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: Present Laplace's Equation.Laplace's equation, a second-order partial differential equation, is expressed as \(abla^2 f = 0\), with \(abla^2\) denoting the Laplacian. In Cartesian form, it reads \(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} = 0\).
Step 2: Describe the Solutions.A scalar function \(f\) satisfying Laplace's equation is termed a harmonic function. For the Laplacian to be valid and zero, the function must be continuous up to its second derivative. These functions are crucial in physics, notably in electromagnetism (electrostatic potential without charges), fluid dynamics, and heat transfer.