Main Results of Vector Analysis

Transcription

1 Main Results of ector Analysis Andreas Wacker Mathematical Physics, Lund University January 5, 26 Repetition: ector Space Consider a d dimensional real vector space with scalar product or inner product v w. Here, the elements of are denoted by bold-face letters. In order to perform calculations, we choose a convenient basis B {e,... e d }, so that each vector can be uniquely written as v d i v ie i. This provides a mapping v v v 2... v d which depends on the particular choice of the basis. For an orthonormal basis ON basis, which satisfies e i e j δ ij, we find the simple rule for scalar product d v w v i w i w v... v d... i w d For real three-dimensional vector spaces, one can further define the cross product or outer product c a b Here, the vector c is determined by its direction c a, b i.e. c a c b with positive orientation right-hand rule: thumb a, index finger b, middle finger c and the length c a 2 b 2 a b 2 which corresponds to the area a b sin a, b of the parallelogram formed by a and b. Beware that the cross product is neither commutative nor associative! For an orthonormal basis with positive orientation i.e. e 3 e 2 e and cyclic the components of the respective vectors satisfy c a 2 b 3 a 3 b 2 c 2 a 3 b a b 3 or c i ɛ ikl a k b l c 3 a b 2 a 2 b kl The triple scalar product only for three-dimensional vector spaces a b c provides the volumes of the parallelepiped formed by the vectors a, b, c. It satisfies a b c b c a c a b The triple cross product satisfies the important BAC-CAB rule a b c ba c ca b Andreas.Wacker@fysik.lu.se This work is licensed under the Creative Commons License CC-BY. It can be downloaded from

2 ector analysis, Andreas Wacker, Lund University, January 5, Multivariable functions We consider an affine space of points P, where for a given origin O each point P is uniquely related to a vector r OP. Here is the associated real vector space. For the elements of P we define scalar functions fp and vector functions FP. If we fix the origin of the point space, they can be written as fr and. Fr. 2. Systems of coordinates In order to perform any calculations, we have to express the points by specific coordinates. Choosing an ON-basis B {e,... e d }, we have r d i r ie i and we may write the scalar function as fr f B r,... r d. In the standard three-dimensional space, the basis vectors are denoted as B {e x, e y, e z } and the corresponding Cartesian coordinates as x, y, z. For problems with a rotational symmetry around the z axis, cylindrical coordinates ρ, ϕ, z are usually a good choice. For systems with spherical symmetry one generally uses spherical coordinates r, ϕ, θ, see Tab. on page for a compilation. Thus the identical function fr can be represented by different expressions. An example is fr r x2 + y 2 + z }{{ 2 } f Cartesian x,y,z ρ2 + z }{{ 2 } f cylindrical ρ,ϕ,z r f spherical r,ϕ,θ 2.2 Examples Elevation in geography: The elevation is a scalar function hr of the points on earth s surface. It can be conveniently parameterized by the longitude ϕ and the latitude θ π/2 so that the poles are at 9 and the equator at Current of a fluid: Here nr, t is the particle density, so that the number of particles dn in the volume element d around the position r at time t is given by dn nr, t d or d 3 r This is a scalar function. The velocity field vr, t denotes the velocity of a test particle following the current. It is a vector function alike the particle current density jr, t nr, tvr, t. 3 Multidimensional derivatives 3. Partial and total derivative In general the derivative of a function f with respect to a quantity t is based on the definition ft+ɛ ft of a mapping t ft. Then the derivative is given by the limit Der t f lim ɛ. ɛ If we consider functions fx, y, z, t, which depend on more than one variable, one needs to specify how all variables depend on the quantity t with respect to which the differentiation is performed. Thus we need to specify functions xt, yt, zt and than we can write Der t f dfx, y, z, t dt lim ɛ fxt + ɛ, yt + ɛ, zt + ɛ, t + ɛ fxt, yt, zt, t ɛ

3 ector analysis, Andreas Wacker, Lund University, January 5, 26 3 which is usually called the total derivative. Frequently, the functions xt, yt, zt are not explicitly stated, if the physical problem suggests a natural choice such as the path of a particle in time. A special choice is that the functions xt, yt, zt are constants. Then one writes Der t f fx, y, z, t t lim ɛ fx, y, z, t + ɛ fx, y, z, t ɛ which is called partial derivative. Note that the partial derivative only exists, if the quantity t is one of the explicit variables of the function. Furthermore the value of the partial derivative may change by coordinate transformations. Thus it is of utmost importance to specify the variables of the function. If there is a natural set of variables, physicists frequently become sloppy here which sometimes causes misunderstandings. The total derivative can be conveniently expressed in terms of partial derivatives via dfx, y, z, t dt fx, y, z, t dxt + dt fx, y, z, t dyt + dt fx, y, z, t dzt fx, y, z, t + dt t and one defines the total differential as dfx, x 2,... x d fx + dx, x 2 + dx 2,... x d + dx d d i fx, x 2,... x d dx i i In the same way second derivatives are defined. An important relation is the symmetry of the second derivatives sometimes also referred to as Schwarz s theorem or Clairaut s theorem fx, x 2,... x d fx, x 2,... x d 2 fx, x 2,... x d i j j i i j provided that the second derivatives are continuous. 3.2 Gradient Consider the change of a scalar function fr due to a change in space dr e x dx + e y dy + e z dz in Cartesian coordinates. We find e x fr dr df f f f dx + dy + dz + e y + e z We define the vector differential operator del which is represented by the symbol nabla by the relation df fr dr In the Cartesian basis it reads e x + e y + e z The vector function Gr fr can be interpreted as follows: Consider fr in the vicinity of r. Then fr + dr fr + Gr dr fr + Gr dr cos Gr, dr

4 ector analysis, Andreas Wacker, Lund University, January 5, 26 4 Thus Gr points in the direction of the largest growth of fr and the absolute value Gr provides the value of the largest growth per distance dr. Thus Gr is also called the gradient of fr at position r. The gradient of a scalar function grad fr fr is a vector function. In Cartesian coordinates it is given by grad fr f xx,y,z f xx,y,z f xx,y,z Examples: to work out In particular grad r a r a r a, grad g r a r a g r a r a grad r a r a r a Divergence, Curl and Laplace operator The Del operator can also be applied to vector functions Fr. There are two different variants. The divergence of a vector function div Fr Fr is a scalar function. In Cartesian coordinates it is given by div Fr F xx, y, z Example: to work out div r 3 + F yx, y, z + F zx, y, z The curl of a vector function only in three dimensions Fr is a new vector function. In Cartesian coordinates Fr F zx,y,z F xx,y,z F yx,y,z Fyx,y,z Fzx,y,z Fxx,y,z Example: to work out r The Laplacian is a second order derivative. It can operate both on scalar and vector functions. In Cartesian coordinates it is given by fr 2 fx, y, z 2 x + 2 fx, y, z 2 y + 2 fx, y, z 2 z

5 ector analysis, Andreas Wacker, Lund University, January 5, Del operator in cylindrical coordinates The relation between cylindrical or polar and Cartesian coordinates is given by: x ρ cosϕ, y ρ sinϕ, z z or ρ x 2 + y 2, ϕ arctany/x This provides dr cosϕe x + sinϕe y dρ + ρ sinϕe x + ρ cosϕe y dϕ + e z dz e ρϕ ρe ϕϕ The vectors e ρ ϕ, e ϕ ϕ, e z are orthonormal. Thus one can write the total differential of Eq. in the form df f cylinderρ, ϕ, z dρ + f cylinderρ, ϕ, z dϕ + f cylinderρ, ϕ, z dz fcylinder ρ, ϕ, z e ρ + f cylinderρ, ϕ, z ρ e ϕ + f cylinderρ, ϕ, z e z dr and one identifies e ρ ϕ + e ϕϕ ρ + e z In order to obtain the divergence in cylindrical coordinates, we consider a vector function Fr e ρ ϕf ρ ρ, ϕ, z + e ϕ ϕf ϕ ρ, ϕ, z + e z F z ρ, ϕ, z. Now some care is needed, as we need to take into account e ρϕ e ϕ ϕ and e ϕϕ e ρ ϕ. Then we find divf F F ρρ, ϕ, z + ρ F ρρ, ϕ, z + F ϕ ρ, ϕ, z + F zρ, ϕ, z. 4 ρ ρ ρfρρ,ϕ,z In the same way, we can obtain the curl and the Laplacian. The results are summarized together with the corresponding expressions for spherical coordinates in Tab. on page Divergence of r/r 3 and the three-dimensional delta function The radial-symmetric function Fr r/r 3 describes, e.g., the electric field of a point charge at the origin. In spherical coordinates we have Fr e r /r 2. Using the expression for the divergence in spherical coordinates see Tab. on page, we find divfr /r 2. Thus, the divergence vanishes for r, while the situation at the origin is unclear due to the singularity. In order to resolve this, we consider the function F ɛ r r r 3 + ɛ 3 e r which becomes the original function in the limit ɛ. we find divf ɛ r r 3 r 2 r r 3 + ɛ 3ɛ 3 3 r 3 + ɛ 3 f ɛr 2 In the limit ɛ the new function f ɛ r vanishes for r, while it becomes infinite at the origin. Integration over the entire three-dimensional space we find d 3 3r 2 ɛ 3 r f ɛ r 4π dr r 3 + ɛ 3 4π dt 2 t + 4π 2

6 ector analysis, Andreas Wacker, Lund University, January 5, 26 6 Thus f ɛ r/4π becomes the three-dimensional delta function δr in the limit ɛ and we may write r div 4πδr. 5 r 3 Note the difference between the three-dimensional delta function δr and the one-dimensional delta function δr. While both are infinite at the origin and zero otherwise, we have d 3 r δr while dr δr This immediately shows that the dimension of δr is inverse volume, while the dimension of δr is inverse length. Actually, in cartesian coordinates, we have δr δxδyδz. 3.6 Nabla-calculus In order to evaluate complicated expression such as fr[gr Hr], one may use the following rules:. Mark which functions shall be affected by the differentiation and apply the product rule of differentiation fr[gr Hr] f G H fg H + f G H + fg H 2. Transform the expression according to the rules of vector algebra where is treated as a conventional vector keep the arrows!. f G H + fh G + fg H Here the triple-scalar-product rule was used for the second and third term. 3. Order the expressions such, that only the marked functions are on the right-hand side of. Then the markings can be finally abandoned. G H f + fh G fg H Examples: to work out r a div r a 3 for r a 6 r a r a 3 for r a 7 4 Integrals in three dimensions 4. Line integrals Example: For a given path of a particle the work acted on the particle reads W dr Fr small path elements r i r i Fr i elements In order to evaluate the line integral we parameterize the path rs with s < s < s 2. Then dr drds and ds s2 W ds dr s ds Frs path

7 ector analysis, Andreas Wacker, Lund University, January 5, 26 7 A special parameterization is the time t. Then dr v is the velocity and v Frt the power, dt such that the total applied work is the time integral of mechanical power. In particular we write for a closed path which is the boundary of an area S W dr Fr 4.2 olume integral S Example: Consider molecules within a volume of a liquid. partial volumes N nr i i d nr all small partial volumes atr i also d 3 r nr Here we need to parameterize the space by three parameters s, t, u such that for s, t, u range, the points rs, t, u are within. The the volume element is d r r r s t u dsdtdu and we find N range dsdtdu r s r t r nrs, t, u u To work out: cylinder d3 r z in Cartesian and cylinder coordinates 4.3 Surface integral Example: Particle flow through a surface S. The surface is decomposed in small surface elements S i, where the vector is pointing perpendicular to the surface and defines the orientation of the surface. surface elements I S i jr i ds jr all surface elements The surface is parameterized by two parameters u, v. If u, v Srange, the points ru, v are within S. Then ds r r and we obtain u v r I dudv u r jru, v v Srange In particular the surface enclosing the volume is written as S and the integral is written as I ds jr S 4.4 Gauss Theorem ds Fr d div Fr

8 ector analysis, Andreas Wacker, Lund University, January 5, 26 8 Proof for a cuboid with edges L x, L y, L z Lx Ly d div Fr dx dy Ly Lz Lz dz Fx + F y + F z dy dz[f x L x, y, z F x, y, z] + 2 further terms ds Fr + ds Fr + 4 further surfaces face at xl x face at x ds Fr face of cuboid Here the orientation of all faces is directed outwards of the cuboid, which compensates the minus sign at x. 4.5 Continuity equation Consider particles within a volume of a fluid. The number of particles N is diminished by a current density through the surface of the volume and increased by sources qr particle production per volume at location r. Thus: dnt dt ds jr, t + With Nt d nr, t and Gauss theorem we find [ ] nr, t d + div jr, t t d qr, t d qr, t As this relation holds for arbitrary volumes, this provides the nr, t continuity equation + jr, t qr, t t For liquids the density n is constant and we find jr, t n vr, t. Thus fields with vanishing divergence are often called incompressible. Another common denotation is solenoidal as the magnetic field satisfies B. 4.6 Stokes Theorem dr Fr ds Fr S S Proof for a rectangle in the x, y-plane i.e. z with edges L x, L y Lx Ly Lx Ly Fy ds Fr dx dy e z Fr dx dy F x S Ly Lx S dy [F y L x, y, F y, y, ] dx F x x,, + dr Fr Ly Lx dy F y L x, y, + dx [F x x, L y, F x x,, ] L x dx F x x, L y, + L y dy F y, y,

9 ector analysis, Andreas Wacker, Lund University, January 5, 26 9 In hydrodynamics a finite integral dr vr denotes a rotation in a fluid. Thus the closed S line integral is called circulation and vr is denoted as circulation density or vorticity. Fields with F are called irrotational. 5 Potentials 5. Scalar Potential Let Fr be an arbitrary vector function, which is defined in the simply connected region i.e. there are no holes of space. Then fr exists with Fr fr in a scalar potential exists Fr in Fr is irrotational The scalar potential fr is uniquely determined by Fr except for an arbitrary constant. Proof: fr f f. Alternatively one may use Cartesian coordinates and apply the symmetry of second derivatives. For a given r we define r fr dr Fr r The integral does not depend on the particular path. Otherwise one can construct a closed line integral from the two paths with non-vanishing circulation, which is excluded by Stokes theorem and Fr. Now r+dr r dfr r dr Fr + dr Fr dr Fr r and we identify fr Fr from Eq.. Thus fr is the scalar potential. 5.2 ector Potential Let Fr be an arbitrary vector function, which is defined in the simply connected region of space. Then Ar exists with Br Ar in a vector potential exists div Br in Br is incompressible/solenoidal There is some freedom to choose the vector potential Ar. Two potential A r and A 2 r provide the identical function Fr if A r A 2 r grad ξr. gauge invariance Proof: Ar A A A.

10 ector analysis, Andreas Wacker, Lund University, January 5, 26 We construct Ar in Cartesian coordinates. A z. Then Due to the gauge invariance we may set B x x, y, z A y A yx, y, z z z z dz B x x, y, z + fx, y B y x, y, z A x A xx, y, z dz B y x, y, z z B z x, y, z A y A z x dz Bx x, y, z + B yx, y, z fx, y + z Bzx,y,z fx, y B z x, y, z B z x, y, z + By choosing fx, y x x dx B z x, y, z the last equation is satisfied. Thus we have constructed the vector potential z A x z dz B y x, y, z A y z z dz B x x, y, z + x x dx B z x, y, z A z 5.3 Defining a vector field by divergence and curl For given functions qr, wr, and hr, the vector field Fr is uniquely defined for r simply connected by the requirements F qr, F wr and n F hr on The construction contains three parts which add up to the vector field Fr: F l d 3 r qr 4π r r F t d 3 r wr 4π r r F bound Ψ with Ψ and n Ψ hr n F l + F t on In Fourier space we find F l q q and F t q q. The term F bound is both irrotational and solenoidal. If is the entire three-dimensional space and we apply the boundary condition F for r, we find F bound, provided that gr and wr are restricted to a finite region in space.

11 ector analysis, Andreas Wacker, Lund University, January 5, 26 Cartesian coordinates polar coordinates spherical coordinates Coordinates qi x, y, z ρ, ϕ, z r, θ, ϕ x x ρ cosϕ r sinθ cosϕ y y ρ sinϕ r sinθ sinϕ z z z r cosθ Unit vector e eq x eρ cos ϕex + sin ϕey er sin θ cos ϕex + sin θ sin ϕey + cos θez Unit vector eq2 e y eϕ sin ϕex + cos ϕey eθ cos θ cos ϕex + cos θ sin ϕey sin θez Unit vector eq3 e z ez eϕ sin ϕex + cos ϕey del operator: ex + e y + e z eρ + e ϕ ρ + e z er r + e θ r θ + e ϕ Gradient: fr ex f + e y f + e z f eρ f + e ϕ ρ f + e z f er f r + e θ r f θ + e ϕ f Divergence: Fr F x + F y + F z Curl: Fr F z F y + Fx F z + F y F x ex Laplacian: f 2 f 2 x + 2 f 2 y + 2 f 2 z ρ ρ e y + ez + ρ ρ Fρ + ρ Fz F ϕ F ρ F ϕ Fϕ F z + F z eρ F ρ ρ f ρ r 2 Fr r 2 r + eϕ + ez + r + 2 f ρ 2 2 ϕ + 2 f 2 z sin θf ϕ θ F θ rf θ Fr rf ϕ r r F r θ olume element: d 3 r dx dy dz ρ dρ dϕ dz sin θr 2 dr dθ dϕ r sin θfθ θ + eϕ er eθ 2 rf 2 r + 2 f r 2 2 θ + cotθ f r 2 θ + r 2 sin 2 θ Fϕ 2 f 2 ϕ Table : ector analysis in different coordinate systems. The unit vectors are defined as eq r q eq, eq2, eq3 form an orthonormal basis with positive orientation in each case. r / q. They are ordered such such that