Vector calculus. Appendix A. A.1 Definitions. We shall only consider the case of three-dimensional spaces.

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1 Appendix A Vector calculus We shall only consider the case of three-dimensional spaces A Definitions A physical quantity is a scalar when it is only determined by its magnitude and a vector when it is determined by its magnitude and direction It is crucial to distinguish vectors from scalars; standard notations for vectors include u u u A unit vector commonly denoted by î î has magnitude one The coordinates of a vector a are the scalars a a 2 and a 3 such that a = a e +a 2 e 2 +a 3 e 3 (a a 2 a 3 a 2 a 3 in the basis {e e 2 e 3 } If {ê ê 2 ê 3 } are orthonormal (ie ê i ê j = δ ij the magnitude of the vector is a = a 2 +a2 2 +a2 3 Scalar or dot product: is a scalar Vector or cross product: a b = a b cosθ = a b +a 2 b 2 +a 3 b 3 a b = (a 2 b 3 a 3 b 2 e +(a 3 b a b 3 e 2 +(a b 2 a 2 b e 3 is a vector with magnitude a b sinθ and a direction perpendicular to both vectors a and b in a right-handed sens Triple scalar product: is a scalar a [abc] = a b c = a b c = b c a Triple vector product: a (b c = (a cb (a bc is a vector 75

2 76 A2 Suffix notation A2 Suffix notation It is often very convenient to write vector equations using the suffix notation Any suffix may appear once or twice in any term in an equation; a suffix that appears just once is called a free suffix and a suffix that appears twice is called a dummy suffix Summation convention Dummy suffices are summed over from to 3 whilst free suffices take the values 2 and 3 Hence free suffices must be the same on both sides of an equation whereas the names of dummy suffices are not important (eg a i b i c k = a j b j c k Tensors used in suffix notation Kronecker Delta: { if i = j δ ij = 0 if i j ( 0 0 δ ij = The Kronecker Delta is symmetric δ ij = δ ji and δ ij a j = a i Alternating Tensor: 0 if any of i j or k are equal; ǫ ijk = if (ijk = (23 (23 or (32; if (ijk = (32 (32 or (23 The Alternating Tensor is antisymmetric ǫ ijk = ǫ jik and it is invariant under cyclic permutations of the indices ǫ ijk = ǫ jki = ǫ kij The tensors δ ij and ǫ ijk are related to each other by ǫ ijk ǫ klm = δ il δ jm δ im δ jl Examples Standard algebraic operations on vectors can be written in a compact form using the suffix notation A scalar product can be written as a b = a b +a 2 b 2 +a 3 b 3 = a j b j and the i th component of a vector product as (a b i = ǫ ijk a j b k A3 Vector differentiation A3 Differential operators in Cartesian coordinates Weconsiderscalarandvectorfieldsf(xandF(x = (F (xf 2 (xf 3 (xrespectivelywhere x = (x x 2 x 3 (xyz are Cartesian coordinates in the orthonormal basis {ê ê 2 ê 3 } {îĵˆk} We also define the vector differential operator ( is pronounced grad nabla or del ( x The gradient of a scalar field f(x x 2 x 3 is given by the vector field ( gradf f = = ê + ê 2 + ê 3 x x f is the vector field with a direction perpendicular to the isosurfaces of f with a magnitude equal to the rate of change of f in that direction

3 Chapter A Vector calculus 77 The divergence of a vector field F is given by the scalar field divf = F = F x + F 2 + F 3 A vector field F is solenoidal if F = 0 everywhere The curl of a vector field F is given by the vector field ( F3 curlf = F = F 2 F F 3 F 2 F x x ( F3 = F ( 2 F ê + F ( 3 F2 ê 2 + F ê 3 x x ê ê 2 ê 3 = x x 3 F F 2 F 3 A vector field F is irrotational if F = 0 everywhere Thedirectional derivative(f isadifferentialoperatorwhichcalculatesthederivative of scalar or vector fields in the direction of F (It is not to be confused with the scalar F The directional derivative of a scalar field is given by the scalar field (F f = F f = F x +F 2 +F 3 If ˆn is a unit vector (ˆn f gives the rate of change of f in the direction of ˆn The directional derivative of a vector field G is given by the vector field (F G = ((F G (F G 2 (F G 3 ( G G G = F +F 2 +F 3 x F G 2 x +F 2 G 2 +F 3 G 2 F G 3 x +F 2 G 3 +F 3 G 3 The Laplacian 2 = 2 or vector fields: is a scalar field and is a vector field is a differential operator which can act on scalar 2 f = 2 f + 2 f + 2 f F = ( 2 F 2 F 2 2 F 3 The Lagrangian derivative of scalar and vector fields are D Dt f(xt = d dt f(x(tt = t f(xt+(u f(xt

4 78 A3 Vector differentiation and D Dt F(xt = d dt F(x(tt = t F(xt+(u F(xt respectively with u(xt = dx/dt and where acts upon the variables (x x 2 x 3 only Notice that if F = F(t then df ( dt = df dt df 2 dt df 3 dt Differential operators in Cartesian coordinates using suffix notation (gradf i = ( f i = x i divf = F = F j x j (curlf i = ( F i = ǫ ijk F k x j (F f = F j x j Notice that the operator / x j cannot be moved around as it acts on everything that follows it A32 Differential operators polar coordinates ThedifferentialoperatorsdefinedaboveinCartesiancoordinatestakedifferentformsfordifferent systems of coordinates eg in cylindrical polar coordinates or spherical polar coordinates Cylindrical polar coordinates Let p and u = u r ê r +u θ ê θ +u z ê z be scalar and vector fields respectively functions of (rθz p = + + rêr r êθ zêz u = r r (ru r+ u θ r + u z z ( u z u = r u ( θ ur ê r + z z u z r (u p = u r r + u θ r +u z z 2 p = ( r + 2 p r r r r p z 2 ê θ + ( r r (ru θ u r ê z

5 Chapter A Vector calculus 79 Spherical polar coordinates Let p and u = u r ê r +u θ ê θ +u ϕ ê ϕ be scalar and vector fields respectively functions of (rθϕ p = + + rêr r êθ rsinθ ϕêϕ u = ( r 2 r 2 u r + r rsinθ (u θsinθ+ u ϕ rsinθ ϕ u = ( rsinθ (u ϕsinθ u θ ê r + ( u r ϕ r sinθ ϕ r (ru ϕ + ( r r (ru θ u r ê ϕ (u p = u r r + u θ r 2 p = r 2 r ( r 2 r + u ϕ rsinθ + r 2 sinθ A33 Vector differential identities ϕ ( sinθ + 2 p r 2 sinθ ϕ 2 The simple vector identities that follow can all be proved with suffix notation Note that these vector identities are true for all systems of coordinates Let F and G be vector fields and ϕ and ψ be scalar fields ê θ ( ϕ = 2 ϕ ( F = 0 ( ϕ = 0 (ϕψ = ϕ ψ +ψ ϕ (ϕf = ϕ F+F ϕ (ϕf = ϕ F+ ϕ F ( F = ( F 2 F (F G = F ( G+G ( F+(F G+(G F (F G = G ( F F ( G (F G = F( G G( F+(G F (F G 2 F = ( F ( F F ( F = 2 (F F (F F A4 Vector integral theorems A4 Alternative definitions of divergence and curl An alternative definition of divergence is given by F = lim δv 0 δv F nds where δv is a small volume bounded by a surface δs which has a normal n pointing outwards δs

6 80 A4 Vector integral theorems An alternative definition of curl is given by n F = lim δs 0 δs δc F dx where δs is a small open surface bounded by a curve δc which is oriented in a righthanded sense A42 Physical interpretation of divergence and curl The divergence of a vector field gives a measure of how much expansion and contraction there is in the field The curl of a vector field gives a measure of how much rotation or twist there is in the field A43 The Divergence and Stokes Theorems The divergence theorem states that V F = S F nds where S is the closed surface enclosing the volume V and n is the outward-pointing normal from the surface Stokes theorem states that S ( F nds = C F dx where C is the closed curve enclosing the open surface S and n is the normal from the surface A44 Conservative vector fields line integrals and exact differentials The following five statements are equivalent in a simply-connected domain: i F = 0 at each point in the domain ii F = φ for some scalar φ which is single-valued in the region iii F dx is an exact differential iv v Q C P F dx is independent of the path of integration from P to Q F dr = 0 around every closed curve in the region If F = 0 then F = A for some A (This vector potential A is not unique