PlanetPhysics/Basic Tensor Theory

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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.

Summation Notation

The summation notation used throughout this section will be of the type:

$\sum _{i=1}^{n}=a_{i}x^{i}=\mathrm {a} _{1}\mathrm {x} _{1}+\mathrm {a} _{2}\mathrm {x} _{2}+\ldots +\mathrm {a} _{\mathrm {n} }\mathrm {x} _{n}$

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.

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:

\includegraphics[scale=0.3]{image010.eps}

Fig. 1-1.

$\mathrm {d} \mathrm {A} =\mathrm {d} \mathrm {x} ==dy$ 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 : ${\frac {\partial \mathrm {x} }{\partial \mathrm {r} }}=\cos \theta =={\frac {\partial \mathrm {x} }{\partial \theta }}=-\mathrm {r} \sin \theta$

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

This set of partial derivatives are used to form the following Jacobian:

$\left|{\begin{matrix}{lll}\cos \theta &-r\sin \theta \\\sin \theta &r\cos \theta \end{matrix}}\right|=r\left(\cos ^{2}\theta +\sin ^{2}\theta \right)=r$

In the same manner, the Jacobian of polar cooordinates with respect to the cartesian coordinates is formed from the following partial derivatives:

${\frac {\partial r}{\partial x}}=\cos \theta =={\frac {\partial r}{\partial y}}=\sin \theta$ ${\frac {\partial \theta }{\partial x}}=-{\frac {1}{r}}\sin \theta =={\frac {\partial \theta }{\partial y}}={\frac {1}{r}}\cos \theta$

This set of partial derivatives are used to form the following Jacobian:

$\left|{\begin{matrix}{ll}\cos \theta &\sin \theta \\-{\frac {1}{r}}\sin \theta &{\frac {1}{r}}\cos \theta \end{matrix}}\right|={\frac {1}{r}}\left(\cos ^{2}\theta +\sin ^{2}\theta \right)={\frac {1}{r}}$

Now, returning to the expression for differential area in cartesian coordinates and polar co-ordinates, the following equation can be written:

$Sdxdy={\overline {S}}drd\theta$ where,

$S=1$

${\overline {S}}=r$

$S$ and ${\overline {S}}$ are called relative tensors, as they are related by the equations:

${S}=\left|{\frac {\partial {y}^{i}}{\partial {x}^{j}}}\right|^{n}=={\overline {S}}$

${\overline {S}}=\left|{\frac {\partial {x}^{i}}{\partial {y}^{j}}}\right|^{n}=={S}$

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:

${\overline {S}}={r}$ $\left|{\frac {\partial {y}^{i}}{\partial {x}^{j}}}\right|=\left|{\begin{matrix}{ll}\cos \theta &\sin \theta \\-{\frac {1}{r}}\sin \theta &{\frac {1}{r}}\cos \theta \end{matrix}}\right|={\frac {1}{r}}$ ${y}^{i}==ranges==from=={i}=1==to=={i}=2$ ${x}^{j}==ranges==from=={i}=1==to=={j}===2$ ${y}^{1}={r},=={x}^{1}={x}$ ${y}^{2}=\theta ,=={x}^{2}={y}$

${S}={\frac {1}{r}}({r})=1$

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.

$dM=\rho dxdydz$

Now the incremental mass in spherical coordinates is written in terms of relative tensor ${\overline {S}}$:

$d{\overline {M}}={\overline {S}}drd\phi d\theta$

${\overline {S}}$ is evaluated by the relative tensor equation:

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

In this example $S=\rho$, where $\rho$ is called the scalar density.

${x}^{1}={x},=={x}^{2}={y},=={x}^{3}={z}$ ${y}^{1}={r},=={y}^{2}=\phi ,=={y}^{3}=\theta$

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:

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

Fig. 1-2.

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

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:

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

Now, Equation 5 can be evaluated:

${\overline {S}}=r^{2}\sin \phi \rho$

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

$d{\overline {M}}=\rho r^{2}\sin \phi drd\phi d\theta$

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:

$dM=Sdxdydz$

where,

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

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 group 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:

$\left|J\right|\neq 0$

Another property of an admissible transformation is:

$\left|{\frac {\partial x^{i}}{\partial y^{j}}}\right|\left|{\frac {\partial y^{i}}{\partial x^{j}}}\right|=1$

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

$\left(r\right)\left({\frac {1}{r}}\right)=1$

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 manifold, 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 $V_{2}$.

Kronecker Deltas

This is a Derivative work from the public domain work of

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