Question

Solution of Laplace's equation, which are continuous through the second derivative, are called

• A. Bessel functions
• B. Odd functions
• C. Harmonic functions
• D. Fundamental functions

Hint:

Laplace's Equation (\(\nabla^2 f = 0\)) and Harmonic Functions are synonymous. If a function satisfies one, it is the other.

Answer 1

The Correct Option is C

Step 1: State Laplace's Equation. Laplace's equation is a second-order partial differential equation given by \(\nabla^2 f = 0\), where \(\nabla^2\) is the Laplacian operator. In Cartesian coordinates, it is \(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} = 0\).
Step 2: Define the solutions. By definition, any scalar function \(f\) that is a solution to Laplace's equation is called a harmonic function. The condition that the function is continuous through the second derivative is a requirement for the Laplacian to be well-defined and equal to zero. These functions are fundamental to many areas of physics, including electromagnetism (electrostatic potential in charge-free regions), fluid dynamics, and heat flow.