Covariant vs absolute derivative

In summary, the conversation discusses the concept of covariant and contravariant derivatives in differential geometry, specifically the use of the term "covariant derivative" to refer to the "derivative along the curve." The author notes that this is incorrect and points to several resources on the topic, including a paper by Izu Vaisman. The conversation also mentions the existence of both covariant and contravariant geometries, connections, and differentiations in differential geometry.
  • #1

pmb

In the online text on differential geometry

http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/pdfs/DiffGeom.pdf [Broken]

The author calls the "derivative along the curve" (aka absolute derivative) the "covariant derivative" which is wrong.

It's on box 8.2 on page 59.

Does anyone else here refer to DP/dtau as the covariant derivative of P?

Pete
 
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  • #2
Waner talks about total and partial covariant derivatives on pp. 59-61 and even covariant differentials on p. 62, with no regard to the business of setting new lower indices at all. There is a clue on p. 62, exercise set 8 #10(b), where a contravariant derivative is suggested but not exhibited. This yields some fruit under web search.

There seem to be covariant AND contravariant differential geometries, covariant AND contravariant affine connections, and covariant AND contravariant differentiations afoot. So, I suppose, that means partial and total derivatives of both kinds.

some found links -->

http://emis.bibl.cwi.nl/proceedings/Coimbra99/pdgloja.pdf [Broken]
contravariant connections on poisson manifolds {Fernandes}

http://www.math.toronto.edu/henrique/keio.pdf [Broken]
poisson vector bundles, contravariant connections and deformations {Bursztyn}

The name Izu Vaisman seems to be important.

{SIGH!}, so be the shifting sands of terminology!

Regards,
 
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1. What is the difference between covariant and absolute derivative?

The covariant derivative is a mathematical operation used to calculate the rate of change of a vector field along a certain direction, while the absolute derivative is a similar operation that takes into account changes in the coordinate system itself. In simpler terms, the covariant derivative measures how a vector changes as it moves along a curved surface, while the absolute derivative measures how the vector changes as the curved surface itself is changing.

2. How are covariant and absolute derivatives related?

The covariant derivative can be thought of as a generalization of the absolute derivative, as it includes the effects of a changing coordinate system. In fact, the absolute derivative can be obtained from the covariant derivative by setting the coordinate system to be constant.

3. What is the significance of covariant and absolute derivatives in science?

Covariant and absolute derivatives are important concepts in fields such as physics and engineering, where they are used to describe the behavior of objects in curved spaces. They are particularly useful in understanding how objects move and change in relativistic and gravitational fields.

4. Can you provide an example of when it is necessary to use the covariant derivative instead of the absolute derivative?

One example is in general relativity, where the curvature of spacetime affects the motion of objects. In this case, the covariant derivative is necessary to properly describe the movement of particles, as the absolute derivative does not take into account the changing curvature of spacetime.

5. Are there any limitations to using covariant and absolute derivatives?

One limitation is that these derivatives can only be applied to vector fields, and not scalar fields. Additionally, they can become quite complex and difficult to calculate in highly curved spaces, making them more challenging to use in certain situations.

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