EE2 Mathematics: Vector Calculus - Imperial College London

1 ee2 mathematics : Vector CalculusJ. D. Gibbon(Professor J. D Gibbon1, Dept of jdgThese notes are not identical word-for-word with my lectures which will be given on a of these notes may contain more examples than the corresponding lecture while in othercases the lecture may contain more detailed working. I willNOTbe handing out copies ofthese notes you are therefore advised to attend lectures and take your The material in them is dependent upon the Vector algebra you were taught at A-leveland your 1st year. A summary of what you need to revise lies inHandout 1 : Thingsyou need to recall about Vector algebra which is also 1 of this Further handouts are :(a)Handout 2 : The role of grad, div and curl in Vector Calculus summarizes mostof the material in 3.(b)Handout 3 : Changing the order in double integration is incorporated in (c)Handout 4 : Green s, Divergence & Stokes Theorems plus Maxwell s Equations summarizes the material in 6, 7 and , while Maxwell s Equations themselves are not in the syllabus, three ofthe four of them arise naturally out of the Divergence & Stokes Theorems andthey connect all the subsequent material with that given from lectures on e/mtheory given in your own not confuse me with Dr J.)

2 Gibbons who is also in the mathematics ( )1 Contents1 Revision : Things you need to recall about Vector Algebra22 Scalar and Vector Fields33 The Vector operators : grad, div and of the gradient operatorr.. of the divergence of a Vector field divB.. of the curl of a Vector field curlB.. Vector identities.. and solenoidal Vector fields..64 Line (path) integrals of Type 1 : C (x,y,z)ds.. integrals of Type 2 : CF(x,y,z) dr.. of path in line integrals of Type 2..135 Double and multiple to evaluate a double integral.. of multiple integration.. of variable and the Jacobian.. the order in double integration..206 Green s Theorem in a plane2272 DDivergence and Stokes Theorems258 Maxwell s Equations2914th/10/10 ( )21 Revision : Things you need to recall about Vector AlgebraNotation:a=a1 i+a2 j+a3 k (a1,a2,a3).1. Themagnitude or length of a vectorais|a|=a=(a21+a22+a23)1/2.

3 ( )2. Thescalar (dot) productof two vectorsa= (a1,a2,a3)&b= (b1,b2,b3)is given bya b=a1b1+a2b2+a3b3.( )Sincea b=abcos , where is the angle betweenaandb, thenaandbareperpendicular ifa b= 0, assuming neitheranorbare Thevector (cross) product2between two vectorsaandbisa b= i j ka1a2a3b1b2b3 .( )Recall thata bcan also be expressed asa b= (absin ) n( )where nis a unit Vector perpendicular to bothaandbin a direction determined by theright hand rule. Ifa b= 0thenaandbare parallel if neither Vector is Thescalar triple productbetween three vectorsa,bandcisa (b c) = a1a2a3b1b2b3c1c2c3 a% &c bCyclic Rule:clockwise +ve( )According to the cyclic rulea (b c) =b (c a) =c (a b)( )One consequence is that ifany two of the three vectors are equal (or parallel)then their scalar product is zero: (b a) =b (a a) = 0. Ifa (b c) = 0and no pair ofa,bandcare parallel then the three vectors must be Thevector triple productbetween three vectors isa (b c) =b(a c) c(a b).

4 ( )The placement of the brackets on the LHS is important: the RHS is a Vector that liesin the same plane asbandcwhereas(a b) c=b(c a) a(c b)lies in the planeofaandb. Thus,a b cwithout brackets is a meaningless statement!2It is acceptable to use the notationa bas an alternative toa ( )32 Scalar and Vector Fields(L1)Our first aim is to step up from single variable Calculus that is, dealing with functionsof one variable to functions of two, three or even four variables. The physics of electro-magnetic (e/m) fields requires us to deal with the three co-ordinates of space(x, y, z)andalso timet. There are two different types of functions of the four variables :1. Ascalar field3is written as = (x,y,z,t).( )Note that one cannot plot as a graph in the conventional sense as takes values atevery point in space and time.

5 A good example of a scalar field is the temperature ofthe air in a room. If the box-shape of a room is thought of as a co-ordinate system withthe origin in one corner, then every point in that room can be labelled by a co-ordinate(x,y,z). If the room is poorly air-conditioned the temperature in different parts mayvary widely : moving a thermometer around will measure the variation in temperaturefrom point to point (spatially) and also in time (temporally). Another example of ascalar field is the concentration of salt or a dye dissolved in a Avector fieldB(x,y,z,t)must have components(B1, B2, B3)in terms of the threeunit vectors( i, j, k)B= iB1(x,y,z,t) + jB2(x,y,z,t) + kB3(x,y,z,t).( )These components can each be functions of(x,y,z,t). Three physical examples ofvector fields are :(a) Anelectric field :E(x,y,z,t) = iE1(x,y,z,t) + jE2(x,y,z,t) + kE3(x,y,z,t),( )(b) Amagnetic field :H(x,y,z,t) = iH1(x,y,z,t) + jH2(x,y,z,t) + kH3(x,y,z,t),( )(c) Thevelocity fieldu(x,y,z,t)in a classic illustration of a three-dimensional Vector field in action is the e/m signal receivedby a mobile phone which can be received anywhere in Vector operators : grad, div and of the gradient operatorrThe gradient operator (grad) is denoted by the symbolrand is defined asr= i x+ j y+ k z.

6 ( )3 The convention is to use Greek letters for scalar fields and bold Roman for Vector ( )4As such it is a Vector form of partial differentiation because it has spatial partial derivatives ineach of the three directions. On its right,rcan operate on a scalar field (x,y,z)r = i x+ j y+ k z.( )Note that while is a scalar field,r itself is a Vector . Example 1) :With =13(x3+y3+z3)r = ix2+ jy2+ kz2.( )As explained above, the RHS is a Vector whereas is a scalar. Example 2) :With =xyzthe vectorr isr = iyz+ jxz+ kxy .( )End of of the divergence of a Vector field divBL2 Because Vector algebra allows two forms of multiplication (the scalar and Vector products)there are two ways of operatingron a vectorB= iB1(x,y,z,t) + jB2(x,y,z,t) + kB3(x,y,z,t).( )The first is through the scalar or dot productdivB=r B=( i x+ j y+ k z) ( iB1+ jB2+ kB3).( )Recalling that i i= j j= k k= 1but i j= i k= k j= 0, the result isdivB=r B= B1 x+ B2 y+ B3 z.

7 ( )Note that divBis a scalar because div is formed through the dot physical explanation that can be given is that divBis a measure of the compression orexpansion of a Vector field through the 3 faces of a cube. If divB= 0the the Vector fieldBis incompressible ; If divB>0the the Vector fieldBis expanding ; If divB<0the the Vector fieldBis compressing .14th/10/10 ( )5 Example 1 :Letrbe the straight line vectorr= ix+ jy+ kzthendivr= 1 + 1 + 1 = 3( )Example 2 :LetB= ix2+ jy2+ kz2thendivB= 2x+ 2y+ 2z( )Note : the usual rule in Vector algebra thata b=b a(that is,aandbcommute)doesn t hold when one of them is an operator. ThusB r=B1 x+B2 y+B3 z6=r B( ) of the curl of a Vector field curlBThe alternative in Vector multiplication is to userin a cross product with a vectorB:curlB=r B= i j k x y zB1B2B3 .( )The best physical explanation that can be given is to visualize in colour the intensity ofBthen curlBis a measure of the curvature in the field lines 1) :Take the Vector denoting a straight line from the origin to a point(x,y,z)denoted byr= ix+ jy+ kz.

8 Thencurlr=r r= i j k x y zx y z = 0.( )Example 2) :ChooseB=12( ix2+ jy2+ kz2)thencurlB=r B= i j k x y z12x212y212z2 = 0.( )Example 3) :Take the Vector denoted byB= iy2z2+ jx2z2+ kx2y2. ThencurlB=r B= i j k x y zy2z2x2z2x2y2 = 2x2 i(y z) 2y2 j(x z) + 2z2 k(x y).( )14th/10/10 ( )6 Example 4) :For the curl of a two-dimensional vectorB= iB1(x,y) + jB2(x,y)we havecurlB=r B= i j k x y zB1(x,y)B2(x,y) 0 = k( B2 x B1 y),( )which points in the vertical direction only because there are no iand of Vector identitiesL3 There are 5 useful Vector identities (see hand out No 2). The proofs of 1), 4) and 5) areobvious : for No 1) use the product rule. 2) and 3) can be proved with a little The gradient of the product of two scalars and r( ) = r + r .( )2. The divergence of the product of a scalar with a vectorbdiv( B) = divB+ (r ) B.

9 ( )3. The curl of the product of a scalar with a vectorBcurl( B) = curlB+ (r ) B.( )4. The curl of the gradient of any scalar curl(r ) =r r = 0,( )because the cross productr ris The divergence of the curl of any vectorBdiv(curlB) =r (r B) = 0.( )The cyclic rule for the scalar triple product in ( ) shows that this is zero for all vectorsBbecause two vectors (r) in the triple are the and solenoidal Vector fieldsConsider identities 4) & 5) above (see also Handout 2 The role of grad, div & curl .. )curl(r ) =r r = 0,( )div(curlB) = 0.( )( ) says that if any vectorB(x,y,z)can be written as the gradient of a scalar (x,y,z)(which can t always be done)B=r ( )14th/10/10 ( )7then automatically curlB= 0. Such Vector fields are called irrotational Vector is equally true that for a given fieldB, then if it is found that curlB= 0, then we canwrite4B= r ( ) is called the scalar potential.

10 Note that not every Vector field has a correspondingscalar potential, but only those that are we now turn to ( ) : Vector fieldsBfor which divB= 0are calledsolenoidal,in which caseBcan be written asB=curlA( )where the vectorAis called a Vector potential .Note that only vectors those that arediv-free have a corresponding Vector :The Newtonian gravitational force between massesmandM(with gravitationalconstantG) isF= GmMrr3,( )wherer= ix+ jy+ kzandr2=x2+y2+ Let us first calculate curlFcurlF= GmMcurl( r),where =r 3.( )The 3rd in the list of Vector identities givescurl( r) = curlr+ (r ) r( )and we already know that curlr= 0. It remains to calculater :r =r{(x2+y2+z2) 3/2}= 3( ix+ jy+ kz)(x2+y2+z2)5/2= 3rr5.( )Thus, from ( ),curlF= GmM(0 3r2r5 r)= 0.( )Thus the Newton gravitational force field is curl-free, which is why a gravita-tional potential can write6F= r.