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Terilien
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I've heard of something called a covariant derivative. what motivates it and what is it?
That's not quite right. When one calculates the directional derivative of a vector you need two things. The vector field and a vector which determines the direction you're interested.Terilien said:Ok this question is stupid, but can't we just use the chain rule to calculate the directional derivative of a tensor field in an arbitrary direction(byt that I mean can the directional derivative be written as a linear combination of the covariant derivative along corrdinate axis)? I heard that you can't but don't know why you wouldn't be able to.
Calculating the Christoffel symbols can be laborious at times but once you've done it a dozen or so times it will become second nature to you.If so how do we calculate it in a rbitrary direction. please don't tear me apart. There's something weird about the covariant derivative. the christoffel symbols make computation seem impossbile.
Terilien said:how does one derive the general formula for the covariant derivative of a tensor field? To be more precise I took out sean carolls book at the library but did not understand equation 3.17 on page 97. Could someone derive it or prove it, or at the very least give me a better hint?
A covariant derivative is a mathematical operation that is used to calculate the rate of change of a vector field or tensor field along a given curve or surface in a curved space. It takes into account the curvature of the space and allows for a more accurate description of how vectors and tensors change as they move through the space.
A covariant derivative is specific to curved spaces and takes into account the curvature of the space when calculating the rate of change. In contrast, a regular derivative is used in flat spaces and does not consider the curvature of the space.
Covariant derivatives have numerous applications in physics and engineering, such as in the study of general relativity, fluid dynamics, and electromagnetism. They are also used in computer graphics to model the behavior of light and in robotics for motion planning and control.
In general, a covariant derivative is written as ∇vu, where v is the vector field or tensor field along which the derivative is being taken, and u is the field being differentiated. However, the specific mathematical expression can vary depending on the type of space and the coordinate system being used.
The intuition behind covariant derivatives is that they take into account the effects of curvature on the rate of change of a vector or tensor field. In a curved space, the direction in which a vector is pointing may change as it moves along a curve, and the covariant derivative captures this change by adjusting for the curvature of the space.