INTRO THEORET PHYSICS
INTRO THEORET PHYSICS PHZ 3113
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Date Created: 09/18/15
4 Multiple integrals vectors 41 Multiple integrals Let s review this subject by doing various examples of integrating a function f x 3 over a region of 2 space EX 1 4 4 1 32 I y dydx dx ydy dx xg 1 region 0 0 0 Figure 1 EX 1 Q how about changing the order of integration We could do the x integral rst obviously But we need to be careful of the order of the limits 2 4 I dyy dx 2 y0 zy2 5 Q Why change the order Sometimes it s important EX 2 lnl6 4 dx 3 10 ye 2 1113 Can t do fdy lny so switch Figure 2 EX 1 So let s draw a new gure always draw a gure if you switchl 4 2111 4 d y I d dy26 5 yl 1113 0 yl 42 Change of variables the Jacobian First let s do a standard example where we don t get into formalities Figure 3 EX 1 EX 3 l V1712 2 2 I 671 ly dydx 6 10 y0 This will certainly be easier in polar coordinates x rcos6 y 7 sin 6 7r2 1 2 71 I 6 1 made 1 16 7 90 7 0 4 Note the measure for the integral in polar coords dx dy rd d0 8 is just the size of the little area element in Fig 3 but can be obtained formally In general the change of variables in 2D is de ned Rfuvdudv fursvrs where R is a region in 2D and R is the transformed region which need not have the same shape J is the Jacobian of transformation 7 1 we 7 Jegt ie the determinant of the partial derivatives as shown If you have forgotten how to take a determinant go ahead and look it up in Boas p 89 Of course the concept can be generalized to higher D transformations 057372 gt uvw but the determinental form remains one just has to do a bit more work Let s specialize now to polar coordinates Take u x 7 cos 6 v y rsin0 so 3573 i W so dxdy rdrdd what we worked out already in more intuitive fashion d7 d5 9 10 694 BT88 cos 0 r sin0 i 2 2 i sing r0086 7 rcos 0 sin 6 i r 11 Here s a slightly tricky example where the shape of the transformed region is quite different EX 5 1 z xy I0 day0 dy 12 Let s make a transformation 05 y gt u v where u yx v x y The choice of v is somewhat straightforward why u is chosen 1 have no idea Anyway 1st task 3 Figure 4 EX 5 Original and transformed regions is to calculate J acobian 06 y 13 2 lt gt 2 a we 05 u7v 95 Imx2vdudvevdudv 15 nice and simple but What are the transformed limits Well starting from the original ones we learn y0gtu0ifx7 0 yxgtu1 16 x1gtv1u yx0gtv0 17 1 1u 1 I du dv e duelu 1 62 e 1 18 0 0 0 measures for cylindrical and spherical coords Derived in Boas7checkl So So Spherical x rsin6cosq5 y rsin0sin zrcos6 x7y7z Jltna gt ie dxdydz gt 7 2 sin 6drd0d 20 7 2 sin 0 19 4 Cylindrical x pcos0 y psin6 z z x7y7z J 21 p0zgtl p l ie dxdydzapddedz 22 NB 0 in spherical coords is not the same 0 as for cylindrical coords Might want to use q for cylindrical coords instead EX 6 Calculate the moment of inertia of a cone of height equal to its base radius h R Take the density of the material to be p0 assumed homogeneous Moment of inertia is then h 2 2w 7Tb I pOdVx2y2 po dz rdr d6 7 2 p0 23 V 20 7 0 0 Note dimensions are correct since p0 MLg so p0h5 ML2 43 Vectors 431 Properties of vectors Vector set of components which transforms under rotation of a coordinate system in the same way as the coordinates of a point F 7 A vector is something that transforms like a vector Huh What does that mean Take the coordinates x1 952 of a point F in a Cartesian coordinate system F X1vX2 X1le2 X1 Figure 5 Transformation of coordinates 1 2 a If we rotate the coordinate axes to a new set 051 952 by an angle q geometry tells US 951 cos 051 sin 052 24 V V an an x2 sin x1 cosq xg W V 021 0227 so we can write this as a general linear transformation 2 xi a xl aigxg E aijxj 28 j1 What happens to the vector F under this transformation F x1 1x2 g 29 953 xQ g 30 31 0512 0522 95 x3 32 33 gt aii 5 31 1 i ai2 5 32 1 34 anon 021022 0 35 The length of the vector is preserved in the new coordinate system the transfor mation is said to be orthogonal We can write the same thing in a nice compact notation using the 7 Kronecker delta 7 symbol 7 1239j 6U0 Z j 36 Z aijaik 61k Z ajiaki 37 If you nd this confusing write this out in terms of indices and see what it means You may also wish to ask yourself how it s related to matrix multiplication as lnverse transformation q gt 896 Z ij i 805 aij Why is this true Some properties of general N D vectors 14f 14111412 6 3M o az laA1aA2aAN 3 A1 A2 aAN J X o al aflb etc Magnitude of a vector N N A2 2A 2A check il il m A A 432 Products of vectors 1 Dot or scalar product A g Z AllBl 2 Cross product A X Z ijkAjBk7 jk 39 40 41 42 Where eijk is the so called Levi CiVita symbol sometimes called the completely antisymmetric tensor 0 if any two indices are the same eijk 1 if indices correspond to an even permutation of 123 1 if indices correspond to an odd permutation of 123 Very useful identity 7 worth memorizingl E ijk lmk 6239Z6jm Sim ji k 7 43 44 4x o x 7 45 ZQT gtlt EMC x D Z ijkAjBk 26247210sz jk m 2 Z Z ijk6izmAjBkCsz ijkim 2 514514721 5jm5k4AjBk04Dm jkim 44340sz AmBngDm m Z AME Bm0mgt Z 34002 AmDmgt Z m E m 46 D o4 46 Some more identities to check using these techniques 4 x6 6x44x o 4 x E x 5 EM 5 0144 44 BAGCAB rule Remarks 0 2 or 3d A g AB cos 6A3 If g 0 two vectors are orthogonal 0A3 7T2 0 unit vectors l g We can choose mutually orthogonal 627 Also note X j Eijk k 0 Expand any vector fl in terms of D orthogonal unit vectors in D dimensions A Ai i o The scalar or dot product of two vectors is invariant under coordinate trans formations 147 By aikBk ZaijaikAjBk i j k 239 ijk Zak13 243 4 E 47 jk k 0 14lgtlt ABSlDOAB 0 meaning of gtlt volume of parallelogram spanned by vectors fl g and 6 7 Curvilinear coordinates Read Boas sec 54 108 109 71 Review of spherical and cylindrical coords First l ll review spherical and cylindrical coordinate systems so you can have them in mind when we discuss more general cases 711 Spherical coordinates Z Figure 1 Spherical coordinate system The conventional choice of coordinates is shown in Fig 1 0 is called the polar angle q the azimuthal angle The transformation from Cartesian coords is xrsin0cos yrsin0sin 27 cos6 1 In the gure the unit vectors pointing in the directions of the changes of the three spherical coordinates r 6 q are also shown Any vector can be expressed in terms of them 41 A3593 AygjA22 M Ae AM 2 Note the qualitatively new element here while both 923 2 and 720265 are three mutually orthogonal unit vectors 023272 are xed in space but 73045 point in different directions according to the direction of vector F We now ask by how large 1 a distance d5 the head of the vector f changes if in nitesimal changes dr d0 dq are made in the three spherical directions dsT d7 7 6159 rdO 7 dng rsin 6d 7 as seen from gure 2 only the and lt25 displacements are shown v 7 jE rsmOdq frsine I Figure 2 Geometry of in nitesimal changes of P So the total change is d cw rdO 7 sin Megs The volume element will be d7 dSTdSstqg r2 sin6 dr d6 digs and the surface measure at constant 7 will be dd 6159 dng f 7 2 sin 6d0d f EX 1 Volume of sphere of radius R spheie dT TR0 90 A2 72 Sin 0dr d6 61 RS3 2 2W More interesting gradient etc in spherical coordinates W nal twain 8m By 82 3i ai ai t 805 87 805 868x assays QCquot and i j and n can be replaced by sin0cos fcos6cos sin q3 sin0sin fcos6sin cos q3 cos6f sin0 P m a 2 4 gWRS 3 Combining all these we nd 78 18 1 8qu W artzwgtmw 39 Similarly we nd a a 1 8 1 8 1 8A A 214T 39 6A V r287 76 l rsin086 sm 6 l rsin6 Bq and 1 f r rsinqu v x141 a a 2 2 1 66 1 T 81116 AT r1419 rsin0A and 1 8 811 1 8 811 1 821 2 7 2 V 1 i r287 lt76 87 l rgsin680 8111680 l 7 2sin268q52 712 Cylindrical coordinates 10 11 12 13 I won t belabor the cylindrical coordinates but just give you the results to have handy I ve written here the cylindrical radial coordinate as called 7 the angle variable 6 like Boas but keep in mind that a lot of books use p and q x rcos0 yrsin6 22 dsT d7 d59 rd6 dsz dz d cw rd0 c122 d7 rdrdddz a quJA 1811A BzJA W WWW a a a 18 181419 8A2 v14 i FarerTlA TE 82 a a 1 f r0 2 VxA 6 5 lt14 713 General coordinate systems With these speci c examples in mind let s go back the the general case and see where all the factors come from We can pick a new set of coordinates q1 qg qg which have isosurfaces which need not be planes nor parallel to each other Let s just assume that among 05 y z and q1q2q3 there are some relations frq1q2q3 yyq1q2q3 22q17q27q3 15 which we can nd and invert to get Q1Q1957372 12CI295737Z 1361395737Z 16 The differentials are then d8xdq8x 3 11 9612 9613 It s very useful to know what the measure of distance or metrlc is in a given 51 01 17 coordinate system Of course in Cartesian coordinates the distance between two points whose coordinates differ by doc dy dz is ds where ds2 doc2 dy2 dzg 18 Your book calls ds the arc length Now if you imagine squaring an equation like 17 you ll get terms like dqg but also terms like dqldqg etc So in general plugging into 18 we expect d5 mm gmdqldqg Zgijdqidqj 19 ii and the 91 are called the metrlc components and g itself is the metrlc tensor ln Einstein s theory of general relativity the metric components depend on the amount of mass nearby Most of the coordinate systems we are interested in are orthogonal ie gij olt 627 Thus we can write at h1 dq12 hg dq22 hgdqg 20 The hi s are called scale factors and are 1 for Cartesian coordinates Now let s look at the change of the position vector r in our new coordinate system when we change the coordinates q by a small amount We have CW CE hid h i1 h2dCI2 i2 hngS f3 21 4 We can de ne the distance changes 51 52 and 53 by 52 E hiqi Let s make con tact with something concrete by comparing with say spherical coordinates The in nitesimal change in the position vector is what s given in 4 so we can identify the scale factors for spherical coordinates as h l he r and hag rsin0 Since we know how to express d now we can immediately say how to do line elements for line integrals 27df ZvihZdqi 22 as well as surface and volume integrals U1d82d83U2d81d83U3d81d82 U1h2h3dCI2dqgU2h1h3d hdqgv3h1h2dq1dQ2 23 and d739 h1h2h3dq1dq2dq37 24 where the stands for the integrand Differential operators in curvilinear coordinates I am not going to develop all of this here it s pretty tedious and is discussed in Boas secs 98 and 99 However the basic idea comes from noting that the gradient is the fastest change of a scalar eld so the ql component is obtained by dotting into Q1 ie 2amp1wa A v 7 25 ql 1 881 hl aql 7 etc Note that we are allowed to do the last step because h is a function of qg and q3 but these are held constant during the partial differentiation Therefore a A 181 W 7 mi 26 Similarly a 1 8 8 8 V vhh vhh vhh 27 h1h2h3 l8Q1 1 2 3 86122 1 3 96133 1 2 l 5 and lt1 61 hlhghg Cf1h1 i 641 h1v1 f2 M i 642 hg U2 Eh i 643 hgvg 28