Intrinsic derivative is a concept in differential geometry that measures the rate of change of a geometric object without reference to an external coordinate system. It is defined using the object's own internal structure. On the other hand, covariant derivative is a concept in differential geometry that takes into account the curvature of the underlying space and adjusts for it in the derivative calculation. In simpler terms, intrinsic derivative only considers the object itself, while covariant derivative takes into account the space it is in.
Covariant derivative can be seen as a generalization of intrinsic derivative. In fact, when the space is flat (i.e. has no curvature), the two derivatives are equivalent. However, in curved spaces, covariant derivative is a more accurate measure of the rate of change for an object as it takes into account the curvature of the space.
The notation for intrinsic derivative is d/dt, where t represents the parameter along the curve. For covariant derivative, the notation is ∇X, where X is the vector field along which the derivative is being taken. Both derivatives can also be written in index notation, using Greek indices for the intrinsic derivative and Latin indices for the covariant derivative.
Regular derivative, or partial derivative, is used to calculate the rate of change of a function with respect to a specific independent variable. It is defined in terms of the coordinate system in which the function is expressed. On the other hand, intrinsic derivative and covariant derivative are independent of any coordinate system and take into account the underlying geometry of the space.
Intrinsic derivative and covariant derivative are essential concepts in differential geometry, which has numerous real-world applications. Some examples include: