Ordinary and covaraint derivative

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In summary, the covariant derivative is a generalization of the classical derivative in Riemann geometry that is independent of its description in a particular coordinate system. It takes into account changes in coordinates through the use of Christoffel symbols. This discussion provides more information and further resources on covariant and contravariant tensors.
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world line
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Hello
what is the meaning of covaraint derivative ?
where the ordinary derivative of a function whit respect to a variable is zero, it means that function doesn't depend on that variable.but what about covaraint derivative ?
for example the metric tensor may depends on coordinate but its covaraint derivative is zero.
 
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The classical (directional) derivative is dependent on your choice of coordinates. The covariant derivative is not; is it covariant in the sense that it is defined in such a way as to be independent of its description in a particular coordinate system. That is why it is written as a directional derivative plus a term that compensates for any changes in coordinates, expressed through the Christoffel Symbols. It is basically a generalization of the classical derivative into Riemann geometry.
 
  • #4

What is the difference between an ordinary derivative and a covariant derivative?

An ordinary derivative is a mathematical operation that measures the rate of change of a function with respect to one independent variable. A covariant derivative, on the other hand, is a more general concept that measures the rate of change of a function with respect to a curve or surface in a higher-dimensional space.

How is a covariant derivative calculated?

The calculation of a covariant derivative involves the use of a connection, which is a mathematical object that describes how to differentiate vectors along a curve or surface. It takes into account the curvature of the space in which the curve or surface is embedded.

What is the significance of covariant derivatives in physics?

Covariant derivatives are essential in the field of physics, particularly in the theory of relativity and in the study of curved spacetime. They are used to describe how physical quantities change as objects move through curved spaces, such as in the case of gravitational fields.

Can an ordinary derivative be written as a covariant derivative?

Yes, an ordinary derivative can be written as a covariant derivative in certain cases. This is known as the intrinsic derivative, which takes into account the natural geometry of a curved space. However, in general, an ordinary derivative and a covariant derivative are different mathematical operations.

Are there any other types of derivatives besides ordinary and covariant derivatives?

Yes, there are other types of derivatives, such as partial derivatives, directional derivatives, and total derivatives. Each type of derivative has its own specific definition and use in mathematics and physics.

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