Question About Curvilinear Cooridnates

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In summary, in curvilinear coordinates, the components of a vector are still the projections on the basis vectors. However, the basis vectors change from point to point, making it challenging to understand the geometric meaning. This only becomes an issue when trying to integrate a vector quantity, but can be accounted for.
  • #1
TonyEsposito
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Hi guys, i can not understand something about curvilnear coordinates...from a "geometric" point of view what dose the componentes of the vector mean? in a non-curvilinear coordinates the components are the "projections" on along the bases but in a curvilinear coordinates the directions of the basis vectors changes from point to pint so i can not grasp the geometrical meaning!
Thanks!
 

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  • #2
The components are still the projection on the basis vectors. In general, it only makes sense to talk about the projection on the basis at the same point as the vector is defined and therefore it is not really an issue that the basis changes from point to point.

Of course, if you want to integrate a vector quantity, it does matter - but can be taken into account fairly easy.
 
  • #3
Ok...thanks! :)
 

FAQ: Question About Curvilinear Cooridnates

1. What are curvilinear coordinates?

Curvilinear coordinates are a set of coordinates used to describe points in space, typically used in three-dimensional systems. Unlike the more familiar Cartesian coordinates, curvilinear coordinates take into account the curvature of the surface on which the points are located.

2. How are curvilinear coordinates useful?

Curvilinear coordinates are useful in situations where the surface being studied is not flat, such as in geography, astronomy, and fluid dynamics. They allow for a more accurate and detailed description of the points in space and can make complex calculations more manageable.

3. What are some examples of curvilinear coordinates?

Some examples of curvilinear coordinates include spherical coordinates, cylindrical coordinates, and polar coordinates. Spherical coordinates are useful for describing points on a sphere, while cylindrical coordinates are often used in cylindrical or circular objects. Polar coordinates are commonly used in two-dimensional systems, such as in graphs and polar plots.

4. How do curvilinear coordinates differ from Cartesian coordinates?

Unlike Cartesian coordinates, which use perpendicular axes to describe points in space, curvilinear coordinates use a combination of curves and angles. This allows for a more flexible and accurate representation of points on curved surfaces.

5. What are the main challenges of working with curvilinear coordinates?

One of the main challenges of working with curvilinear coordinates is that the equations and calculations involved can be more complex compared to Cartesian coordinates. Additionally, different coordinate systems have different limitations and may not be suitable for certain types of problems. It is important to carefully choose the appropriate coordinate system for the problem at hand.

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