Editing Metric tensor (section)

===Metric in coordinates===
A system of {{mvar|n}} real-valued functions {{math|(''x''<sup>1</sup>, ..., ''x''<sup>''n''</sup>)}}, giving a local [[coordinates|coordinate system]] on an [[open set]] {{mvar|U}} in {{mvar|M}}, determines a basis of vector fields on {{mvar|U}}
:<math>\mathbf{f} = \left(X_1 = \frac{\partial}{\partial x^1}, \dots, X_n = \frac{\partial}{\partial x^n}\right) \,.</math>

The metric {{mvar|g}} has components relative to this frame given by
:<math>g_{ij}\left[\mathbf{f}\right] = g\left(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}\right) \,.</math>

Relative to a new system of local coordinates, say
:<math>y^i = y^i(x^1, x^2, \dots, x^n),\quad i=1,2,\dots,n</math>

the metric tensor will determine a different matrix of coefficients,
:<math>g_{ij}\left[\mathbf{f}'\right] = g\left(\frac{\partial}{\partial y^i}, \frac{\partial}{\partial y^j}\right).</math>

This new system of functions is related to the original {{math|''g''<sub>''ij''</sub>('''f''')}} by means of the [[chain rule]]
:<math>\frac{\partial}{\partial y^i} = \sum_{k=1}^n \frac{\partial x^k}{\partial y^i}\frac{\partial}{\partial x^k}</math>

so that
:<math>g_{ij}\left[\mathbf{f}'\right] = \sum_{k,l=1}^n \frac{\partial x^k}{\partial y^i} g_{kl}\left[\mathbf{f}\right]\frac{\partial x^l}{\partial y^j}.</math>

Or, in terms of the matrices {{math|''G''['''f'''] {{=}} (''g''<sub>''ij''</sub>['''f'''])}} and {{math|''G''['''f'''′] {{=}} (''g''<sub>''ij''</sub>['''f'''′])}},
:<math>G\left[\mathbf{f}'\right] = \left((Dy)^{-1}\right)^\mathsf{T} G\left[\mathbf{f}\right] (Dy)^{-1}</math>

where {{mvar|Dy}} denotes the [[Jacobian matrix]] of the coordinate change.