This preview shows pages 1–9. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Various quantities in physics such as temperature, volume and speed can be speciﬁed by a real number.
Such quantities are called scalars. Other quantities such as force, velocity and momentum require for their speciﬁcation a direction as
well as magnitude. Such quantities are called vectors; A vector is represented by an arrow or directed
line segment indicating direction. The magnitude of the vector is determined by the length of the arrow,
using an appropriate unit. 1 seanon“ VE_¢_‘IORS;;T : ‘° A vector is denoted by a bold faced letter such as A [Fig 224]. The magnitude is denoted by [A] or
A. The tail end of the arrow is called the initial point while the head is called the terminal point. f'FUNDAMENTAL.Diyflﬁlﬂbﬂsr  ‘ 1. Equality of vectors. Two vectors are equal if they have the same
magnitude and direction. Thus A = B in Fig. 221. A
2. Multiplication of a vector by a scalar. If m is any real number 3
(scalar), then mA is a vector whose magnitude is times the
magnitude of A and whose direction is the same as or opposite
to A according as m > O or m < 0. If m = 0, then mA = 0 is
called the zero or null vector.
Fig. 221 5» Sums of vectors. The sum or resultant of A and B is a vector C = A+B formed by placing the
initial point of B on the terminal point of A and joining the initial point of A to the terminal point
of B [Fig 222(b)]. This deﬁnition is equivalent to the parallelogram law for vector addition as in—
dicated in Fig. 22~2(c). The vector A —— B is deﬁned as A + (~B). FORMULAS FROM VECTOR ANALYSIS 117 Extensions to sums of more than two vectors are immediate. Thus Fig. 22—3 shows how to obtain
the sum E of the vectors A, B, C and D. (a) (b)
Fig. 223 4. Unit vectors. A unit vector is a vector with unit magnitude. If A is a vector, then a unit vector in
the direction of A is a = A/A where A > 0. .LAws or VECTOR ALGEBRA If A, B, C are vectors and m, n are scalars, then 22.] A + B = B + A Commutative law for addition 22.2 A + (B + C) = (A + B) + C Associative law for addition 22.3 m(nA) = (mn)A = n(mA) Associative law for scalar multiplication
22.4 (’m + n)A = mA + nA Distributive law 22.5 m(A + B) mA + m8 Distributive law . _‘ COMPONENTS OF A vecronj A vector A can be represented with initial point at the
origin of a rectangular coordinate system. If i, j,k are unit
vectors in the directions of the positive x,y,z axes, then 22.6 A = A1i+ A2]: + Agk Where Ali,A2j,A3k are called component vectors of A in the
i, j, k directions and 141,112,113 are called the components of A. Fig. 22—4 no”! OR scALAR PRODUCT 22.7 AB 2 AB coso 0 ll/\
Q:
“A
=1 where 9 is the angle between A and B. 118 FORMULAS FROM VECTOR ANALYSIS Fundamental results are 22.8 A ' B =2 B ' A Commutative law
22.9 A ' (B + C) = A  B + A  C Distributive law A'B 33 A131 +A282+A3Bg where A = A,i+A2j +A3k, B : B1i+sz +B3k. i fthSSORVECTOR §R0b__ucr‘i _ a HA
ll/\ 22.” AXB =ABsin9u 0 0 71 Where 0 is the angle between A and B and u is a
unit vector perpendicular to the plane of A and B
such that A, B, 11 form a right—handed system [i.e. a
rightthreaded screw rotated through an angle less
than 180° from A to B will advance in the direction
of u as in Fig. 22—5]. Fundamental results are i k
22/.12 AXB = A1 A2 A3
B1 82 33] Fig. 225
= (A233“A3Bz)i + (A331 ’A1lel + (A132 _A281)k
22.13 A x B = ——B x A
22.14 A><(B+C) = A><B+A><c
22.15 IA X B! = area of parallelogram having sides A and B “ A1 A2 A3 A ° X : Bi 32 33' : 44.18203 + AzB3C1 + 1433102 '_ 14.38201 — 1123103 "‘ [41336.2
1 01 02/ 03}
22.17 [A ‘ (B X C)! = volume of parallelepiped with sides A, B, C 22.18 AX (B x C) B(A o C) ~ C(A ~ B) B(A' C) — A(B C) H 22.19 (A X B) x C 22.20 (AXB)'(CXD) = (A‘“)(B'D)—(Aﬁ)(B“)
22.21 (AXB)><(C><D) 2 C{A(BXD)}D{A(BXC)} B{A (C x 11)} — A{B° (C x D)} FORMULAS FROM VECTOR ANALYSIS 119 DERIVATIVES OF VECTORS The derivative of a vector function A(u) z A1(u)i + A2(u)j + A3(u)k of the scalar variable u is
given by
dA AM + A“) “ AW) dAl dAz . dAs ‘du' : ALIS/10 Au " du l ‘I‘ du J l d“ k »  ' W  l 4
Partial derivativesﬁgf ﬁctoﬁfﬁtiﬁiﬁA<ﬁw z) are similarly deﬁned. We assume that all derivatives
exist unless otherwise speciﬁed. FORMULAS INVOLVING DERIVATIVES d _ dB dA
22.23 ——du(A B) —_ A du + du B d _ dB (1A
22.24 aZ(A><B) —— Axgz—LerduxB 22.25 flex(me _ g1—‘3~(1.>.><C)+A<dB><c>+A.<B><9l9> du du du du
clA _ dA
22.26 A W _ A57;
22.27 lag:— — 0 ifIAIisaconstant . _ ~ 13: DEL OPERATOR The operator del is deﬁned by  .a .i _a_
22.28 V  1ax+16y+kaz In the results below we assume that U = U(x,y, 2), V = V(x,y,z), A = A(x,y,z) and B = B(x.y,z)
have partial derivatives. was GRADIENT _‘ . _ ‘_ ,_ .a .a a _ 9g. 9g. 5'32
22.29 GradientofU —— gradU —— VU <lax+lay+kaz>Ul 8901+ ay3+ azk ma DIVERGENCEV / \
. t9 . (9' B . . .
«lb—92+ 152!— ‘I‘ (All + A2] '1‘ II
‘3'
<:
{9
II
<1
>
II 22.30 Divergence of A BA 6A 6A
__1 + __2 + .3
8x 63/ 62 Il 120 FORMULAS FROM VECTOR ANALYSIS we can. __ ‘ L 22.31 Curl of A curIA = VXA .6 .8 '6 . .
<1$+Jay+k5>><(A11+A2]+A3k) 1 ] k
_ .8. i .6.
‘ 6.90 63/ az
A1 A2 A3
6A 6A 6A 6A 6A 6A
.. __3__ 2122.1___§J+_.3__1k
8y Oz 82 6x 623 ﬂy 22.32 Laplacian of U = V2U = V (VU) = —— + — + * H
S
:3.
I
l
+
‘
+
l 22.33 Laplacian of A 2‘ 2 . in; B‘IHARMOMC organpk} : 22.34 Biharmonic operator on U = V4U = V2(V2U)
_ a4U a4U , 64U , a4U am a4U
” 6x4 + 6114 ‘ 6z4 T 26x2 61,12 + 2 6212 622 + 2 6x2 622 * :f _:*l?Mitscsumons saunasmoms:'vC 22.35 V(U+V) = vvtvv 22.36 V(A+B) = vA+vB
22.37 VX(A+B) = VXA+V><B
22.38 V(UA) : (VU)A+ U<V~A> H 22.39 V x (UA) (VU) X A + U<V x A) 22.40 V(AXB) = B(VXA)—A'(V><B)
22.4! V ><(A><B) = (BV)A —— B(V~A) — (AV)B + A(\7«B)
22.42 V(AB) = (BV)A+ (AV)B+B><(V><A) +A.><(V><B) 22.43 V X (V U) 3 0, Le. the curl of the gradient of U zero.
22.44 V  (V X A) = 0, Le. the divergence of the curl of A is zero. 22.45 \7><(VXA) = V(VA)—V2A FORMULAS FROM VECTOR ANALYSIS 121 Imaokmsimvow‘me VECTORS d If AW) 2 QEBW), then the indeﬁnite integral of AM) is 214$ f A(u) du 2 8(a) + c c : constant vector The deﬁnite integral of AM) from u = a to it : b in this case is given by
b
22.47 f A(u) du = 3(5) — 8(a)
(1 The deﬁnite integral can be deﬁned as on page 94. LINE INTEGRALS Consider a space curve C joining two points P1011, a2, a3) and
P2011, b2, ()3) as in Fig. 22—6. Divide the curve into n parts by points of subdivision (x1, y1, 21), . . .,(xn21, ynwl, 5411). Then the line integral
of a vector A(x,y, 2) along C is deﬁned as P2 11
22.48 f A . dr 2 Adr = lim 2 Amp, yp, 21,) ' Arp
. C P1 n—mo p=1
Where Arp = Awpi + Aypj + Azp k, Amp 2: mp“  00p, Ayp = ypﬂ —~ yp,
A210 2 2p“ m 22, and where it is assumed that as n —> 00 the largest of the magnitudes {Mp} approaches zero. The result 22.48 is a gen
eralization of the ordinary deﬁnite integral [page 94]. Fig. 226 The line integral 22.48 can also be written 22.49 {Adr = I Aldx + Am, + Agdz
l/C C I using A = Ali+1125 +A3k and dr — dwil— dyj + dzk. ' = pLROPERTlELSQOFlUNE :lNl'iEGRMS‘ 21 L‘ .P2 P1
22.50 A°dr = — A‘dr
P1 P2
P2 "P3 (“P2
22.57 A'dr = J A'a'riJ Adr
Px P1 P3 _ i‘NDEPENDENc‘s orrHa PATH!" _ In general a line integral has a value which depends on the particular path C joining points P1 and P2
in a region R. However, in case A 2 VHS or V X A z 0 where as and its partial derivatives are con r
tinuous in ‘R, the line integral J A ° dr is independent of the path. In such case
C P2
22.52 f Adr = Ant: 2 ¢(P2) ~¢<Po
‘C (PI 122 FORMULAS FROM VECTOR ANALYSIS where ¢(P1) and ¢(P2) denote the values of <35 at P1 and P2 respectively. In particular if C is a closed curve, 22.53 fAdr = §Awlr = 0
C C where the circle on the integral sign is used to emphasize that C is closed. MULTIPLE INTEGRALS Let F(x,y) he a function deﬁned in a region ‘R of the
xy plane as in Fig. 227. Subdivide the region into 71 parts
by lines parallel to the x and y axes as indicated. Let AAp 2
Amp Ayp denote an area of one of these parts. Then the in~
tegral of F(x, 2/) over ‘R is deﬁned as n
22.54 I F(oc,y)dA : lim EF(xp,yp)AAp
‘R nNo p=1 provided this limit exists. In such case the integral can also be written as b f2(x)
22.55 I F(x, y) dy also a ‘ y=f1<£> 1. {fame} F(x,y)dy}dx z y=f1(x) Fig. 227 where y = f1(x) and y : f2(x) are the equations of curves PHQ and PGQ respectively and a and b are
the at: coordinates of points P and Q. The result can also be Written as d. 92(y) d 92(11)
22.56 I f F(x, y) dx dy Z I F(x, y) (156} dy
y=c ‘ x=gl(y) y==c x=g;(y) where so = 91(31), :5 = 92(y) are the equations of curves HPG and HQG respectively and c and d are the y coordinates of H and G.
These are called "ouble integrals or area integrals. The ideas can be similarly extended to triple or
volume integrals or to higher multiple integrals. SURFACE INTEGRALS Subdivide the surface S [see Fig. 22$] into n elements of
area Asp, p = 1,2, . . ._,n. Let Amp, yp, 2p) = Ap where (mp, yp, zp)
is a point P in Asp. Let Np be a unit normal to ASp at P. Then the surface integral of the normal component of A over S is
deﬁned as 22.57 fANdS = lim EApNp/isp
S 'n—vuo p=1 Fig. 228 FORMULAS FROM VECTOR ANALYSIS 123 RELATION BETWEEN SURFACE AND DOUBLE INTEGRALS If 92 is the projection of S on the my plane, then [see Fig. 22—8] . 22.58 . fANdS : f ANdxdy
S (R {Nk} THE DIVERGENCE THEOREM Let S be a closed surface bounding a region of volume V; then if N is the positive (outward drawn)
normal and d5 = N dS, we have [see Fig. 22—9] 22.59 I V'AdV = fA'dS
V s The result is also called Gauss’ theorem or Green’s theorem. Fig. 229 STOKE’S THEOREM Let S be an open twosided surface bounded by a closed non—intersecting curve C [simple closed curve]
as in Fig. 2210. Then 22.60 3,;Adr = f(V><A)dS
C S Where the circle on the integral is used to emphasize that C is closed. GREEN'S THEOREM IN THE PLANE \
22.67: §de+Qdy 2 Jr <%§.«::)dxdy
C R where R is the area bounded by the closed curve C. This result is a special case of the divergence theorem
or Stoke’s theorem. h 126 FORMULAS FROM VECTOR ANALYSIS SPECIAL ORTHOGONAL COORDINATE SYSYEMSL Cylindrical Coordinates (r, 6, 2) [See Fig. 22—12] 6%? 1 6(1) 1 62¢! 62¢ 2 _
2230 V (I) 672 1* 67‘ 7‘2 602 l 822 Fig. 22—12. Cylindrical coordinates. Fig. 22—13. Spherical coordinates. Spherical Coordinates (7*, (9, (1)) [See Fig. 22—13] 22.8? x = rsinacos¢, y = rsinosingb, z = rcoso 22.82 h: : rzsin28 1 52 64> 1 a an 1 62¢
2 : —— —— 2— w ‘ —
22'83 ‘ V 4) r2 6r<y 8r + r2 sin 0 60 <51!” 69 + 7‘2 sinZo 8¢2 22.84 x : l(u2~v2), y = in), z = z 22.85 h? 2 hi : u2+v2, h§ : 1
1 €924) 82(1) 62(1) :5
2 _ l l
22.86 V<1> “2+”? <au2 1 6v2> t 622 The traces of the coordinate surfaces on the my
plane are shown in Fig. 2244. They are confocal
parabolas with a common axis. “\ ...
View
Full
Document
This note was uploaded on 02/15/2012 for the course PHYS 240 taught by Professor Winn during the Winter '11 term at University of Michigan.
 Winter '11
 Winn

Click to edit the document details