# Differential Geometry Consider a region of a [[manifold]] $U \subset M$ that has a [[Coordinate Basis|coordinate system]] $\{x^i\}$ and we wish to introduce new functions $\{y^{i'}\}$ given by the equations: $ y^{i'} = f^{i'}(x^j), i', \quad j = 1, \ldots n $ These equations are a coordinate transformation if the [[Jacobian]] matrix of partial derivatives $\partial y^{i'}/ \partial x^j$ has a nonzero [[determinant]] in $U$ (to guarantee [[Non-singular Matrix|invertability]], and this is also equivalent to saying that the $\{y^{i'}\}$ are [[Linear Independence|linearly independent]]). A given point $P \in U$, can be described by two different sets of numbers, $\{x^i\}$ or $\{y^{i'}\}$. Likewise, at $P$ we have two different [[Coordinate Basis|coordinate vector bases]], $\{\partial /\partial x^i\}$ and $\{\partial /\partial y^{i'}\}$. By the chain rule: $ \frac{\partial}{\partial y^{i'}} = \frac{\partial x^i}{\partial y^{i'}} \frac{\partial}{\partial x^{i}} $ Thus we conclude: $ \Lambda^{i}_{\,\,i'} = \frac{\partial x^{i}}{\partial y^{i'}} $ With the inverse matrix: $ \Lambda^{i'}_{\,\,i} = \frac{\partial y^{i'}}{\partial x^{i}} $ as one can see from: $ \frac{\partial x^{i}}{\partial y^{i'}} \frac{\partial y^{i'}}{\partial y^{k}} = \delta^{i}_{\,k} $