Which of the following equation represents two dimensional Laplace equation?
2 Answers
The two-dimensional **Laplace equation** is a second-order partial differential equation (PDE) that describes the behavior of scalar fields like electrostatic potentials or temperature distributions in two dimensions. It is a special case of the Poisson equation when the source term is zero.
The general form of the two-dimensional Laplace equation is:
\[
\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0
\]
where:
- \(\phi(x, y)\) is the unknown scalar field (e.g., potential or temperature),
- \(\frac{\partial^2 \phi}{\partial x^2}\) is the second partial derivative of \(\phi\) with respect to \(x\),
- \(\frac{\partial^2 \phi}{\partial y^2}\) is the second partial derivative of \(\phi\) with respect to \(y\).
In words, this equation states that the sum of the second derivatives of \(\phi\) in both the \(x\)- and \(y\)-directions must equal zero for the field to be in equilibrium (i.e., harmonic).
This equation appears frequently in physics and engineering, especially in problems involving steady-state heat conduction, electrostatics, and incompressible fluid flow.
The general form of the two-dimensional Laplace equation is:
\[
\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0
\]
where:
- \(\phi(x, y)\) is the unknown scalar field (e.g., potential or temperature),
- \(\frac{\partial^2 \phi}{\partial x^2}\) is the second partial derivative of \(\phi\) with respect to \(x\),
- \(\frac{\partial^2 \phi}{\partial y^2}\) is the second partial derivative of \(\phi\) with respect to \(y\).
In words, this equation states that the sum of the second derivatives of \(\phi\) in both the \(x\)- and \(y\)-directions must equal zero for the field to be in equilibrium (i.e., harmonic).
This equation appears frequently in physics and engineering, especially in problems involving steady-state heat conduction, electrostatics, and incompressible fluid flow.
The two-dimensional Laplace equation is a second-order partial differential equation that is widely used in fields such as electrostatics, fluid dynamics, and heat conduction. The standard form of the **two-dimensional Laplace equation** is:
\[
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
\]
Where:
- \( u(x, y) \) is the unknown function (which could represent potential, temperature, etc.),
- \( x \) and \( y \) are the spatial variables.
This equation essentially states that the sum of the second partial derivatives of the function \( u \) with respect to both \( x \) and \( y \) is equal to zero, meaning that there is no change in the "curvature" of the function in any direction within the domain.
This form of the Laplace equation describes steady-state conditions for potential fields (e.g., electric potential or temperature) in two dimensions.
\[
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
\]
Where:
- \( u(x, y) \) is the unknown function (which could represent potential, temperature, etc.),
- \( x \) and \( y \) are the spatial variables.
This equation essentially states that the sum of the second partial derivatives of the function \( u \) with respect to both \( x \) and \( y \) is equal to zero, meaning that there is no change in the "curvature" of the function in any direction within the domain.
This form of the Laplace equation describes steady-state conditions for potential fields (e.g., electric potential or temperature) in two dimensions.