Covariant derivative)

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Derivative refers to the derivative of a point that represents the change rate of the specified function at a point.

For example, to determine the acceleration of an object's velocity at a specific time point, take a tiny increment at the last and last time points of the timeline, and then evaluate the ratio of the increment of velocity to the increment of time. If the ratio is relatively large, it indicates that the speed change per unit time is large, and vice versa. Note that this ratio is the acceleration of the time point (that is, the derivative of the velocity) only when the limit of the small increment on the timeline tends to be zero ).

We can see that the definition of derivative is inseparable from the concept of limit. The expression of the limit was first provided by the French ferma.

The common derivative is the method for defining the derivative on the fluid.

In the Cartesian coordinate system based on Euclidean space, the method for obtaining the derivative of the vector field is similar to the above, that is, to obtain the points with the same coordinate of the two spaces, and then evaluate the ratio of the vector difference to the location change. If the position change is an infinitely small amount, the derivative of the point can be obtained.

However, on the sphere of a manifold, the calculation of the amount of location change is impractical, because when moving a vector, the results will be different as the path is different. A vector rotates around the sphere. Because the curvature is not zero, it may not be the original vector. In other words, there is no uniform coordinate system at each point on the surface, so we need to consider the changes in the coordinate system. In other words, the derivative of the common variable does not depend on the coordinate system.

Connection)

The contact describes a point in the space, which corresponds to another point in the space conversion. This statement implies some assumptions.

Each point on a surface defines an independent space, called a tangent space ). The tangent space is a space composed of all the tangent vectors of the point. These tangent vectors are vectors perpendicular to the normal direction of the point. Second, the cut vectors defined in different cut spaces cannot be operated on each other, such as addition and subtraction. Because the curvature is not considered, these operations are meaningless.

However, the split spaces of these differences are related, and these connections are called connections. The link can convert the vectors in the infinitely close two tangent spaces to the same tangent space. The contact actually reflects the bending degree of the split space.

There are many ways to achieve contact. However, the premise is that the components of the corresponding vectors in these different cut spaces must correspond to each other.

How do I use Contact definitions to change the derivative? Use the inverted triangle plus two vectors at the same point (such as V, u. You can write DVU and read the vector u along the shared derivative of the vector v. The significance of defining a reference vector V is that the moving U vector should consider the changes in its original reference system (the contact is related to the structural changes of the space ), this is the biggest difference between it and the ordinary derivative.

Covariant derivative)