Laplace’s Equation
Steady-state potentials.
Key idea
Steady-state potentials.
In depth
u_xx + u_yy = 0 describes steady-state phenomena. Solutions (harmonic functions) satisfy the maximum principle: the max and min occur on the boundary, and the value at the centre of a disc equals its average on the circle.
Worked example
u = x² − y² is harmonic; u = xy as well.
Formula
Real-world uses
- Predicting electric potential around wires and steady temperatures in a plate
Common mistake
Applying it to time-dependent problems.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Laplace’s Equation?
Steady-state potentials. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.
What is the formula for Laplace’s Equation?
The key formula is ∇²u = 0.
Where is Laplace’s Equation used in real life?
Real-world examples: Predicting electric potential around wires and steady temperatures in a plate.
What should I learn before Laplace’s Equation?
Make sure you are comfortable with Wave Equation, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Laplace’s Equation?
Applying it to time-dependent problems.
How can I practise Laplace’s Equation?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Laplace’s Equation?
Move on to Separation of Variables for PDEs next.
Related expert topics in Differential Equations, Abstract Algebra & Applied Math
- Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
- Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
- Runge–Kutta Methods – Higher-accuracy numerical solvers.
- Second-Order Homogeneous Equations – Use the characteristic equation.
- Undetermined Coefficients – Guess the particular solution form.
- Variation of Parameters – General method for particular solutions.