Given that the stream function is provided as $\psi = x^2 - y^2$, we want to find the corresponding velocity potential $\phi$, or the real part of the complex potential $w = \phi + i\psi$.

Steps to Find the Velocity Potential:

1. Complex potential definition: The complex potential $w = \phi + i\psi$, where $\phi$ is the velocity potential and $\psi$ is the stream function.

2. Cauchy-Riemann equations: Since $w = \phi + i\psi$ is analytic, it must satisfy the Cauchy-Riemann equations: $\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}, \quad \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}$

3. Given stream function: $\psi = x^2 - y^2$

4. First Cauchy-Riemann equation: $\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}$ Calculating $\frac{\partial \psi}{\partial y}$: $\frac{\partial \psi}{\partial y} = \frac{\partial}{\partial y}(x^2 - y^2) = -2y$ Therefore, $\frac{\partial \phi}{\partial x} = -2y$

5. Second Cauchy-Riemann equation: $\frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}$ Calculating $\frac{\partial \psi}{\partial x}$: $\frac{\partial \psi}{\partial x} = \frac{\partial}{\partial x}(x^2 - y^2) = 2x$ Therefore, $\frac{\partial \phi}{\partial y} = -2x$

6. Solving for $\phi$: Integrating both partial derivatives, we obtain $\phi$.

   • From $\frac{\partial \phi}{\partial x} = -2y$, integrate with respect to $x$: $\phi(x, y) = -2xy + f(y)$
   • From $\frac{\partial \phi}{\partial y} = -2x$, integrate with respect to $y$: $\phi(x, y) = -2xy + g(x)$

   Since the function of integration $f(y)$ or $g(x)$ must be a constant, the velocity potential is: $\phi(x, y) = -2xy$

Thus, the correct answer is:

Option (c): $-2xy$

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Would you like further details on any of the steps, or have any other questions?

Here are 5 related questions to explore further:

1. How are the Cauchy-Riemann equations used in fluid flow problems?
2. What is the physical significance of the velocity potential $\phi$?
3. How can one derive the streamlines from the stream function?
4. What conditions must a complex potential $w$ satisfy to describe a flow?
5. Can the method of complex potential be applied to non-laminar flows?

Tip: In flow problems, the Cauchy-Riemann equations ensure that the velocity field satisfies continuity, implying an incompressible flow.