# Second order pde - on invariant?

• sigh1342
In summary, a second order partial differential equation (PDE) is a mathematical equation with two independent variables and their partial derivatives up to second order. It is used to model physical phenomena in various fields. The significance of invariance in second order PDEs refers to the property that allows for simplification and solving of the equation by considering only the invariant elements. Second order PDEs can be classified based on their coefficients, with common classifications including elliptic, parabolic, and hyperbolic PDEs. Examples of physical systems described by second order PDEs include heat flow, wave propagation, and diffusion, as well as fluid dynamics and electromagnetism. Common numerical methods for solving second order PDEs include finite difference
sigh1342
second order pde -- on invariant?

What the meaning for a second order pde is rotation invariant?
Is all second order pde are rotation invariant? or only laplacian?

It sounds like you've come across the terms as part of a course involving laplacians - in which case, you were shown a proof somewhere that the laplacian is rotation invarient. What was the condition the laplacian had to satisfy for the proof to work? Do all 2ndO PDEs satisfy this condition?

## 1. What is a second order partial differential equation (PDE)?

A second order PDE is a mathematical equation that involves two independent variables and their partial derivatives up to second order. It is often used to model physical phenomena in many fields, including physics, engineering, and economics.

## 2. What is the significance of invariance in second order PDEs?

Invariance refers to the property of a PDE that remains unchanged under certain transformations, such as rotations or translations. This property is important because it allows us to simplify and solve the PDE by considering only the invariant elements.

## 3. How are second order PDEs classified?

Second order PDEs can be classified based on their coefficients, which determine the type of equation and the method used to solve it. Some common classifications include elliptic, parabolic, and hyperbolic PDEs.

## 4. What are some examples of physical systems described by second order PDEs?

Second order PDEs are used to describe a wide range of physical systems, including heat flow, wave propagation, and diffusion. They are also commonly used in the study of fluid dynamics and electromagnetism.

## 5. What are some numerical methods used to solve second order PDEs?

Some commonly used numerical methods for solving second order PDEs include finite difference methods, finite element methods, and spectral methods. These methods involve discretizing the PDE into smaller, simpler equations that can be solved using computer algorithms.

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