Saltwater intrusion

2014-01-07T17:59:05Z

122.164.159.160: /* Groundwater extraction */

In [[mathematics]] and [[theoretical physics]], the '''induced metric''' is the [[metric tensor]] defined on a [[submanifold]] which is calculated from the metric tensor on a larger [[manifold]] into which the submanifold is embedded. It may be calculated using the following formula (written using [[Einstein summation convention]]):

:<math>g_{ab} = \partial_a X^\mu \partial_b X^\nu g_{\mu\nu} (X^\alpha) \ </math>

Here <math>a,b \ </math> describe the indices of coordinates <math>\xi^a \ </math> of the submanifold while the functions <math>X^\mu(\xi^a) \ </math> encode the embedding into the higher-dimensional manifold whose tangent indices are denoted <math>\mu,\nu \ </math>.

==Example - Curve on a torus==

Let
: <math>\begin{align}
\Pi\colon \mathcal{C} &\to \mathbb{R}^3 \\
\tau &\mapsto \left\{\quad\begin{matrix}x^1=(a+b\cos(n\cdot \tau))\cos(m\cdot \tau)\\x^2=(a+b\cos(n\cdot \tau))\sin(m\cdot \tau)\\x^3=b\sin(n\cdot \tau)\end{matrix}\right.
\end{align}</math>

be a map from the domain of the curve <math>\mathcal{C}</math> with parameter <math>\tau</math> into the euclidean manifold <math>\mathbb{R}^3</math>. Here <math>a,b,m,n\in\mathbb{R}</math> are constants.

Then there is a metric given on <math>\mathbb{R}^3</math> as

:<math>g=\sum\limits_{\mu,\nu}g_{\mu\nu}\mathrm{d}x^\mu\otimes \mathrm{d}x^\nu\quad\text{with}\quad
g_{\mu\nu} = \begin{pmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{pmatrix}
</math>.

and we compute

:<math>g_{\tau\tau}=\sum\limits_{\mu,\nu}\frac{\partial x^\mu}{\partial \tau}\frac{\partial x^\nu}{\partial \tau}\underbrace{g_{\mu\nu}}_{\delta_{\mu\nu}} = \sum\limits_\mu\left(\frac{\partial x^\mu}{\partial \tau}\right)^2=m^2 a^2+2m^2ab\cos(n\cdot \tau)+m^2b^2\cos^2(n\cdot \tau)+b^2n^2
</math>

Therefore <math>g_\mathcal{C}=(m^2 a^2+2m^2ab\cos(n\cdot \tau)+m^2b^2\cos^2(n\cdot \tau)+b^2n^2)\mathrm{d}\tau\otimes \mathrm{d}\tau</math>

==See also==
*[[First fundamental form]]

[[Category:Differential geometry]]

{{physics-stub}}

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Saltwater intrusion