13. Partial Differential Equations

Try these first

1. 1.

   What is the general form of the two-dimensional Laplace equation?

   Hint

   Recall the summation of the second partial derivatives.

2. 2.

   Name a property of solutions to the Laplace equation.

   Hint

   Think about where the extremes can occur.

3. 3.

   What does the Laplace equation model?

   • Dynamic systems
   • Steady-state systems
   • Unsteady-state systems

   Hint

   Think about systems that have reached equilibrium.

4. 4.

   True or False: Harmonic functions can have local maxima inside their domain.

   • True
   • False

   Hint

   Recall the properties of harmonic functions.

5. 5.

   Provide a real-life scenario where the two-dimensional Laplace equation can be applied and solve for the temperature distribution given specific boundary conditions.

   Hint

   Look at the conditions provided and ensure they match a physical context.

6. 6.

   Demonstrate how to convert the two-dimensional Laplace equation into polar coordinates and solve it for a circular domain.

   Hint

   Review the polar form of the Laplace equation to guide your transformation.