Question

Consider the vector field F(x,y,z)=⟨−7y,−7x,9z⟩. 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 that vector field F is a gradient vector field, we need to find a scalar function V(x, y, z) such that its partial derivatives with respect to x, y, and z are equal to the given components of the vector field F. The function V that satisfies these conditions is V = -7xy + 4.5z² + K.

Step-by-step explanation:

To show that vector field F is a gradient vector field and find the function V, we can apply the gradient operator ∇ to a scalar function V(x, y, z). Recall that the gradient of a scalar function is defined as ∇V = (∂V/∂x)î + (∂V/∂y)ĵ + (∂V/∂z)k.

In this case, we need to find a scalar function V(x, y, z) such that its partial derivatives with respect to x, y, and z are equal to the given components of the vector field F.

For the x-component of F, we have -7y = (∂V/∂x), which implies V = -7xy + g(y, z), where g(y, z) is a function of y and z only. To find g(y, z), we differentiate V with respect to y: (∂V/∂y) = -7x + (∂g/∂y) = -7x. Integrating this equation gives g(y, z) = -7xy + C(z), where C(z) is a constant with respect to y.

Finally, we differentiate V with respect to z and set it equal to the z-component of F: (∂V/∂z) = 9z + (∂C/∂z) = 9z. Integrating this equation gives C(z) = 4.5z² + K, where K is a constant. Therefore, the function V(x, y, z) that satisfies the conditions is V = -7xy + 4.5z² + K.