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A Gentle Introduction to Tensors - ese.wustl.edu

A Gentle Introduction to TensorsBoaz PoratDepartment of Electrical EngineeringTechnion Israel Institute of 27, 2014 Opening RemarksThis document was written for the benefits of Engineering students, Elec-trical Engineering students in particular, who are curious about physics andwould like to know more about it, whether from sheer intellectual desireor because one s awareness that physics is the key to our understanding ofthe world around us. Of course, anybody who is interested and has somecollege background may find this material useful. In the future, I hope towrite more documents of the same kind. I chose Tensors as a first topic fortwo reasons. First, Tensors appear everywhere in physics, including classi-cal mechanics, relativistic mechanics, electrodynamics, particle physics, andmore. Second, tensor theory, at the most elementary level, requires onlylinear algebra and some calculus as prerequisites.

A Gentle Introduction to Tensors Boaz Porat Department of Electrical Engineering Technion – Israel Institute of Technology boaz@ee.technion.ac.il

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Transcription of A Gentle Introduction to Tensors - ese.wustl.edu

1 A Gentle Introduction to TensorsBoaz PoratDepartment of Electrical EngineeringTechnion Israel Institute of 27, 2014 Opening RemarksThis document was written for the benefits of Engineering students, Elec-trical Engineering students in particular, who are curious about physics andwould like to know more about it, whether from sheer intellectual desireor because one s awareness that physics is the key to our understanding ofthe world around us. Of course, anybody who is interested and has somecollege background may find this material useful. In the future, I hope towrite more documents of the same kind. I chose Tensors as a first topic fortwo reasons. First, Tensors appear everywhere in physics, including classi-cal mechanics, relativistic mechanics, electrodynamics, particle physics, andmore. Second, tensor theory, at the most elementary level, requires onlylinear algebra and some calculus as prerequisites.

2 Proceeding a small stepfurther, tensor theory requires background in multivariate calculus. For adeeper understanding, knowledge of manifolds and some point-set topologyis required. Accordingly, we divide the material into three chapters. Thefirst chapter discusses constant Tensors and constant linear and transformations are inseparable. To put it succinctly, Tensors aregeometrical objects over vector spaces, whose coordinates obey certain lawsof transformation under change of basis. Vectors are simple and well-knownexamples of Tensors , but there is much more to tensor theory than second chapter discusses tensor fields and curvilinear coordinates. It isthis chapter that provides the foundations for tensor applications in third chapter extends tensor theory to spaces other than vector spaces,namely manifolds.

3 This chapter is more advanced than the first two, but allnecessary mathematics is included and no additional formal mathematicalbackground is required beyond what is required for the second ,asopposedtotheformalgeometrical approach. Although this approach is a bit old fashioned, I stillfind it the easier to comprehend on first learning, especially if the learner isnot a student of mathematics or vector spaces discussed in this document are over the fieldRof realnumbers. We will not mention this every time but assume it ,comments,corrections,andcriticisms. Pleasee-mail to 1 Constant Tensors and ConstantLinear Plane VectorsLet us begin with the simplest possible setup: that of plane vectors. Wethink of a plane vector as an arrow having direction and length, as shown length of a physical vector must have physical units; for example: dis-tance is measured in meter, velocity in meter/second, force in Newton, elec-tric field in Volt/meter, and so on.

4 The length of a mathematical vector ,butdirectionmustbemeasuredrelativeto some (possibly arbitrarily chosen) reference direction, and has units of ra-dians (or, less conveniently, degrees). Direction is usually assumed positivein counterclockwise rotation from the reference , by definition, are free to move parallel to themselves anywhere inthe plane and they remain invariant under such moves (such a move is calledtranslation).Vectors are abstract objects, but they may be manipulated numerically andalgebraically by expressing them in bases. Recall that a basis in a plane is2 Figure : A plane vector having length and directionapairofnon-zeroandnon-collinear vectors(e1,e2). When drawing a basis,it is customary to translatee1ande2until their tails touch, as is shown : A basis in the planeThe basis depicted in to beorthonormal; that is, the twovectors are perpendicular and both have unity length.

5 However, a basis neednot be orthonormal. another basis ( e1, e2), whose vectorsare neither perpendicular nor having equal an arbitrary plane vector and let (e1,e2) be expressed in a unique manner as a linear combination of thebasis vectors; that is,x=e1x1+e2x2( )The two real numbers (x1,x2)arecalledthecoordinatesofxin the basis3 Figure : Another basis in the plane(e1,e2). The following are worth are set in bold font whereas coordinates are set in italic basis vectors are numbered by subscripts whereas the coordinatesare numbered by superscripts. This distinction will be important products such ase1x1we place the vector on the left and the scalaron the right. In most linear algebra books the two are reversed thescalar is on the left of the vector. The reason for our convention willbecome clear later, but for now it should be kept in notations from vector-matrix algebra, we may express ( )asx= e1e2 x1x2 ( )For now we will use row vectors to store basis vectors and column vectors tostore coordinates.

6 Later we will abandon expressions such as ( )infavorof more compact and more general of BasesConsider two bases (e1,e2), which we will henceforth call theold basis,and( e1, e2), which we will call thenew basis. See, for example, ,inwhich we have brought the two bases to a common (e1,e2)isabasis,eachofthevectors( e1, e2)canbeuniquelyexpressedas a linear combination of (e1,e2), similarly to ( ): e1=e1S11+e2S21 e2=e1S12+e2S22( )4 Figure : Two bases in the planeEquation ( )isthebasis transformation formulafrom (e1,e2)to( e1, e2).The 4-parameter object Sji,1 i, j 2 is called thedirect transformationfrom the old basis to the new basis. We may also write ( )invector-matrixnotation: e1 e2 = e1e2 S11S12S21S22 = e1e2 S( )The matrixSis thedirect transformation matrixfrom the old basis to thenew basis.

7 This matrix is uniquely defined by the two bases. Note thatthe rows ofSappear as superscripts and the columns appear as subscripts;remember this convention for special case occurs when the new basis is identical with the new basis. Inthis case, the transformation matrix becomes the identity matrixI, whereIii=1andIji=0fori = ( e1, e2)isabasis,eachofthevectors(e1,e2)maybe expressedasalinearcombinationof( e1, e2). Hence the transformationSis perforceinvertible and we can write e1e2 = e1 e2 S11S12S21S22 1= e1 e2 T11T12T21T22 = e1 e2 T( )5whereT=S 1or, equivalently,ST=TS=I. The object Tji,1 i, j 2 is theinverse transformationandTis theinverse transformation summary, with each pair of bases there are associated two we agree which of the two bases is labeled old and which is labeled new,there is a unique direct transformation (from the old to the new) and a uniqueinverse transformation (from the new to the old).

8 The two transformationsare the inverses of each Coordinate Transformation of VectorsEquation ( )expressesavectorxin terms of coordinates relative to a givenbasis (e1,e2). If a second basis ( e1, e2)isgiven,thenxmay be expressedrelative to this basis using a similar formulax= e1 x1+ e2 x2= e1 e2 x1 x2 ( )The coordinates ( x1, x2)differ from (x1,x2), but the vectorxis the situation is depicted in The vectorxis shown in red. Thebasis (e1,e2)anditsassociatedcoordinates(x1,x2 )areshowninblack;thebasis ( e1, e2)anditsassociatedcoordinates( x1, x2) now pose the following question: how are the coordinates ( x1, x2)relatedto (x1,x2)? To answer this question, recall the transformation formulas be-tween the two bases and perform the following calculation: e1 e2 x1 x2 = e1e2 S x1 x2 =x= e1e2 x1x2 ( )Since ( )mustholdidenticallyforanarbitraryvector x, we are led toconclude thatS x1 x2 = x1x2 ( )Or, equivalently, x1 x2 =S 1 x1x2 =T x1x2 ( )6 Figure : A vector in two basesWe have thus arrived at a somewhat surprising conclusion:the coordinatesof a vector when passing from an old basis to a new basis are transformed viathe inverse of the transformation from the old basis to the an example, the direct transformation between the bases in 1 The inverse transformation isT= 1 1 Examination of this result, at least result obtained in the section is important and should be memorized.

9 When a basis is transformed using a direct transformation, the coordinatesof an arbitrary vector are transformed using the inverse transformation. Forthis reasons, vectors are said to becontravariant( they vary in a contrarymanner , in a way of speaking). Generalization to Higher-Dimensional Vec-tor SpacesWe assume that you have studied a course a linear algebra; therefore you arefamiliar with general (abstract) finite-dimensional vector spaces. In particu-lar, ann-dimensional vector space possesses a set ofnlinearly independentvectors, but no set ofn+1linearlyindependentvectors. Abasisforann-dimensional vector spaceVis any ordered set of linearly independent vec-tors (e1,e2,..,en). An arbitrary vectorxinVcan be expressed as a linearcombination of the basis vectors:x=n i=1eixi( )The real numbers in ( ) are called linear coordinates.

10 We will refer to themsimply as coordinates, until we need to distinguish them from curvilinearcoordinates in Chapter2. Note again our preferred convention of writingthe vector on the left of the scalar. If a second basis ( e1, e2,.., en)isgiven,there exist unique transformationsSandTsuch that ei=n j=1ejSji,ei=n j=1 ejTji,T=S 1( )The coordinates ofxin the new basis are related to those in the old basisaccording to the transformation law xi=n j=1 Tijxj( )Equation ( )isderivedinexactlythesamewayas( ). Thus, vectors inann-dimensional space that the rows ofSappear as superscripts and the columns appear assubscripts. This convention is important and should be kept in remark that orthonormality of the bases is nowhere required or evenmentioned. Moreover, we have not even defined the concept of vectors being8orthogonal or normal, although you may know these definitions from previ-ous studies; we will return to this topic later.


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