Editing Covariant transformation (section)

===Basis vectors transform covariantly===
A vector can be expressed in terms of basis vectors. For a certain coordinate system, we can choose the vectors tangent to the coordinate grid. This basis is called the coordinate basis.

To illustrate the transformation properties, consider again the set of points ''p'', identifiable in a given coordinate system <math>x^i</math>  where <math>i = 0, 1, \dots</math> ([[manifold]]). A scalar function ''f'', that assigns a real number to every point ''p'' in this space, is a function of the coordinates <math>f\;\left(x^0, x^1, \dots\right)</math>. 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 <math>dc/d\lambda</math> 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
:<math>\mathbf{v}[f] \ \stackrel{\mathrm{def}}{=}\  \frac{df}{d\lambda} = \frac{d\;\;}{d\lambda} f(c(\lambda))</math>

The parallel between the tangent vector and the operator can also be worked out in coordinates
:<math>\mathbf{v}[f] = \frac{dx^i}{d\lambda} \frac{\partial f}{\partial x^i}</math>

or in terms of operators <math>\partial/\partial x^i</math> 
:<math>\mathbf{v} = \frac{dx^i}{d\lambda} \frac{\partial \;\;}{\partial x^i} = \frac{dx^i}{d\lambda} \mathbf{e}_i</math>
where we have written <math>\mathbf{e}_i = \partial/\partial x^i</math>, the tangent vectors to the curves which are simply the coordinate grid itself.

If we adopt a new coordinates system <math>{x'}^i, \;i=0,1,\dots</math> then for each ''i'', the old coordinate <math>{x^i}</math> can be expressed as function of the new system, so <math>x^i\left({x'}^j\right), j=0,1,\dots</math>
Let <math>\mathbf{e}'_i = {\partial}/{\partial {x'}^i}</math> be the basis, tangent vectors in this new coordinates system. We can express <math>\mathbf{e}_i</math> in the new system by applying the [[chain rule]] on ''x''. As a function of coordinates we find the following transformation
:<math>
  \mathbf{e}'_i = \frac{\partial}{\partial {x'}^i} =
                  \frac{\partial x^j}{\partial {x'}^i}
                    \frac{\partial}{\partial x^j} =
                  \frac{\partial x^j}{\partial {x'}^i} 
                    \mathbf{e}_j
</math>
which indeed is the same as the covariant transformation for the derivative of a function.