Vector Analysis

Appendix AVector AnalysisA.1Vector OperationsA vector is a quantity that has direction as well as magnitude.On the other hand,a quantity that has only magnitude is called a scalar.The magnitude of a vector a is denoted by|a|or a,and its direction is given byˆa,i.e.a=|a|ˆa=aˆa.It is assumed that the reader is familiar with the basic operations of vectors,including addition and scalar multiplication.It is often convenient to express vectors in component form in some par-ticular coordinate system,i.e.presenting a vector by its projections along the coordinate axes.For example,an arbitrary vector a can be expressed in rect-angular coordinates asa=a xˆe x+a yˆe y+a zˆe z(A.1) whereˆe x,ˆe y,andˆe z are unit vectors along the x,y,and z axes,respectively.The vector operations can also be formulated in component forms.For example,addition and scalar multiplication of vectors can be expressed asa+b=(a x+b x)ˆe x+(a y+b y)ˆe y+(a z+b z)ˆe z(A.2)λa=λa xˆe x+λa xˆe y+λa zˆe z(A.3)It should be noted that,though the components of a vector are determined by the particular choice of coordinate system,the results of vector operations are coordinate independent.In other words,once a vector equation is established in rectangular coordinates,it is equally valid in any curvilinear coordinates. Orthogonal Transformation of VectorsA vector can also be defined in terms of the behavior of its components under rotation of the coordinate axes.Let x i,(i=1,2,3)denote the original coordinates, and x i,(i=1,2,3)the coordinates of the same point in terms of the rotated axes.12APPENDIX A.VECTOR ANALYSIS The two sets of coordinates are related to each other as followsx i =3j =1a ij x j (A.4)where the coefficients a ij ,(i,j =1,2,3)satisfy the orthogonality condition3i =1a ij a ik =δjk (A.5)Here,the symbol δjk is the Kronecker delta ,which is unit for j =k and zero otherwise.The above relation can be directly derived by requiring |x |2=|x |2,i.e.the length of a vector remains invariant under rotation.A.1.1Dot and Cross ProductsDot ProductThe dot product or scalar product of two vectors is a scalar defined bya ·b =ab cos θ(A.6)where θis the angle between a and b .Geometrically,a ·b is the projection of b along a ,multiplied by the magnitude of a .And a ·b =0implies that a and b are perpendicular to each other.In rectangular coordinates,the dot product can be expressed asa ·b =a x b x +a y b y +a z b z(A.7)And it is clear that a ·b =b ·a ,i.e.the dot product is commutative .Cross ProductThe cross product or vector product of two vectors is another vector,denoted by a ×b .The magnitude of the resultant vector is given by|a ×b |=ab sin θ(A.8)while its direction is determined by the right-hand rule,i.e.vectors a ,b ,and a ×b form a right-handed system.Geometrically,|a ×b |is the area of the parallelogram generated by a and b .And a ×b =0implies that a and b are parallel to each other.The cross products can be expressed in rectangular coordinates asa ×b =(a y b z −a z b y )ˆe x +(a z b x −a x b z )ˆe y +(a x b y −a y b x )ˆe z = ˆe x ˆe y ˆe z a x a y a z b x b y b z (A.9)It can be easily verified that a ×b =−b ×a ,i.e.the cross product is anti-commutative .A.1.VECTOR OPERATIONS3A.1.2Triple Products of VectorsScalar Triple ProductThe scalar triple product is a scalar of the form a·(b×c),which is the dot product between a and b×c.Geometrically,|a·(b×c)|is the volume of the parallelepiped generated by a,b,and c.And a·(b×c)=0implies that a,b,and c lie in the same plane.In rectangular coordinates,the scalar triple product can be expressed asa·(b×c)=a x a y a zb x b y b zc x c y c z(A.10)And it can be shown that the following relations holda·(b×c)=b·(c×a)=c·(a×b)=−b·(a×c)=−a·(c×b)=−c·(b×a)(A.11)i.e.the scalar triple product remains unvaried after a cyclic permutation of the three vectors,whereas a non-cyclic permutation reverses its sign.Moreover,the dot and the cross in a scalar triple product can be interchangeda·(b×c)=(a×b)·c(A.12)As a result,the scalar triple product is sometimes denoted by(a,b,c).Vector Triple ProductThe vector triple product is a vector of the form a×(b×c).It can be simplified as followsa×(b×c)=(a·c)b−(a·b)c(A.13) By repeated application of the above equation,it is possible to reduce any vector expression into the sum of items each containing one cross product at most.A.1.3DyadicsA dyad is formed by juxtaposing two vectors side by side1,e.g.ab.The order is of consequence and normally ab=ba.There are two types of scalar products between a vector a and a dyad bc. The anterior dot product is defined asa·bc=(a·b)c(A.14a)while the posterior dot product has the formbc·a=b(c·a)(A.14b) 1Dyad is also known as the vector direct product or simply dyadic product.4APPENDIX A.VECTOR ANALYSIS And normally a ·bc =bc ·a .In rectangular coordinates,an arbitrary dyad ab can be expressed asab = 3i =1a i ˆe i 3 j =1b j ˆe j = i,j a i b j ˆe i ˆe j (A.15)where the unit vectors along x ,y ,and z axes have been written as ˆe1,ˆe 2and ˆe3,respectively.As indicated by (A.15),a dyad ab can be viewed as a combination of dyads ˆei ˆe j ,(i,j =1,2,3).Generally,the linear combination of ˆe i ˆe j is called a dyadic ,denoted by ↔T ,i.e.↔T =i,j T ij ˆei ˆe j (A.16)where T ij ,(i,j =1,2,3)are the nine components of dyadic ↔T .A dyadic ↔T canbe viewed as a tensor of second rank.According to (A.14)and (A.16),the anterior and posterior dot products between a vector a and a dyadic ↔T can be explicitly expressed asa ·↔T = i,j a i T ij ˆej (A.17)and ↔T ·a =i,j a j T ij ˆei (A.18)respectively.It is clear that normally a ·↔T =↔T ·a .A dyadic ↔T is said tobe symmetric if a ·↔T =↔T ·a holds true for any vector a ,and antisymmetric if a ·↔T =−↔T ·a .It is straight forward to show that if ↔T is symmetric,then T i j =T ji ,whereas if ↔T is antisymmetric,T i j =−T ji .A unit dyadic or idemfactor is a symmetric dyadic of the form↔I =ˆe 1ˆe 1+ˆe 2ˆe 2+ˆe 3ˆe 3(A.19)And it can be easily verified that the following relation holds for any vector aa ·↔I =↔I ·a =a(A.20)The colon product between two dyadics ↔A and ↔B is defined as 2↔A :↔B = i,jA i jB ji (A.21)2In some literature the colon product is defined as ↔A :↔B = A ij B ij ,and ab :cd =(a ·c )(b ·d ).So it is advisable to make sure which definition is adopted when dealing with a dyadic equation containing colon product.A.2.INTEGRALS OF FIELDS5Consequently,the colon product between two dyads is given byab:cd=(b·c)(a·d)(A.22) It is easy to verify that↔I:↔A=↔A:↔I=iA ii=tr↔A(A.23)and↔I:ab=a·b(A.24) A.2Integrals of FieldsOne of the most important concepts in physics is that of thefield.Many physical quantities can be characterized by scalar functions of position in space,and are termed as scalarfields.On the other hand,there are other quantities that are not only functions of position but also of the nature of vectors.Such quantities are called vectorfields.To characterize the behavior of a vectorfield over a specified region,we often resort to their integral properties.Two types of integral are of particularly importance for the characterization of vectorfields.A.2.1Line IntegralsOne type of integral often encountered in the study of vectorfields is the line integral or path integral of the following formL F(x)·d l=LF(x)·ˆτdlwhereˆτdenotes the unit vector tangential to the curve L.Physically,the the above line integral is often related to the work done by moving a particle along the path L against the force F.Normally,the line integral of a vectorfield between two given end points depends critically on the particular path taken.Afield whose line integral is independent of the path is called conservative.Gravitationalfield and electro-staticfield are two such examples.The path integral of a conservativefield is is solely determined by the end points.For a closed path,the two end points of L coincide.The line integral of a vectorfield along a closed path is called its circulation,denoted byL F(x)·d l=LF(x)·ˆτdlwhere a circle over the integral symbol indicates that the integral is taken along a closed path.33For a closed path L,we always assume that it forms the boundary of a two-sided,smooth surface S,and that the positive direction of L is related to the positive direction of S by the right-hand rule.6APPENDIX A.VECTOR ANALYSISA.2.2Surface IntegralsA frequently exploited concept in the study of vector fields is the flux ,which is the surface integral defined as follows S F (x )·d S = SF (x )·ˆndS where ˆnis the normal unit vector of the surface element d S .Physically,the flux is related to the rate of F ,which can be the flow of fluid,particles,or energy,through a given surface S .4The total flux of F through a closed surface S is denoted by 5S F (x )·d S = SF (x )·ˆndS where a circle over the integral symbol indicates that the integral is taken over a closed surface.A.3Differential Operators of FieldsThe integral properties of a field offer us the information about its global behav-ior over a given region.On the other hand,the local behavior of a field in the vicinity of a particular point in space is provided by the differential properties of the field.A.3.1Gradient,Divergence and CurlDifferential operators grad,div,and curl,which stand for gradient ,divergence and curl ,respectively,are of great importance in the study of fields.The first is used to depict scaler fields,while the other two are used to characterize vector fields.Gradient of a Scalar FieldThe most important feature of a scalar field is how it varies in the neighborhood of a particular point,which can be characterized by the derivatives of the field quantity.However,different from the case of a scalar function with a single variable,the variation of a scalar field in space depends on the direction along which the derivative is taken.So we first introduce the concept of directional derivative of a scalar field.Let f (x )be a scalar field and x 0a fixed point in space,the directionalderivative of the field at x 0along the direction of a unit vector ˆn is defined as ∂f ∂n x =x 0=lim l →0f (x 0+l ˆn )−f (x 0)l (A.25)4Hereafter,we shall always assume that S is a well behaved oriented surface,i.e.S is atwo-sided,smooth surface with prescribed positive direction.5For a closed surface S ,the normal unit vector ˆn is always taken to be pointing outwards.A.3.DIFFERENTIAL OPERATORS OF FIELDS7The gradient of a scalarfield f(x)is a vectorfield,denoted by grad f,whose projection at any point alongˆn is just the directional derivative of f along thesame direction,i.e.(grad f)n=ˆn·grad f=∂f∂n(A.26)It is clear that grad f at any point in space points in the direction along which the directional derivative reaches its maximum.According to(A.26),once the gradient of thefield is known,the directional derivative along any direction can be determined with ease,i.e.the variation of a scalarfield in the neighborhood of afixed point in space can be fully depicted by the gradient of thefield at that point.The differential of f(x)can be expressed in terms of its gradient asd f=grad f·d l(A.27) where d l is the infinitesimal displacement in space.It is clear that the variation of f(x)is the greatest parallel to grad f and zero perpendicular to it.Supposing f(x)is a potentialfield,it can be visualized by equipotential surfaces,i.e.surfaces on which the potential takes the same value.6The geo-metrical implication of(A.27)is that grad f at any point is always normal to the equipotential surface passing that point.Physically,grad f is related to the force associated with the potentialfield f(x).In rectangular coordinates,the gradient offield f(x)can be expressed asgrad f=∂f∂xˆe x+∂f∂yˆe y+∂f∂zˆe z(A.28)Divergence of a Vector FieldFor a vectorfield,we are concerned with not only its magnitude but also its direction at any point in space.A vectorfield can be visualized byfield lines, i.e.curves whose tangent is everywhere along the direction of thefield quantity.We are particular interested to know if thefield lines of a vectorfield“con-verges”or“diverges”at some particular point.Such a property is depicted by the divergence of thefield.The divergence of a vectorfield F(x)is a scalarfield,denoted by div F(x), whose value at afixed point x0is given by the limitdiv Fx=x0=limV→0SF·d SV(A.29)where V is a volume containing x0,and we mean by the limit V→0that the region enclosed by surface S contracts to x0.Physically,supposing that F(x)represents theflow of an incompressible liquid,a positive div F implies a“source”or“faucet”at that point,whereas a negative divergence indicates a“sink”.6Such surfaces are also known as level surfaces or isosurfaces.8APPENDIX A.VECTOR ANALYSIS In rectangular coordinates,the divergence offield F(x)can be expressed asdiv F=∂F x∂x+∂F y∂y+∂F z∂z(A.30)Curl of a Vector FieldAnother feature we are particular to know about a vectorfield is whether the field lines form a“whirlpool”or“vortex”.This feature is characterized by the curl of thefield.The curl of a vectorfield F(x)is a vectorfield,denoted by curl F,whose projection at point x0alongˆn is given bycurl n Fx=x0=ˆn·curl Fx=x0=limS→0LF·d lS(A.31)where S is a surface containing x0,withˆn as its normal unit vector at x0and curve L as its boundary.As a result,curl F at any point in space points along the direction in which(A.31)obtains its maximum.Physically,the concept of curl is related to rotation:Let F(x)=v be the velocity of points on a rigid body spinning around afixed axis,then curl F is just twice the angular velocityω=v×r,i.e.curl F=2ω.7In rectangular coordinates,the curl of a vectorfield can be written ascurl F=∂F z∂y−∂F y∂zˆe x+∂F x∂z−∂F z∂xˆe y+∂F y∂x−∂F x∂yˆe z=ˆe xˆe yˆe z∂x∂y∂zF x F y F z(A.32)A.3.2Operator∇The notation grad,div,and curl introduced in Section A.3.1are rather cum-bersome,and the symbol∇,which is called nabla or del,is often adopted for notational simplification∇f=grad f=∂f∂xˆe x+∂f∂yˆe y+∂f∂zˆe z(A.33a)∇·F=div F=∂F x∂x+∂F y∂y+∂F z∂z(A.33b)∇×F=curl F=ˆe xˆe yˆe z∂x∂y∂zF x F y F z(A.33c)A Warning7Another name for the curl of a vectorfield F(x)is the rotational,denoted by rot F.A.3.DIFFERENTIAL OPERATORS OF FIELDS9The Hamilton operator ∇,defined as∇=ˆe x ∂∂x +ˆe y ∂∂y +ˆe z ∂∂z was originally invented by Hamilton for the notation of ter,it was adopted by Gibbs for the notation of divergence and curl as well.The introduction of ∇not only simplifies the notation of identities involving dif-ferential operators of fields,it offers some hint to their derivation as well.If we put the expressions of dot and cross products of vectors and that of divergence and curl in rectangular coordinates side by side,a ·b =a x b x +a y b y +a z b z a ×b = ˆe x ˆe y ˆe z a x a y a z b x b y b z ∇·F =∂F x +∂F y +∂F z ∇×F = ˆe x ˆe y ˆe z ∂x ∂y ∂z F x F y F zwe could look upon ∇·F or ∇×F as the formal dot or cross product between ∇and F .Such an interpretation is often of great help in the derivation of many identities containing ∇.Here,however,a warning is in place.Although the formal dot or formal cross product interpretation of ∇·F or ∇×F helps the derivation of many identities,it is not based on mathematically rigorous ground,and unwary manipulation of ∇often leads to erroneous results.It is particularly so in curvilinear coordinates other than rectangular coordinates.With the help of ∇,the notation of many identities in vector analysis can be greatly simplified.Here are some most frequently encountered identities containing ∇,∇(fg )=f ∇g +g ∇f(A.34)∇·(F g )=F ·∇g +g ∇·F(A.35)∇×(F g )=∇g ×F +g ∇×F(A.36)∇·(A ×B )=B ·∇×A −A ·∇×B(A.37)∇×(A ×B )=(∇·B )A −(A ·∇)B +(B ·∇)A −(∇·A )B(A.38)∇(A ·B )=A ×(∇×B )+(A ·∇)B +B ×(∇×A )+(B ·∇)A (A.39)The above identities can be proved by expressing them in component form and evaluating the partial derivatives in rectangular coordinates.8However,this method is only adequate for relatively simple identities,and can become quite awkward for complicated formulae such as (A.39).8The quantity (A ·∇)B in (A.38)or (A.39)can be expressed in rectangular coordinates as(A ·∇)B =(A x ∂∂x +A y ∂∂y +A z ∂∂z )B =A x ∂B ∂x +A y ∂B ∂y +A z ∂B ∂zwhere A ·∇=A x ∂∂x +A y ∂∂y +A z ∂∂zcan be looked upon as an operator.10APPENDIX A.VECTOR ANALYSIS The most rigorous and efficient way to work with∇is to adopt the sym-bolic vector introduced by C.T.Tai[1].However,by taking advantage of the vector resemblance of∇as well as its differential nature,it is also possible to get the right result without having to delve into the mathematical details.It must be admitted that such procedure requires special care.Nevertheless,it could be mastered through practise.Students who worry about the mathemat-ical foundation for such manipulations should refer to Refs.[1]and[2]for a comprehensive treatment of∇expressions.To derive identities of∇acting on the product of several variables,e.g.(A.34)–(A.39),the basic procedure consists of the following two steps:1.Write the original∇expression into the sum of a series of partial differ-ential expressions,each containing only one function to be differentiated;2.Rewrite each item in the sum into the desired form by invoking the vec-torial identities,taking∇as an ordinary vector.Special care must be taken during the second step to make sure that the ex-pression be meaningful,as will be clear from the following examples.Example A.1Prove the following identities,(1)∇·(A×B)=B·∇×A−A·∇×B(2)∇×(A×B)=(∇·B)A−(A·∇)B+(B·∇)A−(∇·A)BProof.(1)First,write∇·(A×B)into the sum of two partial differential expres-sions∇·(A×B)=∇A·(A×B)+∇B·(A×B)where∇A denote differential of variable A only,with B taken as a constant vector,and∇B can be similarly interpreted.With the help of(A.11),thefirst item on the right hand side of the above equation can be expressed as∇A·(A×B)=B·(∇A×A)=B·(∇×A)where∇A behaves as if it were an ordinary vector,and use has been made of the fact∇A×A=∇×A.It is important to note that in the above manipulation, we have kept the operator∇A on the left side of its operand A,so as to keep the expression meaningful.Similarly,we can express the second item as∇B·(A×B)=−A·(∇B×B)=−A·(∇×B)Again,the relative position between operator∇B and its operand B is bining the above results,we have∇·(A×B)=B·∇×A−A·∇×BA.3.DIFFERENTIAL OPERATORS OF FIELDS11(2)First,express∇×(A×B)as the sum of partial differentials,i.e.∇×(A×B)=∇A×(A×B)+∇B×(A×B)With the help of(A.13),we can rewrite thefirst item as∇A×(A×B)=(B·∇A)A−(∇A·A)BNote that instead of sticking exactly to the expression given by(A.13),we have adjusted the order of B and∇A in thefirst item on the left hand side of the equation to produce the above result.The reason is that we need an operator before A,and B·∇A is the only operator that can be constructed out of the vectorial identity.As a result,it is found that∇A×(A×B)=(B·∇)A−(∇·A)BSimilarly,we have∇B×(A×B)=(∇·B)A−(A·∇)BConsequently,we arrive at the identity∇×(A×B)=(∇·B)A−(A·∇)B+(B·∇)A−(∇·A)BThe multiple combination of operator∇can be readily interpreted as the successive application of∇.Here are some most important combinations:∇×∇f=0(A.40)∇·∇×F=0(A.41)∇2f≡ f=∇·∇f(A.42)∇2F≡ F=∇(∇·F)−∇×∇×F(A.43) As we shall see,(A.40)and(A.41)are of great importance in the study of vectorfields.Example A.2Show that∇×∇f=0and∇·∇×F=0,where f(x)and F(x)are well behaved scalarfield and vectorfield,respectively.Proof.As∇×∇f=curl grad f,we see that the result is a vectorfield.According to(A.33),we can express(∇×∇f)x in rectangular coordinates as(∇×∇f)x=∂(∇f)z∂y−∂(∇f)y∂z=∂∂y∂f∂z−∂∂z∂f∂y=∂2f∂y∂z−∂2f∂z∂y=0Similarly,(∇×∇f)y=(∇×∇f)z=0.As a result,we conclude that∇×∇f=0.12APPENDIX A.VECTOR ANALYSIS On the other hand,∇·∇×F=div curl F is a scalarfield.Expressing∇·∇×F in rectangular coordinates,we have∇·∇×F=∂(∇×F)x∂x+∂(∇×F)y∂y+∂(∇×F)z∂z=∂∂x∂F z∂y−∂F y∂z+∂∂y∂F x∂z−∂F z∂x+∂∂z∂F y∂x−∂F x∂y=∂2F z∂x∂y−∂2F y∂x∂z+∂2F x∂y∂z−∂2F z∂y∂x+∂2F y∂z∂x−∂2F x∂z∂y=0It should be noted that,though the above proof is given in rectangular coordinates,the results are valid in any coordinate system,for the values of gradient,divergence,and curl are coordinate independent.Special care has to be paid to the Laplacian∇2,sometimes denoted by , as it acts quite differently on a scalar function and on a vector function.The Laplacian of a scalar function can be interpreted as∇2f=div grad f,and can be expressed in rectangular coordinates as∇2f=∇·∇f=∂2f∂x2+∂2f∂y2+∂2f∂z2(A.44)On the other hand,∇2F can only be properly understood with the concept of dyadic discussed in Section A.1.3,and we have∇2F=∇·(∇F)(A.45) where∇F is the gradient of a vector,which can be expressed as a dyadic.9 In rectangular coordinates,∇2F can be expressed as∇2F=(∇2F x)ˆe x+(∇2F y)ˆe y+(∇2F z)ˆe zHowever,for curvilinear coordinates other than rectangular coordinates,we have to resort to(A.43)and interpret the Laplacian of a vectorfield as∇2F= grad div F−curl curl F.But the dyadic interpretation(A.45)holds true in any curvilinear coordinates.A.4Integral Theorems of FieldsAs mentioned in the previous sections,the integral properties,such as theflux and the path integral,offer us the global information about afield,whereas the differential properties,including gradient,divergence,and curl,present us with the local information about the samefield.It is only natural that these two types of properties should be related to each other.Such relationship is reflected by the integral theorems offields.9See Problem A.4and A.5for more detail.A.4.INTEGRAL THEOREMS OF FIELDS13Of the numerous integral theorems offields,three theorems are of crucial importance:one depicting the integral property of the gradient of a scalar field,and the other two concerning the integral properties of the divergence and the curl of a vectorfield.In the following,we shallfirst introduce the three fundamental theorems one by one,and then introduce some other useful integral theorems which can be derived with the help of these fundamental theorems. Fundamental Theorems of Vector CalculusFor a scaler function with one variable,we have the fundamental theorem of calculusf(b)−f(a)=baf (x)dxwhich states that the integration process is just the inverse of the differentiation pro-cedure.The three integral theorems describing the line,surface,and volume integral of the differentiations offields,namely,the gradient,the divergence,and the curl,can be viewed as a generalization of the fundamental theorem of calculus.Consequently, they may as well be called the fundamental theorems of gradient,divergence,and curl, respectively.A.4.1Integral Theorem of GradientFor a scalarfield f(x),we have the following integral theorem:f(b)−f(a)=ba∇f·d l(A.46)which is a direct consequence of(A.27).In the above equation,the line integral is taken along any path that connects a and b,i.e.the line integral of the gradient of a scalarfield is independent of the path.According to the definition given in Section A.2.1,we see that the gradient of a scalarfield is a conservative field.A.4.2Integral Theorem of Divergence:Gauss’s Theorem Thefirst important integral theorem of vectorfields is the Gauss’s theorem,S F·d S=V∇·F dV(A.47)which can be directly obtained from(A.29).Physically,Gauss’s theorem states that the totalflux of F,which may present theflow of an incompressible liquid, through a closed surface S is just the total sum of the sources or sinks within the enclosed volume.14APPENDIX A.VECTOR ANALYSIS Gauss’s theorem can be also cast into some other forms,S d S×F=V∇×F dV(A.48)S f d S=V∇f dV(A.49)Example A.3Use Gauss’s theorem(A.47)to prove the following relationS f d S=V∇f dVwhere f is a scalarfield.Proof.Let c be a non-zero constant vector,and insert F=c f into(A.47).The left hand side is seen to beS F·d S=S(c f)·d S=c·Sfd Swhereas the right hand side isV ∇·F dV=V∇·(c f)dV=c·V∇f dVConsequently,we havec·S fd S=c·V∇f dVOwning to the arbitrariness of c,we conclude thatS fd S=V∇f dVGreen’s TheoremsWith the help of Gauss’s theorem,we can derive a group of integral theorems called Green’s theorems,which are of great use in potential theory.Scalar Green’s Theorem For scalar functionsφ(x)andψ(x),we have the first scalar Green’s theorem:V (φ∇2ψ+∇φ·∇ψ)dV=Sφ∂ψ∂ndS(A.50)which can be obtained by inserting F=φ∇ψinto Gauss’s theorem(A.47).If we exchange the position ofφandψin the above equation,and subtract it from the original equation,we obtain the second scalar Green’s theorem:V (φ∇2ψ−ψ∇2φ)dV=Sφ∂ψ∂n−ψ∂φ∂ndS(A.51)A.5.PROPERTIES OF VECTOR FIELDS15 Vector Green’s Theorem For vector functions A(x)and B(x),we can obtain thefirst vector Green’s theorem by inserting F=A×(∇×B)into the Gauss’s theorem(A.47),V [∇×A·∇×B−A·(∇×∇×B)]dV=SA×(∇×B)·d S(A.52)From the above identity we can easily obtain the second vector Green’s theorem:V[B·(∇×∇×A)−A·(∇×∇×B)]dV=S[A×(∇×B)−B×(∇×A)]·d S(A.53) Gauss’s Theorem for TensorsThe Gauss’s theorem for tensors or dyadics can be written asd S·↔T=dSˆn·↔T=dV∇·↔T(A.54)The order of the dot product in the above equation is of importance,for anterior and posterior dot products between a vector and a tensor are normally different.A.4.3Integral Theorem of Curl:Stokes’TheoremAnother important integral theorem of vectorfields is the Stokes’Theorem,L F·d l=S∇×F·d S(A.55)which can be viewed as a direct result of(A.31).Like Gauss’s theorem,we also have Stokes’theorem in different forms,L d l×F=S(d S×∇)×F(A.56)L f d l=Sd S×∇f(A.57)A.5Properties of Vector FieldsSo far we have seen that a vectorfield can be characterized by its divergence and its curl.Now we would like to know if such information is sufficient.In other words,is a vectorfield uniquely determined,once its divergence and its curl are known everywhere in space?10Before we can tackle this problem,we shallfirst 10In this section,we shall only discuss vectorfields defined in the entire space or in a simply-connected region,i.e.any simple closed curve within the region can be continuously contracted to a point without ever crossing the boundary.。