# What is intrinsic derivative?

1. Jan 18, 2010

### off-diagonal

What is it? and How different between intrinsic(absolute) derivative and covariant derivative?

What is its geometric interpretation?

2. Jan 19, 2010

### atyy

If on a manifold you have two vector fields V and W, then you can take the covariant derivative of V wrt W at every point in the manifold. The vector field W can also be thought of as a family of curves whose tangent vectors form W. The absolute derivative of V at any point on a particular curve in that family is the covariant derivative of V wrt W at that point in the curve.

The geometric idea is that in a vector space, there is an idea of two vectors being parallel. But on a manifold, there is a vector space at each point in the manifold, but there is no predefined notion of vectors at different points on a manifold being parallel. The covariant derivative/absolute derivative defines the notion of "parallel transport" that allows you to say if vectors at two nearby points on a curve are parallel or not.