Differential Equations › Section I · Partial differential equations › Topic I.7
Laplace's equation
Hold the edges of a metal plate at fixed temperatures and wait: the temperature inside settles into a steady pattern. That pattern obeys Laplace's equation, which also governs electric potential and smooth fluid flow, and separation of variables solves it on rectangles and disks.
In this topic
- Steady states in two dimensions
- A rectangle, one side at a time
- Solving with a hot top edge
- Matching the top edge
- Worked example: a single sine on the top edge
- Worked example: a square with one hot side
- Laplace's equation on a disk
- Solving on the disk
- Worked example: a disk with a varying rim
- Two properties of harmonic functions
- Common mistakes to avoid
- Key takeaways
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