Here are some notes on vector and dyadic tensor notation similar to what I will be using in class, with just a couple of changes in notation. Notes = Dowling ∧ = × British notation for Cross-Product vs. American (Dowling’s) A A = Notation for Vector vs. Standard (Dowling’s). B = B Notation for Dyadic Tensor vs. Standard (Dowling’s). With these notes you should be able to make sense of the expressions in the vector identities pages below that involve the dyadic tensor T. (Notes courtesy of Peter Littlewood, University of Cambridge.) The vector identity pages are from the NRL Plasma Formulary – order your very own online or download the entire thing at: http://wwwppd.nrl.navy.mil/nrlformulary/ --JPD VECTOR IDENTITIES4 Notation: f; g; are scalars; A, B, etc., are vectors; T is a tensor; I is the unit dyad. (1) A B C = A B C = B C A = B C A = C A B = C A B · × × · · × × · · × × · (2) A (B C) = (C B) A = (A C)B (A B)C × × × × · − · (3) A (B C) + B (C A) + C (A B) = 0 × × × × × × (4) (A B) (C D) = (A C)(B D) (A D)(B C) × · × · · − · · (5) (A B) (C D) = (A B D)C (A B C)D × × × × · − × · (6) (fg) = (gf) = f g + g f r r r r (7) (fA) = f A + A f r · r · · r (8) (fA) = f A + f A r × r × r × (9) (A B) = B A A B r · × · r × − · r × (10) (A B) = A( B) B( A) + (B )A (A )B r × × r · − r · · r − · r (11) A ( B) = ( B) A (A )B × r × r · − · r (12) (A B) = A ( B) + B ( A) + (A )B + (B )A r · × r × × r × · r · r (13) 2f = f r r · r (14) 2A = ( A) A r r r · − r × r × (15) f = 0 r × r (16) A = 0 r · r × If e1, e2, e3 are orthonormal unit vectors, a second-order tensor T can be written in the dyadic form (17) T = Tij eiej Pi;j In cartesian coordinates the divergence of a tensor is a vector with components (18) ( T )i = (@Tji=@xj ) ∇· Pj [This definition is required for consistency with Eq. (29)]. In general (19) (AB) = ( A)B + (A )B r · r · · r (20) (fT ) = f T +f T r · r · ∇· 4 Let r = ix + jy + kz be the radius vector of magnitude r, from the origin to the point x; y; z. Then (21) r = 3 r · (22) r = 0 r × (23) r = r=r r (24) (1=r) = r=r3 r − (25) (r=r3) = 4πδ(r) r · (26) r = I r If V is a volume enclosed by a surface S and dS = ndS, where n is the unit normal outward from V; (27) dV f = dSf Z r Z V S (28) dV A = dS A Z r · Z · V S (29) dV T = dS T Z ∇· Z · V S (30) dV A = dS A Z r × Z × V S (31) dV (f 2g g 2f) = dS (f g g f) Z r − r Z · r − r V S (32) dV (A B B A) Z · r × r × − · r × r × V = dS (B A A B) Z · × r × − × r × S If S is an open surface bounded by the contour C, of which the line element is dl, (33) dS f = dlf Z × r I S C 5 (34) dS A = dl A Z · r × I · S C (35) (dS ) A = dl A Z × r × I × S C (36) dS ( f g) = fdg = gdf Z · r × r I − I S C C 6 EXAMPLES CLASS IN MATHEMATICAL PHYSICS V Intro duction to Tensors Commentary This is the rst of two sheets on tensors the second will come in class VI I I Tensors have many applications in theoretical physics not only in p olarization prob lems like the ones describ ed below but in dynamics elasticity uid mechanics quantum theory etc The mathematical apparatus for dealing with tensor problems may not yet be familiar so this sheet contains quite a high prop ortion of formal material Section A intro duces the concept of a tensor in this case a linear op erator that transforms one vector into another in the physical contextofstudying the p olarisation of an aspherical molecule Section B summarizes various mathematical and notational ideas that are useful in physics these include sux notation and concepts suchastheouter product of twovectors and pro jection op erators These are mathematical ob jects which take the projection or comp onent of a vector along a particular direction Section C returns to a physical problem also ab out p olarization and leads to a dis cussion of principal axes eigenvectors and eigenvalues of tensor quantities Section D intro duces the prop erties of the antisymmetric thirdrank tensor which ij k can be used to derive many results in vector calculus much more rapidly than by other metho ds This ground will be covered again in sheet VI I I so it do esnt matter if you run out of time A Intro duction Weoften nd vectors that are linearly related to other vectors for instance the p olar induced in a medium by applying a eld E isation P P E and E are parallel Their cartesian comp onents are related to each other by Here P the same constant of prop ortionality for each comp onent Here we have denoted vectors by underlinings as one would do if writing this out by hand Usually b o oks denote vectors by b old typ e in the rest of this sheet we treat the two notations as completely interchangeable It is p ossible to imagine situations where the medium is anisotropic and the resp onse factor in is dierent for comp onents in dierent directions The twovectors P and E are no longer necessarily parallel and what connects them is called a tensor We will use simple mo del of molecular p olarisability to show the underlying physics and intro duce the ideas of tensors A An Electrostatics Problem The diacetylene molecule HCCCCH is highly p olarisable along its long axis p E being the induced dip ole for a eld E k k k k applied along the molecule The resp onse to p erp endicular elds is p E where Let the cylindrical molecule point in the direction and apply a eld in k E x the x direction E x i Prove p E x x k p E y x k p z Hence one sees that p and E are not parallel and in general we must write E p where is the p olarisability tensor Tensors are written with a double underline as matricies often are or in b o oks sometimes as doubly bold b old sans serif typ eface characters In comp onent form is p E E E x xx x xy y xz z p E E E y yx x yy y yz z p E E E z zx x zy y zz z ii What are the co ecients explicitly for the ab ove example Write as a matrix of its ij comp onents Note that although the tensor is the same physical entity in all co ordinate systems it relates p with E in whatever basis these are expressed its comp onents ij dep end on the co ordinate system used is p where there Remark If many molecules are present with the same orientation P are molecules per unit volume A hyp othetical diacetylene solid might consist of such ro ds arranged in an arrayall directed along Then the macroscopic to replace in would neglecting internal eld corrections be B Miscellaneous Ideas and Notations B Sux notation Commonly suces such as i and j are dummy symb ols and when rep eated in an expression they are to b e summed over this is the Einstein Convention For instance to can be succinctly written as p E i ij j which is Eq rewritten in sux notation Note the order of the indices on the right Thus we have written the vector p simply as p and it will be clear from context that a i can be denoted vector is intended and not simply one of its comp onents Likewise ij In sux notation the dot pro duct also called scalar pro duct or inner productoftwo vectors a b is written as a b The op eration of a tensor on a vector also involves an inner i i pro duct sum over rep eated indices as describ ed ab ove The unit tensor by denition ij transforms any vector into itself v v ij j i B The outer pro duct Consider the ob ject Q dened as ij Q a b ij i j where a and b are vectors i Show using sux notation that if p isavector Q p is also a vector nd its magnitude and direction Note that Qp Q p We see that Q is a linear op erator that transforms ij j ij one vector p into another it is therefore a tensor This denes the outer product or dyadic product of two vectors In nonsux notation sometimes called dyadic notation Eq is written as Q ab a b where there is no dot between the two vectors In this notation we have or Q ab c ab c prove this using sux notation if you did not do so already ab ove This notation is used in various areas of mathematics and physics but in general sux notation is more versatile and can always be used instead ii We can write the tensor in Q A in the following form x x x y x z other terms xx xy xz Check that indeed agrees with to by nding the result of applying a eld in x the xdirection E E x p E x E E x xx yx zx x iii Pro jection op erators In dyadic notation the identity tensor is often written ij Consider the action on a vector v of the op erators uu and as the unit op erator uu where u is a unit vector Show that these p erform the job of resolving the vector v into vectors v and v resp ectively parallel and p erp endicular to u Prove that these k pro jection op erators are idempotent The denition of an idemp otent op erator G is that GG G Show that the p olarizability tensor in problem A can be written uu uu k p where u C Use of Principal Axes C A xed ob ject is placed in a uniform electric eld of kV m When the eld is in the x direction it acquires a dip ole momentwith comp onents and C m in the x y z directions For the same eld strengths in the y z directions the dip ole moments are and C m resp ectively i Write down the comp onents of the tensor that describ es the p olarisability of the ob ject using the x y z co ordinate frame p ii What is the torque T p E on the ob ject when it is placed in an electric eld of kVm in the direction Ans Nm iii Supp ose there are eld directions uv w for which there is no torque on the ob ject what relation do E and p then have Show that u v and w are the eigenvectors of iv Show that if we use as a co ordinate system the unit vectors uv w there are no odiagonal