Editing Metric tensor (section)

===Induced metric===
<!--{{main|Induced metric}} Not currently well-written. -->
Let {{mvar|U}} be an [[open set]] in {{math|'''ℝ'''<sup>''n''</sup>}}, and let {{mvar|φ}} be a [[continuously differentiable]] function from {{mvar|U}} into the [[Euclidean space]] {{math|'''ℝ'''<sup>''m''</sup>}}, where {{math|''m'' > ''n''}}. The mapping {{mvar|φ}} is called an [[immersion (mathematics)|immersion]] if its differential is [[injective]] at every point of {{mvar|U}}. The image of {{mvar|φ}} is called an [[immersed submanifold]]. More specifically, for {{math|1=''m'' = 3}}, which means that the ambient Euclidean space is {{math|'''ℝ'''<sup>''3''</sup>}}, the induced metric tensor is called the [[first fundamental form]].

Suppose that {{mvar|φ}} is an immersion onto the submanifold {{math|''M'' ⊂ '''R'''<sup>''m''</sup>}}. The usual Euclidean [[dot product]] in {{math|'''ℝ'''<sup>''m''</sup>}} is a metric which, when restricted to vectors tangent to {{mvar|M}}, gives a means for taking the dot product of these tangent vectors. This is called the '''induced metric'''.

Suppose that {{mvar|v}} is a tangent vector at a point of {{mvar|U}}, say
:<math>v = v^1\mathbf{e}_1 + \dots + v^n\mathbf{e}_n</math>

where {{math|'''e'''<sub>''i''</sub>}} are the standard coordinate vectors in {{math|'''ℝ'''<sup>''n''</sup>}}. When {{mvar|φ}} is applied to {{mvar|U}}, the vector {{mvar|v}} goes over to the vector tangent to {{mvar|M}} given by
:<math>\varphi_*(v) = \sum_{i=1}^n \sum_{a=1}^m v^i\frac{\partial \varphi^a}{\partial x^i}\mathbf{e}_a\,.</math>

(This is called the [[pushforward (differential)|pushforward]] of {{mvar|v}} along {{mvar|φ}}.) Given two such vectors, {{mvar|v}} and {{mvar|w}}, the induced metric is defined by
:<math>g(v,w) = \varphi_*(v)\cdot \varphi_*(w).</math>

It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields {{math|'''e'''}} is given by
:<math>G(\mathbf{e}) = (D\varphi)^\mathsf{T}(D\varphi)</math>

where {{mvar|Dφ}} is the Jacobian matrix:
:<math>D\varphi = \begin{bmatrix}
  \frac{\partial\varphi^1}{\partial x^1} & \frac{\partial\varphi^1}{\partial x^2} &
    \dots  & \frac{\partial\varphi^1}{\partial x^n} \\[1ex]
  \frac{\partial\varphi^2}{\partial x^1} & \frac{\partial\varphi^2}{\partial x^2} &
    \dots  & \frac{\partial\varphi^2}{\partial x^n} \\
  \vdots                                 & \vdots                                 &
    \ddots & \vdots \\
  \frac{\partial\varphi^m}{\partial x^1} & \frac{\partial\varphi^m}{\partial x^2} &
    \dots  & \frac{\partial\varphi^m}{\partial x^n}
\end{bmatrix}.</math>