To find the partial differential equation from $P = e^q$, we need to differentiate this equation with respect to the variables involved.

Let $P$ be a function of two independent variables, say $x$ and $y$, and $q$ is a function of $x$ and $y$ (i.e., $q = q(x, y)$).

Step 1: Differentiate with respect to $x$

Using the chain rule: $\frac{\partial P}{\partial x} = \frac{\partial}{\partial x}(e^q) = e^q \cdot \frac{\partial q}{\partial x}$ Thus: $P_x = e^q q_x$

Step 2: Differentiate with respect to $y$

Similarly, differentiating with respect to $y$: $\frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(e^q) = e^q \cdot \frac{\partial q}{\partial y}$ Thus: $P_y = e^q q_y$

Conclusion:

The partial differential equation that corresponds to $P = e^q$ is expressed through the derivatives of $P$ and $q$. The system of PDEs we get is:

$P_x = P q_x$ and $P_y = P q_y$

Would you like further details or have any questions?

Here are 5 related questions:

1. What if $P$ and $q$ also depend on time?
2. How would you solve this system of PDEs?
3. Can we express $q$ explicitly as a function of $P$?
4. What happens if $P = e^{q(x)} + e^{q(y)}$?
5. How can boundary conditions affect the solution to these PDEs?

Tip: Always check if the variables in the equation are interdependent before differentiating!