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bchui
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So much has been talking about covariant derivative. Anyone knows about contravariant derivative? What is the precise definition and would that give rise to different [tex]\Gamma^{k}_{i,j}[/tex] and other concepts?
A contravariant derivative is a mathematical concept used in differential geometry to describe how a vector field changes along a given direction. It is a directional derivative that takes into account the change in the coordinate system.
A covariant derivative is defined in terms of the change in the components of a vector field, while a contravariant derivative is defined in terms of the change in the direction of the vector field. In other words, a covariant derivative is defined with respect to the basis of the tangent space, while a contravariant derivative is defined with respect to the basis of the cotangent space.
A metric tensor is used to define the distance between points in a given coordinate system. The contravariant derivative is defined with respect to this metric tensor, which allows it to take into account the change in the coordinate system when calculating the directional derivative of a vector field.
Contravariant derivatives are commonly used in the field of differential geometry to study the curvature and topology of differentiable manifolds. They are also used in physics, particularly in the study of general relativity, to describe the behavior of gravitational fields and other physical phenomena.
The contravariant derivative of a vector field can be calculated using the Christoffel symbols, which are defined in terms of the metric tensor. These symbols represent the connection between the tangent and cotangent spaces, and are used in the formula for the contravariant derivative.