basic tensor theoryBasicTensorTheory2009-02-05 03:12:532009-02-06 01:39:10Topic% this is the default PlanetMath preamble. as your knowledge
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\section{BASIC TENSOR THEORY}
Tensor analysis is the study of invariant objects, whose properties must be independent of the coordinate systems used to describe the objects. A tensor is represented by a set of functions called components. For an object to be a tensor it must be an invariant that transforms from one acceptable coordinate system to another by the tensor rules.
Several examples of tensors are velocity vector, base vectors, metric coefficients for the length of a line, Gaussian curvature, and the Newtonian gravitation potential.
Many of the important differential equations for physics, engineering, and applied mathematics can also be written as tensors. Examples of differential equations that can be written in tensor form are Lagrange's equations of motion and Laplace's equation. When an equation is written in tensor form it is in a general form that applies to all admissible coordinate systems.
\subsection{Summation Notation}
The summation notation used throughout this section will be of the type:
\begin{equation}
\sum_{i=1}^n = a_i x^i =\mathrm{a}_{1}\mathrm{x}_{1}+\mathrm{a}_{2}\mathrm{x}_{2}+\ldots+\mathrm{a}_{\mathrm{n}}\mathrm{x}_{n}
\end{equation}
The superscripts on $\mathrm{x}$ are not powers; they are used to distinguish between the various $\mathrm{x}' \mathrm{s}$. In rectangular cartesian coordinates and vector notation, Equation 1 would be:
$$ \sum_{i=1}^{3} a_i x^i $$
where,
$\mathrm{x}^{1}=\mathrm{x}, \mathrm{x}^{2}=\mathrm{y}, \mathrm{x}^{3}=\mathrm{z}$
$\mathrm{a}_{1}=\mathrm{i}, \mathrm{a}_{2}=\mathrm{j}, \mathrm{a}_{\mathrm{3}}=\mathrm{k}$
With this interpretation of Equation 1.1 and the specific values for $\mathrm{a}_{\mathrm{i}}$ and $\mathrm{x}^{\mathrm{i}}$ as noted, sum $\mathrm{S}$ would be:
$$
\mathrm{S}=\mathrm{i}\mathrm{x}+\mathrm{j}\mathrm{y}+\mathrm{k}\mathrm{z}
$$
For additional simplification, Einstein dropped the $\sum$ in Equation 1.1 and the summation is then expressed
$$
\mathrm{S}=\mathrm{a}_{i}\mathrm{x}^{i}
$$
This short cut is referred to as Einstein notation or Einstein summation convention. Further, a superscript index will indicate a contravariant tensor, while a subscript index will indicate a covariant tensor.
The rank of a tensor is the sum of the covariant and contravariant indexes.
\subsection{Relative Tensors}
The term relative tensor is used to describe scalars that are transformed from one co-ordinate system to another by means of the functional determinate known as the Jacobian. To illustrate this concept, the differential increment of area$(\mathrm{d}\mathrm{A})$ is indicated in Fig. 1-1.
In cartesian coordinates $(\mathrm{x},\ \mathrm{y})$ it is:
\begin{center}
\includegraphics[scale=0.3]{image010.eps}
Fig. 1-1.
\end{center}
$$
\mathrm{d}\mathrm{A}=\mathrm{d}\mathrm{x}\text{ }dy
$$
In polar coordinates $(\mathrm{r},\ \mathrm{\theta})$ it is :
$$
\mathrm{d}\mathrm{A}=\mathrm{r}\mathrm{d}\theta dr
$$
Now, the connection between the $\mathrm{x}, \mathrm{y}$ cartesian coordinates and the $\mathrm{r}, \theta$ polar coordinates is:
$$
\mathrm{x}=\mathrm{r}\cos \theta
$$
$$
\mathrm{y}=\mathrm{r}\sin \theta
$$
$$
\mathrm{r}=(\mathrm{x}^{2}+\mathrm{y}^{2})^{1/2}
$$
$$
\theta =\tan^{-1} \frac{y}{x}
$$
The Jacobian of the cartesian coordinates with respect to the polar coordinates is formed from the following partial derivatives :
$$
\frac{\partial \mathrm{x}}{\partial \mathrm{r}}=\cos \theta \text{ } \frac{\partial \mathrm{x}}{\partial \theta}=-\mathrm{r}\sin \theta
$$
$$
\frac{\partial \mathrm{y}}{\partial \mathrm{r}}=\sin \theta \text{ } \frac{\partial \mathrm{y}}{\partial \theta}=\mathrm{r}\cos \theta
$$
This set of partial derivatives are used to form the following Jacobian:
\begin{equation}
\left|\begin{array}{lll}
\cos \theta & -r \sin \theta \\
\sin \theta & r \cos \theta
\end{array}\right| =r\left(\cos^{2}\theta +\sin^{2}\theta \right)= r
\end{equation}
In the same manner, the Jacobian of polar cooordinates with respect to the cartesian coordinates is formed from the following partial derivatives:
$$
\frac{\partial r}{\partial x}= \cos \theta \text{ }\frac{\partial r}{\partial y}=\sin \theta
$$
$$
\frac{\partial \theta}{\partial x}=-\frac{1}{r}\sin \theta \text{ }\frac{\partial \theta}{\partial y}=\frac{1}{r}\cos \theta
$$
This set of partial derivatives are used to form the following Jacobian:
\begin{equation}
\left|\begin{array}{ll}
\cos \theta & \sin \theta \\
-\frac{1}{r}\sin \theta & \frac{1}{r}\cos \theta
\end{array}\right|=\frac{1}{r} \left( \cos^2 \theta +\sin^2 \theta \right) = \frac{1}{r}
\end{equation}
Now, returning to the expression for differential area in
cartesian coordinates and polar co-ordinates, the following
equation can be written:
$$
S dx dy =\overline{S} dr d \theta
$$
where,
$S=1$
$\overline{S}=r$
$S$ and $\overline{S}$ are called relative tensors, as they are related by the equations:
\begin{equation}
{S}=\left|\frac{\partial {y}^{ {i}}}{\partial {x}^{ {j}}}\right|^{ {n}}\text{ }\overline{ {S}}
\end{equation}
\begin{equation}
\overline{ {S}}=\left|\frac{\partial {x}^{ {i}}}{\partial {y}^{ {j}}}\right|^{ {n}}\text{ } {S}
\end{equation}
Exponent $n$ in Equations 4 and 5 is used to determine the weight of a relative scalar. The examples in this section are relative scalars having a weight equaling one; therefore, $n = 1$. An absolute scalar has a weight of zero; i.e., $n = 0$. To illustrate Equation 4, we use the values:
$$
\overline{ {S}}= {r}
$$
$$
\left|\frac{\partial {y}^{ {i}}}{\partial {x}^{ {j}}}\right|=\left | \begin{array}{ll}
\cos \theta & \sin \theta \\
-\frac{1}{ {r}}\sin\theta & \frac{1}{ {r}}\cos \theta \end{array}\right|=\frac{1}{ {r}}
$$
$$
{y}^{ {i}}\text{ }ranges\text{ }from\text{ } {i}=1\text{ }to\text{ } {i}=2
$$
$$
{x}^{ {j}}\text{ }ranges\text{ }from\text{ } {i}=1\text{ }to\text{ } {j}\text{ }=2
$$
$$
{y}^{1}= {r},\text{ } {x}^{1}= {x}
$$
$$
{y}^{2}=\theta ,\text{ } {x}^{2}= {y}
$$
\begin{equation}
{S}=\frac{1}{ {r}}( {r})=1
\end{equation}
Equation 6 is the desired result.
Now the notion of relative tensors can be extended to volumes and mass. To illustrate this concept, we start with the equation for an incremental mass in orthogonal cartesian coordinates.
\beginn{equation}
dM = \rho dx dy dz
\end{equation}
Now the incremental mass in spherical coordinates is written in terms of relative tensor $\overline{S}$:
$$
d \overline{M} = \overline{S} dr d\phi d\theta
$$
$\overline{ {S}}$ is evaluated by the relative tensor equation:
\begin{equation}
\overline{S} = \left | \frac{\partial x^i}{\partial y^j} \right | S
\end{equation}
In this example $S = \rho$, where $\rho$ is called the scalar density.
$$
{x}^{1}= {x},\text{ } {x}^{2}= {y},\text{ } {x}^{3}= {z}
$$
$$
{y}^{1}= {r},\text{ } {y}^{2}=\phi,\text{ } {y}^{3}= \theta
$$
The geometrical relationship between the cartesian coordinates and the spherical co-ordinates is indicated in Fig. 1-2. The corresponding mathematical relationship between the coordinates is:
\subsection{Admissible Transformations}
\subsection{N Dimensional Space}
\subsection{Contravariant Tensors}
\subsection{Covariant Tensors}
\subsection{Higher Rank and Mixed Tensors}
\subsection{Metric Tensors and the Line Element}
\subsection{Base Vectors}
\subsection{Associated Tensors and the Inner Product}
\subsection{Kronecker Deltas}
This is a Derivative work from the public domain work of
"Principles and Applications of Tensor Analysis"
By MATTHEW S. SMITH