Questions: Determine whether the vector field is conservative. If it is, find a potential function for the vector field. (If the vector field is not conservative, enter DNE.) F(x, y) = sin(y) i + x cos(y) j f(x, y) =

Solution Steps

To determine if a vector field is conservative, we need to check if its curl is zero. For a 2D vector field \(\mathbf{F}(x, y) = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j}\), the curl is given by \(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\). If the curl is zero, the vector field is conservative, and we can find a potential function \(f(x, y)\) by integrating \(P\) with respect to \(x\) and \(Q\) with respect to \(y\), ensuring consistency between the two results.

We have the vector field \(\mathbf{F}(x, y) = \sin(y) \mathbf{i} + x \cos(y) \mathbf{j}\). To determine if this vector field is conservative, we calculate the curl given by:

\[ \text{curl} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \]

where \(P = \sin(y)\) and \(Q = x \cos(y)\).

Calculating the partial derivatives, we find:

\[ \frac{\partial Q}{\partial x} = \cos(y) \quad \text{and} \quad \frac{\partial P}{\partial y} = \cos(y) \]

Thus, the curl is:

\[ \text{curl} = \cos(y) - \cos(y) = 0 \]

Since the curl is zero, the vector field is conservative.

To find the potential function \(f(x, y)\), we integrate \(P\) with respect to \(x\):

\[ f(x, y) = \int P \, dx = \int \sin(y) \, dx = x \sin(y) + g(y) \]

where \(g(y)\) is an arbitrary function of \(y\).

Next, we differentiate \(f(x, y)\) with respect to \(y\) and set it equal to \(Q\):

\[ \frac{\partial f}{\partial y} = x \cos(y) + g'(y) \]

Setting this equal to \(Q\):

\[ x \cos(y) + g'(y) = x \cos(y) \]

This implies:

\[ g'(y) = 0 \implies g(y) = C \]

where \(C\) is a constant. Therefore, the potential function is:

\[ f(x, y) = x \sin(y) + C \]

Final Answer