Question

Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = vf. (If the vector fieid is not conservative, enter DNE.) F(x,y,z)=1+sin(z)⌋+ycos(z)k f(x,y,z)=

Answer 1

A force is conservative if its vector field meets the condition that the curl of the field is zero. This is tested by checking if the partial derivatives of the force components satisfy certain conditions like (dFx/dy) = (dFy/dx). The question regarding the magnitude at x = y = 1 m relates to evaluating the force's components at that point.

Step-by-step explanation:

To determine whether a vector field is conservative, one must check whether the work done by the force represented by this field is independent of the path taken. For a force to be conservative, the curl of the force vector field must be zero, which means that the partial derivatives of the components must satisfy certain conditions. Specifically, the derivative of the x-component with respect to y must equal the derivative of the y-component with respect to x. If these conditions are met, the force is conservative and is related to the gradient of a potential energy function. This relationship means the force can be expressed as the negative gradient of this potential energy function.

In the context of the question provided, a conservative force field in two dimensions should satisfy the condition (dFx/dy) = (dFy/dx). If the student's vector field satisfies this condition, then it is conservative. As an example, if we have (dFx/dy) = (dFy/dx) = (4 N/m³)xy, then the force is conservative. The magnitude of this force field at the point x = y = 1 m can be calculated by evaluating the force components at that point.

In the case of non-conservation, such as when (dFx/dy) = 5N/m and (dFy/dx) = 10N/m, the force does not satisfy the condition for a conservative field, indicating it is a non-conservative force.

Answer 2

If F(x, y, z) = i + sin(z) j + y cos(z) k is conservative, then there exists a scalar function f(x, y, z) such that grad(f) = F, which means

∂f/∂x = 1

∂f/∂y = sin(z)

∂f/∂z = y cos(z)

Integrating each each of these equations gives

∫ ∂f/∂x dx = ∫ dx ⇒ f(x, y, z) = x + α(y, z)

∫ ∂f/∂y dy = ∫ sin(z) dy ⇒ f(x, y, z) = y sin(z) + β(x, z)

∫ ∂f/∂z dx = ∫ y cos(z) dz ⇒ f(x, y, z) = y sin(z) + γ(x, y)

It follows that α(y, z) = y sin(z) and β(x, z) + γ(x, y) = x + C where C is a constant. So

f(x, y, z) = x + y sin(z) + C

and F is indeed conservative.

Step-by-step explanation: