Does metric determines derivatives?

In summary, a metric can be used to define a derivative on a metric space, while a Riemannian manifold uses a covariant derivative defined by an affine connection that is compatible with the metric. This allows for the definition of a unique covariant derivative on the manifold.
  • #1
eljose79
1,518
1
If we have a metric could we define a derivative?..in fact the derivative would be Lim D(f(x+h),f(x))
D(0.h)
 
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  • #2
Covariant derivative, metric compatibility, Levi-Civita connections, etc.
 
  • #3
I am not sure what you are exactly meant by "metric" here.
If a vector space is equipped with "length" (I am using more intuitive notion.) it is called metric space. If you consider a set of nice functions over some domain they usually form a vector space. If this space has notion of norm, (and being complete) then it is called Banach space. With this norm you can define a derivative by usual calculus textbook way. ( In usual book way absolute value of real number plays the roll of norm). In metric space you can use this metric to define a norm.

Metric on Riemannian manifold is slightly different concept even though they are of course related. You can define a covariant derivative with any affine connections on Riemannian manifold. Fumdamental theorem of Riemannian geometry says, then, there is a unique covariant derivative that is compatible with metric. i.e. you can use metric to fix your affine connection to define covariant derivative.

Instanton
 

1. Does the choice of metric affect the calculation of derivatives?

Yes, the choice of metric can affect the calculation of derivatives. The metric determines the distance and angles in a space, which in turn affects the definition of derivatives. Different metrics can result in different values for derivatives.

2. How does the metric influence the calculation of derivatives?

The metric influences the calculation of derivatives through its effect on the curvature of the space. The curvature, in turn, affects the definition of derivatives and the resulting values.

3. Can metric determine the existence of derivatives?

No, the existence of derivatives is determined by the continuity and differentiability of the function, not by the metric. However, the metric can affect the values of derivatives.

4. In what situations does the choice of metric not matter for derivatives?

The choice of metric does not matter for derivatives in flat spaces, where the metric is simply the Euclidean metric. In these cases, the curvature is zero, and the metric does not affect the calculation of derivatives.

5. How can we choose the appropriate metric for calculating derivatives?

The appropriate metric for calculating derivatives depends on the specific problem and the properties of the space in which the function is defined. In general, the metric should be chosen to accurately reflect the curvature of the space and maintain consistency with the underlying geometry.

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