Mathematical tools M3. Tensor fields in curvilinear coordinate systems AleˇsJanka office Math 0.107 [email protected] http://perso.unifr.ch/ales.janka/mechanics November 24, 2010, Universit´ede Fribourg Mathematical tools M3. Tensor fields in curvilinear coordinates 1. Curvilinear coordinate system Position vector of point M (with respect to the origin): −−→OM = x Let x : R3 R3 be a smooth bijective mapping: → x :(ξ1, ξ2, ξ3)T x(ξ1, ξ2, ξ3) 7→ Curvilinear coordinates of a point: ξ1, ξ2, ξ3. Coordinate curves through a point: parametric curves given by g 2 x( 1, , 3 ) 2 3 ξ β ξ x1(α): α x(α, ξ , ξ ) 7→ 1 3 M g x2(β): β x(ξ , β, ξ ) x 1 7→ 1 2 x3(γ): γ x(ξ , ξ , γ) 7→ O x( α , ξ 2 , ξ 3 ) 1 2 3 1 2 3 all 3 curves pass through x(ξ , ξ , ξ ) (=point M with ξ , ξ , ξ fixed). Mathematical tools M3. Tensor fields in curvilinear coordinates 1. Curvilinear coordinate system: local basis Local basis: composed of tangents to the coordinate curves: ∂x g = i ∂ξi 3 We suppose here that g1, g2 and g3 are linearly independent in R . Covariant local basis: differential of the position vector x: ∂x dx = dξi = g dξi ∂ξi i “How much x(ξ1, ξ2, ξ3) changes if we perturb ξi by dξi .” Contravariant basis is induced as before so that g gj = δj i · i Metric tensor: g = g g ... (as before) ij i · j Huge difference with what we have seen so far: 1 2 3 gi = gi (x(ξ , ξ , ξ )) ie. the basis is not constant for all points x R3! ∈ Mathematical tools M3. Tensor fields in curvilinear coordinates 1.1 Curvilinear coordinate system: infinitensimal volume Infinitensimal volume due to coordinate change: +d1 1 3 dV = (g g ) g dξ1 dξ2 dξ3 x( ξ ξ , β , ξ ) | 1 × 2 · 3| g2 1 3 √g x( ξ , β , ξ ) | {z } g T 1 with g = det[gij ] = det(F F) = Ωx 2 det (F), where F = g g g . +d2 2 3 1| 2| 3 x( α , ξ ξ , ξ ) i O 2 In cartesian coordinates: dx = dx ei x( α , ξ , ξ 3 ) and dV = dx1 dx2 dx3. Volume integral of a scalar field f (x) in curvilinear coords: f (x) dx1 dx2 dx3 = f (x(ξ1, ξ2, ξ3)) √g dξ1 dξ2 dξ3 dV ΩZx ΩZξ dV | {z } | {z } with Ωx = x(Ωξ). Mathematical tools M3. Tensor fields in curvilinear coordinates 2. Differential (resp. gradient) of a scalar field Scalar field: f : x Ω R3 R ∈ ⊂ 7→ Gradient of f (x) (with respect to the position x): it is a vector f R3 such that for all dx R3: ∇ ∈ ∈ f (x + dx) = f (x) + f (x) dx +o(dx) ∇ · df here, df is the differential of f (x) along| {zdx.} Gradient and the directional derivative of f (x) The gradient of f (x) is such a vector f R3 for which ∇ ∈ d 3 [ f (x)] d = f (x + αd) d R . ∇ · dα ∀ ∈ α=0 The definition is independent of the choice of basis g1, g2, g3 Hence, f (x) is a field of tensors of order N = 1 (vector field). ∇ Mathematical tools M3. Tensor fields in curvilinear coordinates 2.1 Coordinates of f in the local basis ∇ f (x+dx) = f (x) + f (x) dx + o(dx) ... definition of gradient ∇ · ∂f = f (x) + dξi + o(dξk ) ... f (x) as a function of ξi ∂ξi Xk ∂f = f (x) + δi dξj + o(dξk ) ∂ξi j Xk ∂f = f (x) + gi g dξj +o(dx) ... cf. the first line ∂ξi · j dx f (x) ∇ | {z } ∂f | {z } Hence, f = gi Covariant components of f (x) are: ∇ dξi ⇒ ∇ ∂f ( f ) = ... They coincide with ∂f ! ∇ i ∂ξi ∂ξi f =( f ) = ∂f is named “the covariant derivative of scalar field f ” ∇i ∇ i ∂ξi The differential df then expressed “in coordinates”: df = f dξi ∇i Mathematical tools M3. Tensor fields in curvilinear coordinates 3. Differential (resp. gradient) of a vector field Vector field u(x) : (e.g. velocity, displacements, el. current, . ) field of tensors of order N = 1: 3 3 u : x Ω R R ∈ ⊂ → Components of u in the local basis g : h i i i i u(x) = u gi . both u and gi depend on x! Differential du: change in u going from x to x+dx (up to o(dx)): ∂u u(x+dx) u(x) + du = u(x) + dξj = u(x) + u dx ≈ ∂ξj ∇ · i i i From u = u gi , by chain rule (both u and gi depend on x, ie. ξ !): ∂ui ∂g du = dξj g + ui i dξk (1) ∂ξj i ∂ξk Change due to changing local coordinates ui of u Change due to the curvature of the coordinate system Mathematical tools M3. Tensor fields in curvilinear coordinates 3.1 Differential of a vector field: contravariant components Contravariant components of du: du` = du g`, du = du` g · ` with respect to the local basis g at the point x (not at x+dx!). h i i ∂ui ∂g du = dξj g + ui i dξk g` ∂ξj i ∂ξk · Hence, the contravariant components of du: ∂ui ∂g du` = du g` = dξj g g` +ui i g` dξk · ∂ξj i · ∂ξk · ` δi ∂u` ∂g | {z } ∂u` = dξj + ui i g` dξj = + Γ` ui dξj ∂ξj ∂ξj · ∂ξj ij ` Γij where we define | {z } ∂g Γ` = i g`, ij ∂ξj · the Christoffel symbols of the second kind (not a tensor!). Mathematical tools M3. Tensor fields in curvilinear coordinates 3.2 Differential of a vector field: covariant components i Analogously to (1), from u = ui g , by chain rule: ∂u ∂gi du = i dξj gi + u dξk ∂ξj i ∂ξk Change due to changing local coordinates ui of u Change due to the curvature of the coordinate system Aside differentiation to get rid of the contravariant basis gi (which is less used): ∂ gi g = δi · ` ` ∂ξj ∂gi ∂g g + gi ` = 0 ∂ξj · ` · ∂ξj ∂gi ∂g Hence, g = gi ` ∂ξj · ` − · ∂ξj Mathematical tools M3. Tensor fields in curvilinear coordinates 3.2 Differential of a vector field: covariant components Covariant components of du: du = du g , du = du g`: ` · ` ` ∂u ∂gi du = i dξj gi + u dξk g ∂ξj i ∂ξk · ` Hence, by applying the aside differentiation: ∂u ∂gi du = du g = i dξj gi g +u g dξk ` · ` ∂ξj · ` i ∂ξk · ` i δ` ∂u` ∂g` | {z } ∂u` = dξj u gi dξk = u Γi dξj ∂ξj − i ∂ξk · ∂ξj − i `j i Γ`k | {z } Mathematical tools M3. Tensor fields in curvilinear coordinates 3.3 Differential of a vector field and covariant derivatives Perturbations of the position x(ξ1, ξ2, ξ3) by dξ1, dξ2, dξ3 du: −→ ∂u dx = dξi g , du = dξj = u dx = du` g = du g` i ∂ξj ∇ · ` ` Contravariant and covariant coordinates of the differential du: ∂u` du` = + Γ` ui dξj ∂ξj ij u` ∇j | {z } ∂u du = ` Γi u dξj ` ∂ξj − `j i u ∇j ` | {z } where u` is the covariant derivative of contravariant tensor ∇j and u is the covariant derivative of covariant tensor ∇j ` Mathematical tools M3. Tensor fields in curvilinear coordinates 3.4 Covariant derivatives and gradient of a vector field Differential du using covariant derivatives resp. du = u dx: ∇ · ∂u` du = du` g = + Γ` ui g dξj ` ∂ξj ij ` u` ∇j = u` g δj dξk = u` g gj g dξk ∇j ` |k {z ∇j } ` ⊗ · k gj g u dx · k ∇ Hence, the gradient u|{z}is a 2nd order| tensor{z with:} | {z } ∇ ∂u` u = + Γ` ui g gj = u` g gj ∇ ∂ξj ij ` ⊗ ∇j ` ⊗ the covariant derivative u` is in fact the tensor u in ⇒ ∇j ∇ mixed components u ` !! ∇ , j Similarly, u are the 2 covariant components u of u: ⇒ ∇j ` × ∇ `,j ∇ ∂u` i ` j ` j u = Γ ui g g = j u` g g ∇ ∂ξj − `j ⊗ ∇ ⊗ h i h i Mathematical tools M3. Tensor fields in curvilinear coordinates 4. Covariant derivatives of higher order tensor T , N 2 ≥ Remember: covariant deriv. of scalar field = its partial deriv.: ∂f f = ∇k ∂ξk We can exploit it: Multiply T by N arbitrary vector-fields a, b, . to form a scalar f . Example for 2nd-order tensor T : ij f = T ai bj Apply the “covariant = partial” trick on f : ∂ T ij a b = T ij a b ∇k i j ∂ξk i j ∂T ij ∂a ∂b = a b + T ij i b + T ij a j ∂ξk i j ∂ξk j i ∂ξk For the arbitrary vector fields a, b, . we know how to make a covariant derivative: ∂a ∂a a = ` Γi a i.e. i = a + Γm a ∇j ` ∂ξj − `j i ∂ξk ∇k i ik m Mathematical tools M3. Tensor fields in curvilinear coordinates 4. Covariant derivatives of higher order tensor T , N 2 ≥ In (T ij a b ), replace ∂ of a, b,. by terms containing : ∇k i j ∂ξk ∇k ∂T ij ∂a ∂b T ij a b = a b + T ij i b + T ij a j ∇k i j ∂ξk i j ∂ξk j i ∂ξk ∂T ij = a b + T ij ( a + Γm a ) b + ∂ξk i j ∇k i ik m j +T ij a b + Γm b i ∇k j jk m ∂T ij = + Γi T `j + Γj T i` a b + ∂ξk `k `k i j +T ij a b + T ij a b ∇k i j i ∇k j Here, we re-indexed conveniently dummy indices in order to regroup terms with “ai bj ”. By analogy with (a b c) = a bc + ab c + abc , the term in brackets is T ij : · · 0 0 0 0 ∇k ∂T ij T ij = + Γi T `j + Γj T i` ∇k ∂ξk `k `k Mathematical tools M3.