Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product orthogonalty Drectonal cosnes for coordnate transformaton Vector product Velocty due to rgd body rotatons Trple scalar product Trple vector product Second order tensors Examples of second order tensors Scalar multplcaton and addton Contracton and multplcaton The vector of an antsymmetrc tensor Canoncal form of a symmetrc tensor Readng Assgnment: Chapter 2 of Ars, Appendx A of BSL The algebra of vectors and tensors wll be descrbed here wth Cartesan coordnates so the student can see the operatons n terms of ts components wthout the complexty of curvlnear coordnate systems. Defnton of a vector Suppose x,.e., (x 1, x 2, x 3 ), are the Cartesan coordnates of a pont P n a frame of reference, 0123. Let 0123 be another Cartesan frame of reference wth the same orgn but defned by a rgd rotaton. The coordnates of the pont P n the new frame of reference s x j where the coordnates are related to those n the old frame as follows. x = l x = l x + l x + l x j j 1j 1 2 j 2 3j 3 x = l x = l x + l x + l x j j 1 1 2 2 3 3 where l j are the cosne of the angle between the old and new coordnate systems. Summaton over repeated ndces s understood when a term or a product appears wth a common ndex. Defnton. A Cartesan vector, a, n three dmensons s a quantty wth three components a 1, a 2, a 3 n the frame of reference 0123, whch, under rotaton of the coordnate frame to 0123, become components a1, a2, a 3, where a j = lja 2-1
Examples of vectors In Cartesan coordnates, the length of the poston vector of a pont from the orgn s equal to the square root of the sum of the square of the coordnates. The magntude of a vector, a, s defned as follows. a = ( aa) 1/2 A vector wth a magntude of unty s called a unt vector. The vector, a/ a, s a unt vector wth the drecton of a. Its components are equal to the cosne of the angle between a and the coordnate axs. Some specal unt vectors are the unt vectors n the drecton of the coordnate axs and the normal vector of a surface. Scalar multplcaton If α s a scalar and a s a vector, the product αa s a vector wth components, αa, magntude α a, and the same drecton as a. Addton of vectors Coplanar vectors If a and b are vectors wth components a and b, then the sum of a and b s a vector wth components, a +b. The order and assocaton of the addton of vectors are mmateral. a+ b= b+ a ( a+ b) + c= a+ ( b+ c) The subtracton of one vector from another s the same as multplyng one by the scalar (-1) and addng the resultng vectors. If a and b are two vectors from the same orgn, they are colnear or parallel f one s a lnear combnaton of the other,.e., they both have the same drecton. If a and b are two vectors from the same orgn, then all lnear combnaton of a and b are n the same plane as a and b,.,e., they are coplanar. We wll prove ths statement when we get to the trple scalar product. Unt vectors The unt vectors n the drecton of a set of mutually orthogonal coordnate axs are defned as follows. 1 0 0 e(1) = 0, (2) 1, (3) = 0 e = e 0 0 1 2-2
The suffxes to e are enclosed n parentheses to show that they do not denote components. A vector, a, can be expressed n terms of ts components, (a 1, a 2, a 3 ) and the unt vectors. a= a e + a e + a e 1 (1) 2 (2) 3 (3) Ths equaton can be multpled and dvded by the magntude of a to express the vector n terms of ts magntude and drecton. a a a a= a e + e + e a a a 1 2 3 (1) (2) (3) ( λ 1 (1) λ 2 (2) λ 3 (3)) = a e + e + e where λ are the drectonal cosnes of a. A specal unt vector we wll use often s the normal vector to a surface, n. The components of the normal vector are the drectonal cosnes of the normal drecton to the surface. Scalar product Orthogonalty The scalar product (or dot product) of two vectors, a and b s defned as a b= a b cosθ where θ s the angle between the two vectors. If the two vectors are perpendcular to each other,.e., they are orthogonal, then the scalar product s zero. The unt vectors along the Cartesan coordnate axs are orthogonal and ther scalar product s equal to the Kronecker delta. e e =δ () ( j) j 1, = = 0, j j The scalar product s commutatve and dstrbutve. The cosne of the angle measured from a to b s the same as measured from b to a. Thus the scalar product can be expressed n terms of the components of the vectors. ( a1 (1) a2 (2) a3 (3) ) ( b1 (1) b2 (2) b3 (3) ) a b= e + e + e e + e + e = abδ j j = ab 2-3
The scalar product of a vector wth tself s the square of the magntude of the vector. a a= a a cos 0 = a 2 a a= aa = a 2 The most common applcaton of the scalar product s the projecton or component of a vector n the drecton of another vector. For example, suppose n s a unt vector (e.g., the normal to a surface) the component of a n the drecton of n s as follows. a n= a cosθ Drectonal Cosnes for Coordnate Transformaton The propertes of the drectonal cosnes for the rotaton of the Cartesan coordnate reference frame can now be easly llustrated. Suppose the unt vectors n the orgnal system s e () and n the rotated system s e ( j). The components of the unt vector, e ( j), n the orgnal reference frame s l j. Ths can be expressed as the scalar product. e(j) = l1je(1) + l2 je(2) + l3 je(3), j = 1,2,3 e () e = l,, j = 1,2,3 Snce e ( j) s a unt vector, t has a magntude of unty. (j) j e (j) e = 1 = l l = l l + l l + l l, j =1,2,3 (j) ( j) ( j) 1( j) 1( j) 2( j) 2( j) 3( j) 3( j) Also, the axs of a Cartesan system are orthorgonal. 0, f j e() e(j) = 1, f = j thus e e = δ () (j) j 2-4
e () e(j) = lk lkj = l1 l1j + l2 l2 j + l3 l3 j,, j = 1,2,3 = δ j Vector Product The vector product (or cross product) of two vectors, a and b, denoted as a b, s a vector that s perpendcular to the plane of a and b such that a, b, and a b form a rght-handed system. (.e., a, b, and a b have the orentaton of the thumb, frst fnger, and thrd fnger of the rght hand.) It has the followng magntude where θ s the angle between a and b. a b = a b snθ The magntude of the vector product s equal to the area of a parallelogram two of whose sdes are the vectors a and b. Snce the vector product forms a rght handed system, the product b a has the same magntude but opposte drecton as a b,.e., the vector product s not commutatve, b a= a b The vector product of a vector wth tself or wth a parallel vector s zero or the null vector,.e., a a=0. A quantty that s the negatve of tself s zero. Also, the angle between parallel vectors s zero and thus the sne s zero. Consder the vector product of the unt vectors. They are all of unt length and mutually orthogonal so ther vector products wll be unt vectors. Rememberng the rght-handed rule, we therefore have e e = e e = e (2) (3) (3) (2) (1) e e = e e = e (3) (1) (1) (3) (2) e e = e e = e (1) (2) (2) (1) (3) The components of the vector product can be expressed n terms of the components of a and b and applyng the above relatons between the unt vectors. ( ) ( a b= a e + a e + a e b e + b e + b e 1 (1) 2 (2) 3 (3) 1 (1) 2 (2) 3 (3) ( ) ( ) ( = ab ab e + ab ab e + ab ab)e 2 3 3 2 (1) 3 1 1 3 (2) 1 2 2 1 (3) The permutatons of the ndces and sgns n the expresson for the vector product may be dffcult to remember. Notce that the expresson s the same as that for the expanson of a determnate of the matrx, ) 2-5
e e e (1) (2) (3) a a a 1 2 3 b b b 1 2 3. Expanson of determnants are aded by the permutaton symbol, ε jk. ε jk 0, f any two of, j, k are the same = + 1, f jk s an even permutaton of 1, 2, 3 1, f jk s an odd permutaton of 1, 2, 3 The expresson for the vector product s now as follows. a b=ε abe jk j ( k ) Velocty due to rgd body rotatons We wll show that the velocty feld of a rgd body can be descrbed by two vectors, a translaton velocty, v (t), and an angular velocty, ω. A rgd body has the constrant that the dstance between two ponts n the body does not change wth tme. The translaton velocty s the velocty of a fxed pont, O, n the body, e.g., the center of mass. Now consder a new reference frame (coordnate system) wth the orgn at pont O that s translatng wth respect to the orgnal reference frame wth the velocty v (t). The rotaton of the body about O s defned by the angular velocty, ω,.e., wth a magntude ω and a drecton of the axs of rotaton, n, such that the postve drecton s the drecton that a rght handed screw advances when subject to the rotaton,. ω=ω n. Consder a pont P not on the axs of rotaton, havng coordnates x n the new reference frame. The velocty of P n the new reference frame has a magntude equal to the product of ω and the radus of the pont P from the axs of rotaton. Ths radus s equal to the magntude of x and the sne of the angle between x and ω,.e., x snθ. The velocty of pont P n the new reference frame can be expressed as v= ω x v = ω x snθ The velocty feld of any pont of the rgd body n the orgnal reference frame s now () t ( ) v = v + ω x x o where x o s the coordnates of pont O n the orgnal reference frame. Snce ths equaton s vald for any par of ponts n the rgd body, the relatve velocty v between a par of ponts separated by x can be expressed as follows. 2-6
v= ω x Conversely, f the relatve velocty between any par of ponts s descrbed by the above equaton wth the same value of angular velocty, then the moton s due to a rgd body rotaton. Trple scalar product The trple scalar product s the scalar product of the frst vector wth the vector product of the other two vectors. It s denoted as (abc) or [abc]. ( abc) a ( b c) Recall that b c has a magntude equal to the area of a parallelogram wth sdes b and c and a drecton normal to the plane of b and c. The scalar product of ths normal vector and the vector a s equal to the alttude of the parallelepped wth a common orgn and sdes a, b, and c. The trple scalar product has a magntude equal to the volume of a parallelepped wth a common orgn and sdes a, b, and c. The sgn of the trple scalar product can be ether postve or negatve. If a, b, and c are coplanar, then the alttude of the parallelepped s zero and thus the trple scalar product s zero. The trple scalar product can be expressed n terms of the components by usng the earler defntons of the vector product and scalar product. b c= ε bc e ( ) a= a jk j ( k ) m e ( m) a b c = ε a b c e e = ε a b c δ = ε a b c jk m j ( m) ( k) jk m j mk jk k j = ε abc jk j k From the defnton of the permutaton symbol, the trple scalar product s unchanged by even permutatons of a, b, and c but have the opposte algebrac sgn for odd permutatons. Also, f any two of a, b, and c are dentcal, then permutaton of the two dentcal vectors results n a trple scalar products that are dentcal and also opposte n sgn. Ths mples that the trple scalar product s zero f two of the vectors are dentcal. Trple vector product The trple vector product of vectors a, b, and c results from the repeated applcaton of the vector product,.e., a (b c). Snce b c s normal to the plane of a and b and a (b c) s normal to b c, a (b c) must be n the plane of a and b. It s left as an exercse to show that a ( b c) = ( a c) b ( a b) c 2-7
Second order tensors A second order tensor can be wrtten as a 3 3 matrx. A A A A 11 12 13 = A21 A22 A 23 A A A 31 32 33 A tensor s a physcal entty that s the same quantty n dfferent coordnate systems. Thus a second order tensor s defned as an entty whose components transform on rotaton of the Cartesan frame of reference as follows. Apq = lp ljq Aj If A j =A j the tensor s sad to be symmetrc and a symmetrc tensor has only sx dstnct components. If A j =-A j the tensor s sad to be antsymmetrc and such a tensor s characterzed by only three nonzero components for the dagonal terms, A, are zero. The tensor whose j th element s A j s called the transpose A of A. The determnant of a tensor s the determnant of the matrx of ts components. A = ε det jk A1 A2 j A3k Examples of second order tensors A second order tensor we have already encountered s the Kronecker delta δ j. Of ts nne components, the sx off-dagonal components vansh and the three dagonal components are equal to unty. It transforms as a tensor upon transformng ts components to a rotated frame of reference. δ = l l pq p jq j = l l p = δ pq q δ because of the orthogonalty relaton between the drectonal cosnes l j. In fact, the components of δ j n all coordnates reman the same. δ j s called the sotropc tensor for that reason. The transport coeffcents (e.g., thermal conductvty) of an sotropc medum can be expressed as a scalar quantty multplyng δ j. If a and b are two vectors, the set of nne products, a b j =A j, s a second order tensor, for ( ) A = a b = l a l b = l l a b pq p q p jq j p jq j = l l A p jq j. 2-8
An mportant example of ths s the momentum flux tensor. If ρ s the densty and v s the velocty, ρ v s the th component n the drecton O. The rate at whch ths momentum crosses a unt area normal to Oj s the tensor, ρ v v j. Scalar multplcaton and addton If α s a scalar and A a second order tensor, the scalar product of α and A s a tensor αa each of whose components s α tmes the correspondng component of A. The sum of two second order tensors s a second order tensor each of whose components s the sum of the correspondng components of the two tensors. Thus the j th component of A+B s A j + B j. Notce that the tensors must be of the same order to be added; a vector can not be added to a second order tensor. A lnear combnaton of tensors results from usng both scalar multplcaton and addton. αa + βb s the tensor whose j th component s αa j + βb j. Subtracton may therefore be defned by puttng α = 1, β = -1. Any second order tensor can be represented as the sum of a symmetrc part and an antsymmetrc part. For 1 1 A = A + A + A A 2 2 ( ) ( ) j j j j j and changng and j n the frst factor leaves t unchanged but changes the sgn of the second. Thus, 1 1 A= A+ A' + A A' 2 2 ( ) ( ) represents A as the sum of a symmetrc tensor and antsymmetrc tensor. Contracton and multplcaton As n vector operatons, summaton over repeated ndces s understood wth tensor operatons. The operaton of dentfyng two ndces of a tensor and so summng on them s known as contracton. A s the only contracton of A j, A = A + A + A 11 22 33 and ths s no longer a tensor of the second order but a scalar, or a tensor of order zero. The scalar A s known as the trace of the second order tensor A. The notaton tr A s sometmes used. The contracton operaton n computng the trace of a tensor A s analogous to the operaton n the calculaton the magntude of a vector a,.e.,. a 2 = a a = a a = a 1 a 1 + a 2 a 2 + a 3 a 3. If A and B are two second order tensors, we can form 81 numbers from the products of the 9 components of each. The full set of these products s a 2-9
fourth order tensor. Contracted products result n second order or zero order tensors. We wll not have an occason to use products of tensors n our course. The product A j a j of a tensor A and a vector a s a vector whose th component s A j a j. Another possble product of these two s A j a I. These may be wrtten A a and a A, respectvely. For example, the dffusve flux of a quantty s computed as the contracted product of the transport coeffcent tensor and the potental gradent vector, e.g., q = -k T. The vector of an antsymmetrc tensor We showed earler that a second order tensor can be represented as the sum of a symmetrc part and an antsymmetrc part. Also, an antsymmetrc tensor s characterzed by three numbers. We wll later show that the antsymmetrc part of the velocty gradent tensor represents the local rotaton of the flud or body. Here, we wll develop the relaton between the angular velocty vector, ω, ntroduced earler and the correspondng antsymmetrc tensor. Recall that the relatve velocty between a par of ponts n a rgd body was descrbed as follows. v= ω x We wsh to defne a tensor Ω that also can determne the relatve velocty. v= ω x = x Ω The followng relaton between the components satsfes ths relaton. Ω = ε ω j jk k 1 ω = ε Ω 2 k jk j Wrtten n matrx notaton these are as follows. ω1 0 ω3 ω2 ω = ω, Ω= ω 0 ω 2 3 1 ω 3 ω2 ω1 0 The notaton vec Ω s sometmes used for ω. In summary, an antsymmetrc tensor s completely characterzed by the vector, vec Ω. Canoncal form of a symmetrc tensor We showed earler that any second order tensor can be represented as a sum of a symmetrc part and an antsymmetrc part. The symmetrc part s determned by 6 numbers. We now seek the propertes of the symmetrc part. A 2-10
theorem n lnear algebra states that a symmetrc matrx wth real elements can be transformed by ts egenvectors to a dagonal matrx wth real elements correspondng the egenvalues. (see Appendx A of Ars.) If the egenvalues are dstnct, then the egenvector drectons are orthogonal. The egenvectors determne a coordnate system such that the contracted product of the tensor wth unt vectors along the coordnate axs s a parallel vector wth a magntude equal to the correspondng egenvalue. The surface descrbed by the contracted product of all unt vectors n ths transformed coordnate system s an ellpsod wth axes correspondng to the coordnate drectons. The egenvalues and the scalar nvarants of a second order tensor can be determned from the characterstc equaton. det 2 3 ( A j λδ j ) =Ψ λ Φ+ λ Θ λ where Θ= A + A + A = tra Ψ= det A 11 22 33 Φ= A A A A + A A A A + A A A A Assgnment 2.1 22 33 23 32 33 11 31 13 11 22 12 21 a) Relatve velocty of ponts n a rgd body. If x and y are two ponts nsde a rgd body that s translatng and rotatng, determne the relaton between the relatve velocty of these two ponts as a functon of ther relatve postons. If x and y are ponts on a lne parallel to the axs of rotaton, what s ther relatve velocty? If x and y are ponts on opposte sdes of the axs of rotaton but wth equal dstance, r, what s ther relatve velocty? Draw dagrams. b) Prove that: a (b c) = (a b) c c) Show a (b c) vanshes dentcally f two of the three vectors are proportonal of one another. d) Show that f a s coplanar wth b and c, then a (b c) s zero. a b c = a c b a b c e) Prove that: ( ) ( ) ( ) f) Prove that the contracted product of a tensor A and a vector a, A a, transforms under a rotaton of coordnates as a vector. g) Show that you get the same result for relatve velocty whether you use ω or Ω for the rotaton of a rgd body. 2-11