# tensor

Figure 1.

Figure 2.

Figure 3.

Figure 4.

A tensor is a generalization of a vector. Tensors originated in the 19th century as an abstract mathematical concept. A tensor is specified in terms of a set of coordinates, so that its form changes if a different set of mathematical coordinates is chosen. However, tensors have the property that any equation involving them that is true in one set of coordinates remains true when the equation is written in the same form in any other set of coordinates. A tensor can be thought of as providing information about the rate at which things are changing at a point.

An ordinary vector can be described as a tensor of the first rank and a 2-dimensional matrix as a tensor of the second rank.

## A gentle introduction to tensors

by Lillian R. Lieber from her book *The Einstein Theory
of Relativity* [reproduced with permission; original, easy-to-read layout
retained]

The reader is no doubt familiar

with the words "scalar" and "vector."

A scalar is a quantity which

has magnitude only,

whereas

a vector has

both magnitude and direction.

Thus,

if we say that

the temperature at a certain place

is 70° Fahrenheit,

there is obviously NO DIRECTION

to this temperature,

and hence

TEMPERATURE is a SCALAR.

But

if we say that

an airplane has gone

one hundred miles east,

obviously its displacement

from its original position

is a VECTOR,

whose MAGNITUDE is 100 miles,

and whose DIRECTION is EAST.

Similarly,

a person's AGE is a SCALAR,

whereas

the VELOCITY with which an object moves

is a VECTOR,

and so on;

the reader can easily

find further examples

of both scalars and vectors.

We shall now discuss

some quantities

which come up in our experience

and which are

neither scalars nor vectors,

but which are called

TENSORS.

And,

when we have illustrated and defined these,

we shall find that

a SCALAR is a TENSOR whose RANK is ZERO,

and

a VECTOR is a TENSOR whose RANK is ONE,

and we shall see what is meant by

a TENSOR of RANK TWO, or THREE, etc.

Thus "TENSOR" is a more inclusive term

of which "SCALAR" and "VECTOR" are

SPECIAL CASES.

Before we discuss

the physical meaning of

a tensor of rank two,

let us consider

the following facts about vectors.

Suppose that we have

any vector, *AB*, in a plane,

and suppose that

we draw a pair of rectangular axes,

*X* and *Y* (see Figure 1).

Drop a perpendicular *BC*

from *B* to the *X*-axis.

Then we may say that

*AC* is the *X*-component of *AB*;

for,

as we know from

the elementary law of "The parallelogram
of forces,"

if a force *AC* acts on a particle

and *CB* also acts on it,

the resultant effect is the same

as the of a force *AB* alone.

And that is why

*AC* and *CB* are called

the "components" of *AB*.

Of course if we had used

the dotted lines as axes instead,

the components of *AB* would now be *AD* and *DB*.

In other words,

the vector *AB* may be broken up

into components

in various ways,

depending on our choice of axes.

Similarly,

if we use THREE axes in SPACE

rather than two in a plane,

we can break up a vector

into THREE components

as shown in Figure 2.

By dropping the perpendicular *BD*

from *B* to the *XY*-plane,

and then drawing

the perpendiculars *DC* and *DE*

to the *X* and *Y* axes, respectively,

we have the three components of *AB*,

namely,

*AC*, *AE*, and *DB*,

and, as before,

the components depend upon

the particular choice of axes.

Let us now illustrate

the physical meaning

of a tensor of rank two.

Suppose we have a rod

at every point of which

there is a certain strain

due to some force acting on it.

As a rule the strain

is not the same at all points,

and, even at any given point,

the train is not the same in

all directions.

Now, if the STRESS at the point *A*

(that is, the FORCE causing the strain at *A*)

is represented

both in magnitude and direction

by *AB* (see Figure 3)

and if we are interested to know

the effect of this force upon

the surface *CDEF* (through *A*),

we are obviously dealing

with a situation which depends

not on a SINGLE vector

but on TWO vectors.

Namely,

one vector, *AB*,

which represents the force in question,

and another vector

(call it *AG*),

whose direction will indicate

the ORIENTATION of this surface *CDEF*,

and whose magnitude will represent

the AREA of *CDEF*.

In other words,

the effect of a force upon a surface

depends NOT ONLY on the force itself

but ALSO on the

size and orientation of the surface.

Now, how can we indicate

the orientation of a surface

by a line?

If we draw a line through *A*

in the plane *CDEF*,

obviously we can draw this line

in many different directions,

and there is no way

of choosing any one of these

to represent the orientation of this surface.

BUT,

if we take a line through *A*

PERPENDICULAR to the plane *CDEF*,

such a line is UNIQUE

and CAN therefore be used

to specify the orientation

of the surface *CDEF*.

Hence, if we draw a vector, *AG*,

in a direction perpendicular to *CDEF*

and of such a length that

it represents the magnitude of

the area of *CDEF*,

then obviously

this vector *AG*

indicates clearly

both the SIZE and ORIENTATION

of the surface *CDEF*.

Thus,

the STRESS at *A*

upon the surface *CDEF*

depends upon the TWO vectors,

*AB* and *AG*,

and is called a TENSOR of RANK TWO.

Let us now find a convenient way

of representing this tensor.

And, in order to do so,

let us consider the stress, *F*,

upon a small surface, *dS*,

represented in the figure below

by *ABC* (= *dS*).

Now if *OG*, perpendicular to *ABC*,

is the vector which represents

the size and orientation of *ABC* (see Figure 4),

then,

it is quite easy to see

that the *X*-component of *OG*

represents in magnitude and direction

the projection *OBC* of *ABC* upon the *YZ*-plane.

And similarly,

the *Y* and *Z* components of *OG*

represent the projections

*OAC* and *OAB*, respectively.

Now, if the force *F*,

which is itself a vector,

acts on *ABC*,

we can examine its total effect

by considering separately

the effects of its three components

upon EACH of the three projectionsf,_{X}f, and_{Y}f_{Z}

Let us designate these projectionsOBC,OAC, andOAB.

by

*dS*,

_{X}*dS*, and

_{Y}*dS*, respectively.

_{Z}

Now,

since *f _{X}*

(which is the

*X*-component of

*F*)

acts upon EACH one of the three

above-mentioned projections,

let us designate the pressure

due to this component alone

upon the three projections

*p*,

_{XX}*p*,

_{XY}*p*,

_{XZ}

respectively.

We must emphasize

the significance of this notation:

In the first place,

the reader must distinguish between

the "pressure" on a surface

and the "force" acting on the surface.

The "pressure" is

the FORCE PER UNIT AREA.

So that

the TOTAL FORCE is obtained by

MULTIPLYING

the PRESSURE by the AREA of the surface.

Thus the product

*p*.

_{XX}*dS*

_{X}

gives the force acting upon

the projection *dS _{X}*

due to the action of

*f*ALONE.

_{X}Note the DOUBLE subscripts in

*p*,

_{XX}*p*,

_{XY}*p*:

_{XZ}

The first one obviously refers to the fact

that

these three pressures all emanate

from the component *f _{X}* alone;

whereas,

the second subscript designates

the particular projection upon which

the pressure acts.

Thus

*p*means

_{XY}the pressure due to

*f*

_{X}upon the projection

*dS*,

_{Y}etc.

It follows therefore that

*f*=

_{X}*p*.

_{XX}*dS*+

_{X}*p*.

_{XY}*dS*+

_{Y}*p*.

_{XZ}*dS*

_{Z}

And, similarly,

*f*=

_{Y}*p*.

_{YX}*dS*+

_{X}*p*.

_{YY}*dS*+

_{Y}*p*.

_{YZ}*dS*

_{Z}

and

*f*=

_{Z}*p*.

_{ZX}*dS*+

_{X}*p*.

_{ZY}*dS*+

_{Y}*p*.

_{ZZ}*dS*

_{Z}

Hence the TOTAL STRESS, *F*,

on the surface *dS*,

is

*F*=

*f*+

_{X}*f*+

_{Y}*f*

_{Z}

*F*=

*f*=

_{X}*p*.

_{XX}*dS*+

_{X}*p*.

_{XY}*dS*+

_{Y}*p*.

_{XZ}*dS*

_{Z}+

*f*=

_{Y}*p*.

_{YX}*dS*+

_{X}*p*.

_{YY}*dS*+

_{Y}*p*.

_{YZ}*dS*

_{Z}+

*f*=

_{Z}*p*.

_{ZX}*dS*+

_{X}*p*.

_{ZY}*dS*+

_{Y}*p*.

_{ZZ}*dS*

_{Z}

Thus we see that

stress is not just a vector,

with three components in

three-dimensional space

but has NINE components

in THREE-dimensional space.

Such a quantity is called

a TENSOR OF RANK TWO.

It is obvious that

if we were dealing with a plane

instead of with

three-dimensional space,

a tensor of rank two would then have

only FOUR components instead of nine,

since each of the two vectors involved

has only two components in a plane,

and therefore,

there would now be only

2 × 2 components for the tensor

instead of 3 × 3 as above.

And, in general,

if we are dealing with

*n*-dimensional space,

a tensor of rank two

has *n*^{2} components

which are therefore conveniently written

in a SQUARE array

as was done above.

Whereas,

in *n*-dimensional space,

a VECTOR has only *n* components.

Thus,

a VECTOR in a PLANE

has TWO components;

in THREE-dimensional space it has

THREE components,

and so on.

Hence,

the components of a VECTOR

are therefore written

in a SINGLE ROW;

instead of in a SQUARE ARRAY

as in the case of a TENSOR of RANK TWO.

Similarly,

in *n*-dimensional space

a TENSOR of rank THREE has *n*^{3} components,

and so on.

To sum up:

In *n*-dimensional space,

a VECTOR has *n* components,

a TENSOR of rank TWO has *n*^{2} components,

a TENSOR of rank THREE has *n*^{3} components,

and so on.