Differential operators in arbitrary coordinate systems?

However, in some cases, it may not be obvious which coordinate system to choose. In these situations, an approach known as the method of arbitrary metrics can be used. This involves representing the differential operator in terms of an unknown metric tensor, and then using the boundary conditions to determine the best coordinate system for solving the problem. This method can also be used to determine the optimal coordinate system for finding eigenmodes of a drumhead, such as using skew coordinates for a parallelogram-shaped drumhead or spherical coordinates for a round drumhead. Therefore, by using the method of arbitrary metrics, one can find the most efficient coordinate system for solving a boundary value problem. In summary, the method of arbitrary metrics can be used to determine the optimal coordinate
  • #1
lordkelvin
22
0
Hi, physics undergraduate here. I don't know much about differential geometry yet, but I'm curious about this idea:

Say I encounter a boundary value problem, and I'm not sure what coordinate system would be 'easiest' to solve the problem in. Is there some way to put the differential operator in terms of an unknown metric tensor, then impose some conditions stemming from the boundary values of the problem onto the arbitrary metric tensor in order to select some 'best' coordinate system?

Say I wanted to find the eigenmodes of a parralelogram-shaped drumhead. I'm basically curious if there is some way for me to have mathematics tell me I'd be best off using skew coordinates. Same thing with spherical coordinates on a round drumhead.
 
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  • #2
Often, it is best to choose a coordinate system such that the coordinates on the boundaries are constants. This makes it much easier to apply the boundary conditions.
 

1. What are differential operators in arbitrary coordinate systems?

Differential operators are mathematical operators used in the study of differential equations. In arbitrary coordinate systems, these operators are used to describe how functions change in different directions and are essential in solving problems involving derivatives.

2. What is the difference between differential operators in Cartesian coordinates and arbitrary coordinates?

In Cartesian coordinates, the differential operators are simpler and only involve partial derivatives with respect to the coordinates. In arbitrary coordinates, the operators can be more complex and involve other factors such as scale factors and metric tensors.

3. How are differential operators in arbitrary coordinate systems used in physics?

Differential operators in arbitrary coordinate systems are used in physics to describe the behavior of physical quantities that vary in different directions. They are particularly useful in fields such as fluid mechanics, electromagnetism, and general relativity.

4. What is the importance of understanding differential operators in arbitrary coordinate systems?

Understanding differential operators in arbitrary coordinate systems is important in many areas of mathematics and physics. They allow for the description and solution of complex problems involving derivatives in multiple dimensions, and are essential in many advanced theoretical and applied sciences.

5. Are there any applications of differential operators in arbitrary coordinate systems in engineering?

Yes, there are many applications of differential operators in arbitrary coordinate systems in engineering. They are used in fields such as fluid dynamics, heat transfer, and structural analysis to model and solve problems involving complex systems and multiple dimensions.

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