Vector calculus in curvilinear coordinates D. Jaksch Goals: • Understand the difference between coordinates and vector components • Understand the implications of basis vectors in curvilinear coordinates not being constant • Learn how to use curvilinear coordinate systems in vector calculus Coordinate systems and vector fields Coordinate systems A point in coordinate space r is often represented as r = (x; y; z)T with x, y, and z the distances along the three coordinate axes. We can equally introduce cylindrical polar coordinates which we will use here as the prime example for curvilinear coordinate systems. They are defined through the relations x = ρ cos('); y = ρ sin('); and z = z: The point in space is now written as r = (ρ cos('); ρ sin('); z)T where the meaning of the components is still that they give the distances along the three coordinate axes. Vector fields A vector field assigns a vector to each point r and is usually denoted as F(r) or simply F. The vector field is often defined through components Fi(r) which are the projections of the vector onto the three coordinate axes. For instance F = (−y; x; 0)T =px2 + y2 assigns vectors as indicated in figure 1a). Using cylindrical polar coordinates this vector field is given by F = (− sin('); cos('); 0)T . Here we have rewritten the vector field in different coordinates but not changed the meaning of its components. a) b) c) 푒̂휑 퐹푥 푒휌̂ 휑 퐹푦 푒̂푦 퐹 z y z y z y 푒̂푥 푒̂휑 퐹푥 퐹휑 퐹푦 푒̂푦 x x x 휌 푒̂푥 푒̂ Figure 1: (a) Vector field F = (−y; x; 0)T =px2 + y2 (red arrows) and showing cartesian (b) and cylindrical polar (c) components for two of its vectors. Basis vectors Each point in coordinate space has a vector space associated with it where the vectors of vector fields live. A coordinate system fxig allows us to define bases for all of these vector spaces through @r ei = : @xi ^ ^ ^ For cartesian coordinates the normalized basis vectors are e^x = i, e^y = j, and e^z = k pointing along the three coordinate axes. They are orthogonal, normalized and constant, i.e. their direction does not change with the point r1. Next we calculate basis vectors for a curvilinear coordinate systems using again cylindrical polar coordinates. They are given by @r 1 @r e^ = = cos(')e^ + sin(')e^ ; e^ = = − sin(')e^ + cos(')e^ ; and e^ = e^ : ρ @ρ x y ' ρ @' x y z z The 1/ρ in the definition of e^' is required for the vector to be properly normalized to 1. These basis vectors are mutually orthogonal and normalized. However, they are not constant, their direction changes with position. By inverting this set of linear equations for the basis vectors we find e^x = cos(')e^ρ − sin(')e^' ; e^y = sin(')e^ρ + cos(')e^' ; and e^z = e^z : T We can now write a vector field F(r) = (Fx(r);Fy(r);Fz(r))c ≡ Fx(r)e^x + Fy(r)e^y + Fz(r)e^z in new components using the above relations between the basis vectors for different coordinate systems. This is still T the same vector field but now written as F(r) = (Fρ(r);F'(r);Fz(r))p ≡ Fρ(r)e^ρ + F'(r)e^' + Fz(r)e^z. The new components are projections on the basis vectors e^ρ, e^', and e^z. We use indices c and p to make explicit which components are used. In detail, the change of components is carried out by 0 1 Fx F = @ Fy A = Fxe^x + Fye^y + Fze^z = Fx[cos(')e^ρ − sin(')e^'] F z c 0 1 Fx cos(') + Fy sin(') +Fy[sin(')e^ρ + cos(')e^'] + Fze^z = @ −Fx sin(') + Fy cos(') A F z p T T Cartesian and cylindrical polar components for the vector field F = (− sin('); cos('); 0)c = (0; 1; 0)p are shown T T in figures 1b) and 1c) respectively. Produce a similar figure for the vector field F = r = (x; y; z)c = (ρ, 0; z)p . The most commonly used coordinate systems produce orthogonal right handed bases. This means that scalar and vector products between the basis vectors obey the familiar relations e^i · e^j = δij and e^i × e^j = e^kijk, where δij is the Kronecker delta and ijk the totally antisymmetric tensor. For cylindrical polar coordinates (ρ, '; z) we explicitly have e^ρ · e^ρ = e^' · e^' = e^z · e^z = 1 and e^ρ · e^' = e^' · e^z = e^z · e^ρ = 0 ; e^ρ × e^ρ = e^' × e^' = e^z × e^z = 0; e^ρ × e^' = e^z ; e^' × e^z = e^ρ and e^z × e^ρ = e^' ; Vector Calculus Partial derivatives The partial derivatives with respect to the coordinates are found using the chain rule2 @ρ = cos(')@x + sin(')@y ;@' = −ρ sin(')@x + ρ cos(')@y ; and @z = @z ; and sin(') cos(') @ = cos(')@ − @ ;@ = sin(')@ + @ ; and @ = @ : x ρ ρ ' y ρ ρ ' z z r-operator We now rewrite the r-operator by using these relations 0 1 0 1 @x @ρ e^' 1 r = @ = e^ @ + e^ @ + e^ @ = e^ @ + @ + e^ @ = @' @ y A x x y y z z ρ ρ ρ ' z z @ ρ A @ @ z c z p 1This might seem obvious but needs to be revisited in relativity and has far reaching consequences for the physical nature of space. 2 The calculation is identical to working out the basis vectors but replacing @r=@i ! @i. The partial derivatives can thus directly be read off from the relations between the basis vectors. Gradient The gradient of a scalar field U in cylindrical polar coordinates is now given by 0 @U 1 @ρ 1 @U e^' gradU = rU = @ ρ @' A = e^ρ@ρU + @'U + e^z@zU: @U ρ @z p The expression for the r-operator in cylindrical polar components is thus indirectly given on the data-sheet3. There you will find an expression for rU and the del-operator is found by simply leaving out U in this expression. Divergence When working out the divergence we need to properly take into account that the basis vectors are not constant in general curvilinear coordinates. For cylindrical polar coordinates we have two nonzero derivatives @'e^' = − cos(')e^x − sin(')e^y = −e^ρ and @'e^ρ = − sin(')e^x + cos(')e^y = e^' : P The divergence will thus in general not be given by r · F(r) = i @iFi(r) which is only true for an orthogonal coordinate system whose basis vectors are constant in space. Using the product rule we find e^ e^ r · F = [e^ @ + ' @ + e^ @ ] · [F e^ + F e^ + F e^ ] = @ F + ' · [@ (F e^ ) + @ (F e^ )] + @ F ρ ρ ρ ' z z ρ ρ ' ' z z ρ ρ ρ ' ρ ρ ' ' ' z z e^ @ (ρF ) @ F = @ F + ' · [e^ @ F + F e^ + e^ @ F − e^ F ] + @ F = ρ ρ + ' ' + @ F : ρ ρ ρ ρ ' ρ ρ ' ' ' ' ρ ' z z ρ ρ z z Rotation The calculation is similar to working out the divergence e^ r × F = [e^ @ + ' @ + e^ @ ] × [F e^ + F e^ + F e^ ] = e^ × [e^ @ F + e^ @ F ] + ρ ρ ρ ' z z ρ ρ ' ' z z ρ ' ρ ' z ρ z e^ ' × [e^ @ F − e^ F + e^ @ F ] + e^ × [e^ @ F + e^ @ F ] ρ ρ ' ρ ρ ' z ' z z ρ z ρ ' z ' @ F @ (ρF ) @ F = e^ ' z − @ F + e^ [−@ F + @ F ] + e^ ρ ' − ' ρ : ρ ρ z ' ' ρ z z ρ z ρ ρ Laplace operator We obtain the Laplace operator by replacing F ! r in the expression for the divergence 1 1 ∆ = r · r = @ (ρ∂ ) + @2 + @2 : ρ ρ ρ ρ2 ' z Example A velocity field describing a vortex is given by u = Ae^'/ρ with A a constant. Its divergence and rotation are given by e^ Ae^ e^ Ae^ A r · u = e^ @ + ' @ + e^ @ · ' = 0 ; r × u = e^ @ + ' @ + e^ @ × ' = (e^ − e^ ) = 0 : ρ ρ ρ ' z z ρ ρ ρ ρ ' z z ρ ρ2 z z The acceleration of a parcel of fluid is given by (this term appears in the Navier Stokes equation) A Ae^ A (u · r)u = @ ' = − e^ : ρ2 ' ρ ρ3 ρ Work out these expressions using cartesian components. 3Also spherical polar coordinates can be found on the data sheet. Summary Cylindrical polar coordinates (ρ, '; z) • Relation to cartesian coordinates (care is required to obtain ' in the correct quadrant) x = ρ cos(') ; y = ρ sin(') and z = z p ρ = x2 + y2 ;' = arctan(y=x) and z = z • Basis vectors e^ρ = @ρr, ρe^' = @'r, e^z = @zr e^ρ = cos(')e^x + sin(')e^y ; e^' = − sin(')e^x + cos(')e^y and e^z = e^z e^x = cos(')e^ρ − sin(')e^' ; e^y = sin(')e^ρ + cos(')e^' and e^z = e^z • Non-zero derivatives of basis vectors @'e^ρ = e^' ;@'e^' = −e^ρ • r-operator e^ r = e^ @ + ' @ + e^ @ ρ ρ ρ ' z z • Divergence @ (ρF ) @ F r · F = ρ ρ + ' ' + @ F ρ ρ z z • Rotation @ F @ (ρF ) @ F r × F(r) = e^ ' z − @ F + e^ [−@ F + @ F ] + e^ ρ ' − ' ρ ρ ρ z ' ' ρ z z ρ z ρ ρ • Laplace 1 1 ∆ = @ (ρ∂ ) + @2 + @2 ρ ρ ρ ρ2 ' z Spherical polar coordinates (r; θ; ') • Relation to cartesian and cylindrical coordinates (care is required to obtain ' in the correct quadrant) x = r cos(') sin(θ) ; y = r sin(') sin(θ) ; z = r cos(θ) and ρ = r sin(θ) p r = x2 + y2 + z2 ;' = arctan(y=x) and θ = arccos(z=r) = arctan(ρ/z) • Basis vectors e^r = @rr, re^θ = @θr, r sin(θ)e^' = @'r e^r = cos(θ)e^z + sin(θ)e^ρ ; e^θ = − sin(θ)e^z + cos(θ)e^ρ and e^' = e^' e^z = cos(θ)e^r − sin(θ)e^θ ; e^ρ = sin(θ)e^r + cos(θ)e^θ and e^' = e^' e^x = sin(θ) cos(')e^r + cos(θ) cos(')e^θ − sin(')e^' and e^y = sin(θ) sin(')e^r + cos(θ) sin(')e^θ + cos(')e^' • Non-zero derivatives of basis vectors @θe^r = e^θ ;@θe^θ = −e^r ;@'e^r = sin(θ)e^' ;@'e^θ = cos(θ)e^' and @'e^' = −e^ρ • r-operator e^ e^ r = e^ @ + θ @ + ' @ r r r θ r sin(θ) ' • Divergence @ (r2F ) @ (sin(θ)F ) @ F r · F = r r + θ θ + ' ' r2 r sin(θ) r sin(θ) • Rotation e^ e^ @ F e^ r × F = r [@ (F sin(θ)) − @ F ] + θ −@ (rF ) + ' r + ' [@ (rF ) − @ F ] r sin(θ) θ ' ' θ r r ' sin(θ) r r θ θ r • Laplace 1 2 1 1 2 ∆ = @r(r @r) + @θ(sin(θ)@θ) + @ r2 r2 sin(θ) r2 sin2(θ) '.