Trying to understand covariant derivative, conceptually and visually. Help please?

In summary, the speaker is looking for a visual understanding of the covariant derivative and is seeking clarification on their current understanding. They have some background knowledge in summation notation and tensors, but are not familiar with the subject of tensor calculus. The speaker is studying on their own and is looking for a visual representation of the covariant derivative. They do not want to learn by rote and are seeking a conceptual understanding of the subject. They have provided a simple example and are asking for feedback on their understanding.
  • #1
Einsteinsqd
Hello!

I registered here today because I'm quite curious about the covariant derivative, and although I've consulted several texts on the subject (and wikipedia, and other locations), I've found it somewhat difficult to piece together a visual understanding of the covariant derivative. The material I've found, while somewhat helpful, leaves much to be desired in terms of simplifying the subject.

I've come here to ask you if I at least sort of have it right, and below are pictures which illustrate the best that I've been able to understand from the material I've read so far.

I am eighteen years old and have no grounding in differential geometry, advanced analysis, or any extremely rigorous mathematics. I'm sure there are plenty of other subjects I should tackle in advance of tensor calculus, but I've set my sights on it and I figure that I'll learn anything I need to along the way. I understand the meaning of summation notation, and at this point, I think I've gotten to the point where I understand what tensors are and what meaning they have (hard-fought victories, I assure you). I am studying on my own, and I am not personally familiar with anyone qualified to teach the material.

Now, I am not trying to deal with the matter of covariant derivative computation just yet. I've seen the formula, and I know what Christoffel symbols look like; I could learn it by rote if necessary. But I've never been satisfied learning by rote - without a visual and conceptual understanding of a subject, I can't feel comfortable carrying out computations.

So, for this simple example I've imagined, can you tell me if I've gotten the gist of the covariant derivative, and if not (which I imagine is more probable than my having understood it), could you please show me a good visual representation of the covariant derivative?

100_0125.jpg

100_0126.jpg

(The division at the end isn't mean to express real computation, just the gist of the change with respect to the variable.)
 
Physics news on Phys.org

1. What is a covariant derivative?

A covariant derivative is a mathematical tool used in differential geometry to measure how a vector field changes as it moves along a surface or manifold. It takes into account the curvature of the surface and the changing orientation of the coordinate system.

2. How does a covariant derivative differ from a regular derivative?

A regular derivative measures the rate of change of a function with respect to a single variable. A covariant derivative, on the other hand, measures the rate of change of a vector field with respect to a curved surface or manifold. It takes into account the changing basis vectors and curvature of the surface.

3. What are some real-world applications of covariant derivatives?

Covariant derivatives are used in various fields of science, such as physics, engineering, and computer graphics. They are particularly useful in general relativity for describing the curvature of space-time and in fluid dynamics for analyzing fluid flow on curved surfaces.

4. How can one visualize a covariant derivative?

One way to visualize a covariant derivative is to imagine a vector field on a curved surface. As the vector field moves along the surface, the covariant derivative measures how the vector changes in magnitude and direction due to the curvature of the surface. Another way is to picture a particle moving along a curved path and the covariant derivative measures how its velocity changes as it moves along the path.

5. Are there any helpful resources for understanding covariant derivatives?

Yes, there are many online resources, textbooks, and research papers that explain covariant derivatives in detail. Some helpful resources include Khan Academy, MIT OpenCourseWare, and the book "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo.

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