Calculate Killing Vectors in 3-D Euclidean Space

In summary, a Killing vector in 3-D Euclidean Space is a vector field that satisfies the Killing equation, which describes a symmetry of the space. They are calculated by determining the metric of the space and solving for the components of the vector field using the Killing equation. In physics, Killing vectors have many applications in understanding symmetries and conservation laws. They can also exist in spaces other than 3-D Euclidean Space, but the form of the Killing equation and method for calculating them may vary. Additionally, Killing vectors are closely related to isometries as they represent symmetries that preserve the metric of a space.
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BrunoSantos
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Homework Statement



My problem is to calculate to calculate killing vectors in 3-D euclidean space(flat space).

Homework Equations



The relevante equations are killing equation : d_a*V_b+d_b*V_a=0

The Attempt at a Solution


I found the solution in Ray D'Inverno and that is (d_x,d_y,d_z) but I can't find how this is calculate.
I hope you can help me .
Thanks
 
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1. What is a Killing vector in 3-D Euclidean Space?

A Killing vector in 3-D Euclidean Space is a vector field that satisfies the Killing equation, which is a differential equation that describes a symmetry of the space. In other words, a Killing vector is a vector that, when applied to a point in 3-D Euclidean Space, leaves the metric of the space invariant. This means that the distance between any two points remains the same when the space is transformed by the vector.

2. How do you calculate Killing vectors in 3-D Euclidean Space?

To calculate Killing vectors in 3-D Euclidean Space, you first need to determine the metric of the space. This can be done using the Pythagorean theorem for flat space. Once you have the metric, you can use the Killing equation to solve for the components of the vector field. This typically involves taking partial derivatives and setting them equal to each other to satisfy the Killing equation.

3. What is the significance of Killing vectors in physics?

Killing vectors have many applications in physics, particularly in the study of symmetries and conservation laws. In general relativity, Killing vectors play a crucial role in understanding the behavior of space-time and the conservation of energy and momentum. They are also used in other areas of physics, such as quantum mechanics and electromagnetism.

4. Can Killing vectors exist in spaces other than 3-D Euclidean Space?

Yes, Killing vectors can exist in any space that has a metric. This includes spaces with different dimensionalities and non-Euclidean geometries. However, the form of the Killing equation and the method for calculating Killing vectors may differ depending on the space.

5. How do Killing vectors relate to isometries?

Isometries are transformations that preserve distances and angles in a space. Killing vectors are closely related to isometries, as they represent symmetries of a space that leave the metric invariant. In fact, the set of all Killing vectors for a given space is equivalent to the set of all isometries for that space.

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