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:
- is algebraically linear in so
- is additive in so
- 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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