# Singular points in 3-dim space

• firenze
In summary, singular points in 3-dim space are points where a function or surface is not well-defined or differentiable. They are identified by finding points where the partial derivatives are equal to zero or do not exist, and can be classified as isolated, non-isolated, or essential. Studying singular points has applications in various fields and they are related to critical points, but not all critical points are singular points. In some cases, singular points can be avoided or removed, but in many cases they are essential and cannot be eliminated without significantly altering the function or surface.
firenze
For a linearized system I have eigenvalues $$\lambda_1, \lambda_2 = a \pm bi \;(a>0)$$ and $$\lambda_3 < 0$$,
then it should be an unstable spiral point. As $$t \to +\infty$$ the trajectory will lie in the plane which is parallel with the plane spanned by eigenvectors $$v_1,v_2$$ corresponding to $$\lambda_1, \lambda_2$$.

Right? I am just not very sure.

Yes, that is correct.

Thanks

## 1. What are singular points in 3-dim space?

In mathematics, a singular point in 3-dim space is a point where a function or surface is not well-defined or differentiable. This means that the function or surface has a discontinuity or sharp change at that point.

## 2. How are singular points identified and classified?

Singular points are identified by finding the points where the partial derivatives of a function or surface are equal to zero or do not exist. They can be classified as isolated, non-isolated, or essential, depending on the behavior of the function or surface near the point.

## 3. What are the applications of studying singular points in 3-dim space?

Studying singular points in 3-dim space is important in many fields of science and engineering, including physics, chemistry, and computer graphics. It allows for a better understanding of the behavior of functions and surfaces, and can help in solving optimization problems and modeling real-world phenomena.

## 4. How are singular points related to critical points?

Singular points are a type of critical point, where the first derivatives of a function are equal to zero or do not exist. However, not all critical points are singular points, as some may have well-defined derivatives but still have interesting behavior at that point.

## 5. Can singular points be avoided or removed?

In some cases, singular points can be avoided or removed by using a different coordinate system or by making small changes to the function or surface. However, in many cases, singular points are essential and cannot be eliminated without significantly altering the function or surface.

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