Editing Metric tensor (section)

===Length and angle===
Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of [[tangent vector]]s to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute the [[dot product]] of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surface {{mvar|M}} can be written in the form

:<math>\mathbf{p} = p_1\vec{r}_u + p_2\vec{r}_v</math>

for suitable real numbers {{math|''p''<sub>1</sub>}} and {{math|''p''<sub>2</sub>}}. If two tangent vectors are given:

:<math>\begin{align}
  \mathbf{a} &= a_1\vec{r}_u + a_2\vec{r}_v \\
  \mathbf{b} &= b_1\vec{r}_u + b_2\vec{r}_v
\end{align}</math>

then using the [[bilinear form|bilinearity]] of the dot product,

:<math>\begin{align}
  \mathbf{a} \cdot \mathbf{b}
    &= a_1 b_1 \vec{r}_u\cdot\vec{r}_u + a_1b_2 \vec{r}_u\cdot\vec{r}_v + a_2b_1 \vec{r}_v\cdot\vec{r}_u + a_2 b_2 \vec{r}_v\cdot\vec{r}_v \\[8pt]
    &= a_1 b_1 E + a_1b_2 F + a_2b_1 F + a_2b_2G. \\[8pt]
    &= \begin{bmatrix} a_1 & a_2 \end{bmatrix}
         \begin{bmatrix} E & F \\ F & G \end{bmatrix}
         \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \,.
\end{align}</math>

This is plainly a function of the four variables {{math|''a''<sub>1</sub>}}, {{math|''b''<sub>1</sub>}}, {{math|''a''<sub>2</sub>}}, and {{math|''b''<sub>2</sub>}}. It is more profitably viewed, however, as a function that takes a pair of arguments {{math|'''a''' {{=}} [''a''<sub>1</sub> ''a''<sub>2</sub>]}} and {{math|'''b''' {{=}} [''b''<sub>1</sub> ''b''<sub>2</sub>]}} which are vectors in the {{mvar|uv}}-plane. That is, put

:<math>g(\mathbf{a}, \mathbf{b}) = a_1b_1 E + a_1b_2 F + a_2b_1 F + a_2b_2G \,.</math>

This is a [[symmetric function]] in {{math|'''a'''}} and {{math|'''b'''}}, meaning that

:<math>g(\mathbf{a}, \mathbf{b}) = g(\mathbf{b}, \mathbf{a})\,.</math>

It is also [[bilinear form|bilinear]], meaning that it is [[linear functional|linear]] in each variable {{math|'''a'''}} and {{math|'''b'''}} separately. That is,

:<math>\begin{align}
  g\left(\lambda\mathbf{a} + \mu\mathbf{a}', \mathbf{b}\right) &= \lambda g(\mathbf{a}, \mathbf{b}) + \mu g\left(\mathbf{a}', \mathbf{b}\right),\quad\text{and} \\
  g\left(\mathbf{a}, \lambda\mathbf{b} + \mu\mathbf{b}'\right) &= \lambda g(\mathbf{a}, \mathbf{b}) + \mu g\left(\mathbf{a}, \mathbf{b}'\right)
\end{align}</math>

for any vectors {{math|'''a'''}}, {{math|'''a'''′}}, {{math|'''b'''}}, and {{math|'''b'''′}} in the {{mvar|uv}} plane, and any real numbers {{mvar|μ}} and {{mvar|λ}}.

In particular, the length of a tangent vector {{math|'''a'''}} is given by

:<math> \left\| \mathbf{a} \right\| = \sqrt{g(\mathbf{a}, \mathbf{a})}</math>

and the angle {{mvar|θ}} between two vectors {{math|'''a'''}} and {{math|'''b'''}} is calculated by

:<math>\cos(\theta) = \frac{g(\mathbf{a}, \mathbf{b})}{ \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| } \,.</math>