Covariant Derivative

A covariant derivative of a vector field in the direction of the vector denoted is defined by the following properties for any vector v, vector fields u, w and scalar functions f and g:

  1. is algebraically linear in so
  2. is additive in so
  3. obeys the product rule, i.e. where is defined above.

Note that at point p depends on the value of v at p and on values of u in a neighbourhood of p because of the last property, the product rule.

Read more about Covariant Derivative:  Coordinate Description, Examples, Notation, Derivative Along Curve, Relation To Lie Derivative

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