## tensorA 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 tensorsby Lillian R. Lieber from her bookThe 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,thus: Drop a perpendicular BCfrom 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 particleand 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 calledthe "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 upinto 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 the diagram below. By dropping the perpendicular BDfrom B to the XY-plane,and then drawing the perpendiculars DC and DEto 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 ABand 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 Ain 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 APERPENDICULAR 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 CDEFand of such a length that it represents the magnitude of the area of CDEF,then obviously this vector AGindicates clearly both the SIZE and ORIENTATION of the surface CDEF. Thus, the STRESS at Aupon the surface CDEFdepends 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,then, it is quite easy to see that the X-component of OGrepresents in magnitude and direction the projection OBC of ABC upon the YZ-plane.And similarly, the Y and Z components of OGrepresent 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 projections Let us designate these projections 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 by 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 gives the force acting upon the projection dS_{X}due to the action of f ALONE._{X}Note the DOUBLE subscripts in The first one obviously refers to the fact that these three pressures all emanate from the component f alone;_{X}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 And, similarly, and Hence the TOTAL STRESS, F,on the surface dS,is or 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} componentswhich 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 spacea 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. ## Related category• ALGEBRA | |||||||||

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