To illustrate the transformation properties, consider again the set of points "p", identifiable in a given coordinate system formula_9 where formula_10 (manifold). A scalar function "f", that assigns a real number to every point "p" in this space, is a function of the coordinates formula_11. A curve is a one-parameter collection of points "c", say with curve parameter λ, "c"(λ). A tangent vector v to the curve is the derivative formula_12 along the curve with the derivative taken at the point "p" under consideration. Note that we can see the tangent vector v as an operator (the directional derivative) which can be applied to a function