Appendix A. Nabla and Friends

Transcription

1 Appedix A Nabla ad Frieds A1 Notatio for Derivatives The partial derivative u u(x + he i ) u(x) (x) = lim x i h h of a scalar fuctio u : R R is writte i short otatio as Similarly we have for the higher derivatives xi u(x) = u x i (x) x 2 i u(x) = 2 u x 2 (x), xi xj u(x) = 2 u (x), i x i x j A vector α =(α 1,,α ) T of oegative itegers α i is called a multiidex of order For a give multiidex α we set α = α i For a give oegative iteger k α u(x) = α 1 x 1 α x u(x) = α u x α (x) 1 1 xα D k u(x) ={ α u(x) : α = k} deotes the ordered set of all partial derivatives of order k at the poit x Note that D k u(x) has k elemets, ie xi xj u(x) ad xj xi u(x) are differet elemets although they have the same value For the special cases k =1ad k =2we idetify D 1 u(x) with the gradiet u(x) ad D 2 u(x) with the Hessia matrix 2 u(x) (see below for the defiitio of gradiet ad Hessia) I the case of a fuctio u(x, y), u : R R R, we write D 1 xu or D 2 yu to idicate the variable with respect to which differetiatio is to be applied A2 Vector Differetial Calculus The whole presetatio treats the differetial operators oly i cartesia coordiates 47

2 Appedix A Nabla ad Frieds A21 Nabla Operator The abla operator formally is a row or colum vector of partial derivatives with respect to all variables of its argumet: =( 1,, ) T (A1) (whe we assume that the argumet has variables) A22 Gradiet Gradiet of a Scalar Nabla applied to a scalar fuctio u(x 1,,x ) i variables gives a vector called gradiet of the fuctio: u =( 1 u,, u) T (A2) We ca imagie to be a colum vector i this case applied to a scalar which gives a vector The gradiet of a scalar fuctio i poit x is a vector which is perpedicular to the level set l(c) ={y : u(y) =c} for c = u(x) poitig i the directio of the steepest icrease of the fuctio u Gradiet of a Vector-valued Fuctio Nabla applied to a vector-valued fuctio u(x) =(u 1 (x 1,,x ),,u m (x 1,,x )) T with m compoets i variables gives a matrix called the Jacobia of the fuctio: u = ( u 1 ) T ( u m ) T 1 u 1 u 1 1 u m u m or ( u) i,j = j u i (A3) If we wish to view the gradiet as a colum vector ad the fuctio u also as acolum vector (of possibly differet size) the we formally have: u := u T T (A4) Here u T as a outer product producig a matrix I the case of a scalar fuctio u the matrix u = 2 u is called the Hessia matrix A23 Divergece Divergece of a Vector Field The scalar product of abla with a vector-valued fuctio gives a scalar called the divergece of the fuctio: u = i u i 48

3 A2 Vector Differetial Calculus Divergece of a Matrix-valued Fuctio The divergece operator applied to a matrix-valued fuctio σ 1 σ 1,1 (x) σ 1, (x) σ(x 1,,x )= σ m σ m,1 (x) σ m, (x) i variables is defied to yield the divergece for each row of the matrix Note that σ eeds to have as may colums as there are variables It produces a vector-valued fuctio: σ = σ 1 σ m jσ 1,j jσ m,j or ( σ) i = j σ i,j (A5) If we regard the divergece as a row vector ad σ a m matrix with also the umber of variables, the we ca formally write σ := (σ T ) T (A6) Here the ier product (σ T ) produces a row vector Note the similarity to the formula (A4) A24 Curl The curl (also called rot, which is exactly the same thig) of a vector field is defied as u = 2u 3 3 u 2 3 u 1 1 u 3 1 u 2 2 u 1 (A7) which correspods to the vector (cross) product a b =(a 2 b 3 a 3 b 2,a 3 b 1 a 1 b 3,a 1 b 2 a 2 b 1 ) T As stated, it makes oly sese for u : R R 3 ad there is o obvious extesio of the curl operator to dimesios However, the related Stokes theorem (see below) ca be exteded to arbitrary dimesios A25 Covectio Term i Navier-Stokes Equatios For a vector-valued fuctio u, the covectio term i the Navier-Stokes equatios is writte as u u which is formally defied as u u =( u)u = u 1 u u u u i i u 1 u i i u (A8) Note that the scalar product of a vector with a matrix ( u is a matrix!) is defied as a vector where each compoet is the scalar multiplicatio of the vector with a row of the matrix 49

4 Appedix A Nabla ad Frieds A26 Laplacia The Laplacia takes secod order derivatives of a scalar fuc- u = u = i 2 u (A9) Laplacia of a scalar fuctio tio ad is defied as Laplace of Vector-valued fuctio The defiitio of the Laplacia is exteded to vectorvalued fuctios by applyig it to each compoet, ie the Laplacia of a vector-valued fuctio is agai a vector-valued fuctio I agreemet with the covetios above we have: u 1 u 1 u = u = (A10) u u A3 Vector Itegral Calculus A31 Matrix Product Let T,S be two m matrices, the we defie T : S = m T i,j S i,j (A11) Applied to two vector-valued fuctios u, v with m compoets i variables we have with the defiitios from above: m u : v = u i v i (A12) Now let T,S,Q be matrices The the followig holds: T :(QSQ T )=(Q T TQ):S (A13) This ca be show as follows: T :(QSQ T )= T i,j (e T i QSQ T e j ) = T i,j Q i,k S kl Q T l,j k=1 l=1 = S kl T i,j Q i,k Q T l,j k=1 l=1 = S kl Q T k,i k=1 l=1 =(Q T TQ) T i,j Q j,l 50

5 A3 Vector Itegral Calculus A32 Itegratio by Parts Gree s formula for sufficietly smooth scalar fuctios u, v ad a suitable bouded domai is ( i u)v = u i v + uv i (A14) where i is the i-th compoet of the outer uit ormal vector For a vector-valued fuctio u ad a scalar fuctio v we the have ( u)v = u v + u v (A15) For a matrix-valued fuctio T ad a vector valued fuctio v oe shows the correspodig formula ( T) v = T : v + (T ) v (A16) which is eeded i the variatioal formulatio of the Navier-Stokes equatios Ideed usig the defiitios above oe obtais: ( T) v = j T i,j v i = = T i,j j v i + T i,j v i j T i,j ( v) i,j + T i,j j v i 51