Question

Determine whether the vector field \(F\) is conservative. If it is, find a potential function for the vector field. $$ \mathbf{F}(x, y, z)=\frac{1}{y} \mathbf{i}-\frac{x}{y^{2}} \mathbf{j}+(2 z-1) \mathbf{k} $$

Short answer

First, calculate the curl of the given vector field. If the curl equals zero, the field is conservative, and one can proceed to calculate the potential function by integrating the components of the vector field and adding them together with an arbitrary constant.

Step-by-step solution

Compute the Curl of the Vector Field

The first step is to calculate the curl of the vector field. The curl is given by: \n\n\[\text{curl}\, \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} - \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right)\mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k}\]

Verify if the Curl is Zero

In this step, substitute \(P = \frac{1}{y}\), \(Q = -\frac{x}{y^{2}}\), and \(R = 2z-1\) into the curl expression and simplify. If the result is zero vector, the field is conservative.

Find the Potential Function

To find the potential function when the vector field is conservative, we need to calculate three integrals: \(\int P dx\), \(\int Q dy\), and \(\int R dz\). The potential function is then given by the sum of these three integrals plus an arbitrary constant \(C\).