Euler equations for multiple integrals
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1 Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus Vector ifferentiation Matrix ifferentiation Multiimensional integration
2 This part eals with multivariable variational problems that escribe equilibria an ynamics of continua, optimization of shapes, etc. We consieration of a multivariable omain x R an coorinates x = (x 1,..., x ). The variational problem requires minimization of an integral over of a Lagrangian L(x, u, Du) where u = u(x) is a multivariable vector function an u of minimizers an matrix Du = u of graients of these minimizers. The optimality conitions for these problems inclue ifferentiation with respect to vectors an matrices. First, we recall several formulas of vector (multivariable) calculus which will be commonly use in the following chapters. 1 Reminer of multivariable calculus 1.1 Vector ifferentiation We remin the efinition of vector erivative or erivative of a scalar function with respect to a vector argument a R n. Definition 1.1 If φ(a) is a scalar function of a column vector argument a = (a 1,..., a n ) T, then the erivative φ a is a row vector ( φ φ a =,..., φ ) if a = a 1... (1) a 1 a n a n assuming that all partial erivatives exist. This efinition comes from consieration of the ifferential φ of φ(a): φ(a) = φ(a + a) φ(a) = φ(a) a a + o( a ) Inee, the left-han sie is a scalar an the secon multiplier in the right-han sie is a column vector, therefore the first multiplier is a row vector efine in (1) Examples of vector ifferentiation The next examples show the calculation of erivative for several often use functions. The results can be checke by straightforwar calculations. We assume here that a R n. 1. If φ(a) = a 2 = a a 2 n, the erivative is a a 2 = 2a T 2. The vector erivative of the eucliean norm a of a nonzero vector a is a row vector b, b = a 2 = at a = at a 2 a Observe that b is coirecte with a an has unit length. 2
3 3. The erivative of a scalar prouct c a, where c is an n-imensional vector, c R n, is equal to c: a ct a = c Similarly, if C is a k n matrix, erivative of a prouct Ca equals C T, Ca = CT a 4. Derivative of a quaratic form a T Ca where C is a symmetric matrix, equals a at Ca = 2a T C = 2(Ca) T. Directional erivative Let φ ν be a irectional erivative of a scalar function φ in a irection ν: φ ν = φ ν. Partial erivative of F ( φ) with respect to φ ν is efine as: = φ ν φ ν (2) Graient of a vector If u = (u 1,..., u n ) is a vector function, u j = u j (x 1,... x ), x R, then the graient of u is efine as a n matrix enote u or Du, u = j x i, or, in elements, 1 2 x 1 x 1... n x 1 u = Du = = ( u 1 u 2... u n ) (3) 1 2 x x... The columns of this matrix are graients of the components of the vector function u. 1.2 Matrix ifferentiation Similarly to the vector ifferentiation we efine matrix ifferentiation consiering a scalar function φ(a) of a matrix argument A. As in the vector case, the efinition is base on the notion of scalar prouct. Definition 1.2 The scalar prouct a.k.a. the convolution of the n m matrix A an m n matrix B is efine as following One can check the formula A : B = n n x i=1 j=1 m a ij b ji. A : B = Tr (A B) (4) 3
4 that brings the convolution into the family of familiar matrix operations. The convolution allows us to for calculate the increment of a matrix-ifferentiable function of a matrix argument cause by variation of this argument: φ(a) = φ(a + A) φ(a) = φ(a) A an to give the efinition of the matrix-erivative. : A + o( A ). Definition 1.3 The erivative of a scalar function φ by an n m matrix argument A is an m n matrix D = φ A with elements where a ij is the ij-element of A. D ij = φ a ji In element form, the efinition becomes φ φ φ A = a 11 a φ a 1n φ a 2n... φ a m1 φ a mn (5) Examples of matrix ifferentiation Next examples show the erivatives of several often use functions of matrix argument. 1. As the first example, consier φ(a) = Tr A = n i=1 a ii. Obviously, { φ 1 if i = j = a ij 0 if i j, therefore the erivative of the trace is the unit matrix, A Tr A = I. 2. Using efinition of the erivative, we easily compute the erivative of the scalar prouct or convolution of two matrices, (A : B) A = A Tr (A BT ) = B. 3. Assume that A is a square n n matrix. The erivative of the quaratic form x T Ax = n i,j=1 x ix j a ij with respect to matrix A is an n n ya matrix (x T Ax) = xx T A 4
5 4. Compute the erivative of the eterminant of matrix A for A R n n. Notice that the eterminant linearly epens on each matrix element, et A = a ij M ij + constant(a ij ) where M ij is the minor of the matrix A obtaine by eliminating the ith row an the jth column; it is inepenent of a ij. Therefore, et A a ij = M ij an the erivative of et A is the matrix M of minors of A, A et A = M = M M 1n M n1... M nn Recall that the inverse matrix A 1 can be conveniently expresse through these minors as A 1 = 1 et AM, an rewrite the result as et A = (et A)A 1 A We can rewrite the result once more using the logarithmic erivative x log f(x) = f (x) f(x). The erivative becomes more symmetric, A (log et A) = A 1. Remark 1.1 If A is symmetric an positively efine, we can bring the result to a perfectly symmetric form (log et A) = I log A Inee, we introuce the matrix logarithmic erivative similarly to the logarithmic erivative of a real positive argument, f log x = x f x, which reas f log A = A f A. Here, log A is the matrix that has the same eigenvectors as A an the eigenvalues equal to logarithms of the corresponing eigenvalues of A. Notice that log et A is the sum of logarithms of the eigenvalues of A, log et A = Tr log A. Notice also that when the matrix A is symmetric an positively efine which means that the eigenvalues of A are real an positive, the logarithms of the eigenvalues are real. 5
6 5. Using the chain rule, we compute the erivative of the trace of the inverse matrix: A Tr A 1 = A Similarly, we compute the erivative of the quaratic form associate with the inverse matrix: A xt A 1 x = xa 2 x T. Remark 1.2 (About the notations) The historical Leibnitz notation g = z for partial erivative is not the most convenient one an can even be ambiguous. Inee, the often use in one-variable variational problems partial becomes in multivariable problem the partial of the partials x. Since there is no conventional analog for the symbol in partial erivatives, we nee a convenient way to express the fact that the argument z of ifferentiation can itself be a partial erivative like z = 1 x 2. If we were substitute this expression for z into z, we woul arrive at an a bit awkwar expression g = 1 x 2 (still use in Gelfan & Fomin) which replaces the expression use in onevariable variational problem. There are several ways to fix the inconvenience. To keep analogy with the onevariable case, we use the vector of partials ( u) in the place of. If neee, we specify a component of this vector, as follows [ ] g = ( u 1 ) Alternatively, we coul rename the partial erivatives of u with a single inexe array D ij arriving at the formula of the type g = or use comma to show the erivative g = D 12, where D 12 = 1 x 2. 1,2, where u 1,2 = 1 x 2. The most raical an logical solution (which we o not are to evelop in the textbook) replaces Leibnitz notation with something more convenient, namely with Newton-like or Maple-like notation g = D(f, D(u 1, x 2 )) 2 6
7 Remark 1.3 (Ambiguity in notations) A more serious issue is the possible ambiguity of partial erivative with respect to one of inepenent coorinates. The partial x means the erivative upon the explicitly given argument x of a function of the type F (x, u). If the argument x is one of the inepenent coorinates, an if u is a function of these coorinates, in particular of x (as it is common in calculus of variations problems), the same partial coul mean x + x. To fix this, we nee to specify whether we consier u as a function of x, u = u(x), or as an inepenent argument, which coul make the notations awkwar. For this reason, we always assign the symbol x for a vector of inepenent variables (coorinates). When ifferentiation with respect to inepenent coorinates is consiere, we use the graient notations as u. Namely, the vector is introuce F (u, x) = x 1.. x + where x k always means the erivative upon explicit variable x. The partials correspons to components of this vector. If necessary, we specify the argument of the graient, as follows ξ. 1.3 Multiimensional integration Change of variables Consier the integral I = f(x) x an assume that x = x(ξ), or in coorinates x 1.. x x i = x i (ξ 1,..., ξ ), i = 1,..., In the new coorinates, the omain is mappe into the omain ξ an the volume element x becomes et J ξ where et J is the Jacobian an J is the matrix graient J = ξ x = {J ij }, J ij = x i ξ j, i, j = 1,...,. The integral I becomes I = f(x(ξ))(et ξ x) x ξ (6) The change of variables in the multivariable integrals is analogous to the oneimensional case. 7
8 Green s formula The Green s formula is a multivariable analog of the Leibnitz formula a.k.a. the funamental theorem of calculus. For a ifferentiable in the omain vector-function a(x), it has the form a x = a ν s (7) Here, ν is the outer normal to an is the ivergence operator, a = a 1 x , + a n x n. If = 1, omain becomes an interval [c, ], the normal show the irection along this interval, an (7) becomes c a x = a() a(c). x Integration by parts We will use multivariable analogs of the integration by parts. Suppose that b(x) is a scalar ifferentiable function in an a(x) is a vector ifferentiable fiel in. Then the following generalization of integration by parts hols (a b) x = (b a) x + (a ν)b s (8) The formula follows from the ifferential ientity (ifferentiation of a prouct) an Green s formula a b + b a = (ba) ( c)x = (c ν) s A similar formula hols for two ifferentiable in vector fiels a an b: (a b) x = (b a) x (a b ν) s (9) It immeiately follows from the Green s formula an the ientity (a b) = b a a b 8
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