Talk:PlanetPhysics/Basic Tensor Theory

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(\htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} in progress)

\section{BASIC TENSOR THEORY}

\htmladdnormallink{Tensor}{http://planetphysics.us/encyclopedia/Tensor.html} analysis is the study of invariant \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, whose properties must be independent of the coordinate \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} used to describe the objects. A tensor is represented by a set of \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} 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 \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html}, base vectors, \htmladdnormallink{metric}{http://planetphysics.us/encyclopedia/MetricTensor.html} coefficients for the length of a line, Gaussian curvature, and the Newtonian gravitation potential.

Many of the important \htmladdnormallink{differential equations}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} for physics, engineering, and applied mathematics can also be written as tensors. Examples of differential equations that can be written in tensor form are \htmladdnormallink{Lagrange's equations}{http://planetphysics.us/encyclopedia/LagrangesEquations.html} of \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} and \htmladdnormallink{Laplace's equation}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html}. 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 \htmladdnormallink{summation notation}{http://planetphysics.us/encyclopedia/EinsteinSummationNotation.html} used throughout this \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} will be of the \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html}:

\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 \htmladdnormallink{powers}{http://planetphysics.us/encyclopedia/Power.html}; 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, \htmladdnormallink{Einstein}{http://planetphysics.us/encyclopedia/AlbertEinstein.html} 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 \htmladdnormallink{scalars}{http://planetphysics.us/encyclopedia/Vectors.html} that are transformed from one co-ordinate system to another by means of the functional determinate known as the Jacobian. To illustrate this \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html}, 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 \htmladdnormallink{volumes}{http://planetphysics.us/encyclopedia/Volume.html} and \htmladdnormallink{mass}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}. To illustrate this concept, we start with the equation for an incremental mass in orthogonal cartesian coordinates.

\begin{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:

\begin{center} \includegraphics[scale=0.6]{Figure1-2.eps}

Fig. 1-2. \end{center}

$$ x = r \sin \phi \cos \theta $$ $$ y = r \sin \phi \sin \theta $$ $$ {z}= {r}\cos\phi $$

The partial derivatives for the Jacobian $\left| \frac{\partial x^i}{\partial y^j}\right|$ are:

$$ \frac{\partial x}{\partial r} = \sin \phi \cos \theta $$ $$ \frac{\partial x}{\partial \phi} = r\cos \phi \cos \theta $$

$$ \frac{\partial x}{\partial \theta} = -r \sin \phi \sin \theta $$

$$ \frac{\partial y}{\partial r} = \sin \phi \sin \theta $$ $$ \frac{\partial y}{\partial \phi} = r\cos \phi \sin \theta $$

$$ \frac{\partial y}{\partial \theta} = r \sin \phi \cos \theta $$

$$ \frac{\partial z}{\partial r} = \cos \theta $$ $$ \frac{\partial z}{\partial \phi} = -r\sin \phi $$

$$ \frac{\partial z}{\partial \theta} = 0 $$

Using these values, the resultant determinate is:

\begin{equation} \left | \begin{array}{lll} \sin \phi \cos \theta & r \cos \phi \cos \theta & -r \sin \phi \sin \theta \\ \sin \phi \sin \theta & r \cos \phi \sin \theta & r \sin \phi \cos \theta \\ \cos \phi & -r \sin \phi & 0 \end{array} \right | = r^2 \sin \phi \end{equation}

Now, Equation 5 can be evaluated:

\begin{equation} \overline{S} = r^2 \sin \phi \rho \end{equation}

and the equation for $d \overline{M}$ is:

\begin{equation} d \overline{M} = \rho r^2 \sin \phi dr d\phi d\theta \end{equation}

Equation 11 is the desired result for $d \overline{M}$. If the value for $d \overline{M}$ had been given initially in spherical coordinates, the corresponding value in cartesian coordinates could be found by the equation:

\begin{equation} dM = S dx dy dz \end{equation}

where,

\begin{equation} S = \left | \frac{\partial y^i}{\partial x^j} \right | \overline{S} \end{equation}

\subsection{Admissible Transformations}

From the previous examples, it has been demonstrated that relative tensors transform from one coordinate system to another by means of the functional determinate known as the Jacobian. Since a relative tensor is defined to be a function of the Jacobian, a necessary and sufficient condition for an admissible transformation of coordinates is that it is a member of a set in which the Jacobian does not vanish. This condition is also necessary and sufficient for absolute tensors. Therefore, the set of all admissible transformations of co-ordinates form a \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} with non vanishing Jacobians. If notation $\left| J \right |$ is used for the Jacobian, the definition for an admissible transformation of co-ordinates can be expressed:

\begin{equation} \left| J \right | \ne 0 \end{equation}

Another property of an admissible transformation is:

\begin{equation} \left| \frac{\partial x^i}{\partial y^j} \right | \left| \frac{\partial y^i}{\partial x^j} \right | = 1 \end{equation}

An example of Equation 15 can be found in Jacobian Equation 2 and 3.

\begin{equation} \left( r \right ) \left ( \frac{1}{r}\right) = 1 \end{equation}

\subsection{N Dimensional Space}

In general terms a coordinate system represents a one-to-one correspondence of a point or object with a set of numbers. To measure distance, we can use a rectangular cartesian coordinate system. This is called a metric \htmladdnormallink{manifold}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}, or space of $V_3$.

Now, a space or manifold of $N$ dimensions is expressed by the symbol $V_N$; and it is a coordinate system of $N$ dimensions, if for each set of N numbers there is one corresponding point or object.

A sub space $V_M$ where $M = N - 1$ is called a hypersurface. An example of a hypersurface in Euclidean space is a plane. It is a hypersurface of $V_2$.

\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 \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} work of

"Principles and Applications of Tensor Analysis" By MATTHEW S. SMITH

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