first part. This confirms -4xy is correct and doesn't add any new function of z.
3. Integrate 5z with respect to z, we get V = (5/2)z2 + g(x,y)

Merging the parts, we get V = -4xy + (5/2)z2. As V(0,0,0)=0 by the problem statement, we do not need to add any constants.

Therefore, the scalar potential function V is given by V(x,y,z) = -4xy + (5/2)z2, satisfying F = ∇V.