Misplaced Pages

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34-488: DDX may refer to: d d x {\displaystyle {\mathrm {d} \over \mathrm {d} x}} , a common notation for the differential operator with respect to a variable x DD(X) , former program name of a class of U.S. Navy destroyers Differential diagnosis (DDx or D/Dx), a systematic method used to identify unknowns Device Dependent X , graphics device drivers supporting 2D acceleration in

68-537: A {\displaystyle x\to a} and x → b {\displaystyle x\to b} , one can also define the adjoint of T by T ∗ u = ∑ k = 0 n ( − 1 ) k D k [ a k ( x ) ¯ u ] . {\displaystyle T^{*}u=\sum _{k=0}^{n}(-1)^{k}D^{k}\left[{\overline {a_{k}(x)}}u\right].} This formula does not explicitly depend on

102-1187: A α : C n → C {\displaystyle f,g,a_{\alpha }\colon \mathbb {C} ^{n}\to \mathbb {C} } (or R n → R {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} } ). If α , β ∈ N 0 n {\displaystyle \alpha ,\beta \in \mathbb {N} _{0}^{n}} are multi-indices and x = ( x 1 , … , x n ) {\displaystyle x=(x_{1},\ldots ,x_{n})} , then ∂ α x β = { β ! ( β − α ) ! x β − α if   α ≤ β , 0 otherwise. {\displaystyle \partial ^{\alpha }x^{\beta }={\begin{cases}{\frac {\beta !}{(\beta -\alpha )!}}x^{\beta -\alpha }&{\text{if}}~\alpha \leq \beta ,\\0&{\text{otherwise.}}\end{cases}}} The proof follows from

136-731: A α ( x ) {\displaystyle a_{\alpha }(x)} is a function on some open domain in n -dimensional space. The operator D α {\displaystyle D^{\alpha }} is interpreted as D α = ∂ | α | ∂ x 1 α 1 ∂ x 2 α 2 ⋯ ∂ x n α n {\displaystyle D^{\alpha }={\frac {\partial ^{|\alpha |}}{\partial x_{1}^{\alpha _{1}}\partial x_{2}^{\alpha _{2}}\cdots \partial x_{n}^{\alpha _{n}}}}} Thus for

170-752: A α ( x ) D α   , {\displaystyle P=\sum _{|\alpha |\leq m}a_{\alpha }(x)D^{\alpha }\ ,} where α = ( α 1 , α 2 , ⋯ , α n ) {\displaystyle \alpha =(\alpha _{1},\alpha _{2},\cdots ,\alpha _{n})} is a multi-index of non-negative integers , | α | = α 1 + α 2 + ⋯ + α n {\displaystyle |\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}} , and for each α {\displaystyle \alpha } ,

204-461: A k ( x ) D k u {\displaystyle Tu=\sum _{k=0}^{n}a_{k}(x)D^{k}u} the adjoint of this operator is defined as the operator T ∗ {\displaystyle T^{*}} such that ⟨ T u , v ⟩ = ⟨ u , T ∗ v ⟩ {\displaystyle \langle Tu,v\rangle =\langle u,T^{*}v\rangle } where

238-804: A function f ∈ F 1 {\displaystyle f\in {\mathcal {F}}_{1}} : P f = ∑ | α | ≤ m a α ( x ) ∂ | α | f ∂ x 1 α 1 ∂ x 2 α 2 ⋯ ∂ x n α n {\displaystyle Pf=\sum _{|\alpha |\leq m}a_{\alpha }(x){\frac {\partial ^{|\alpha |}f}{\partial x_{1}^{\alpha _{1}}\partial x_{2}^{\alpha _{2}}\cdots \partial x_{n}^{\alpha _{n}}}}} The notation D α {\displaystyle D^{\alpha }}

272-426: A function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science ). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as

306-408: Is a bundle map , symmetric on the indices α. The k order coefficients of P transform as a symmetric tensor whose domain is the tensor product of the k symmetric power of the cotangent bundle of X with E , and whose codomain is F . This symmetric tensor is known as the principal symbol (or just the symbol ) of P . The coordinate system x permits a local trivialization of

340-450: Is attributed to Louis François Antoine Arbogast in 1800. The most common differential operator is the action of taking the derivative . Common notations for taking the first derivative with respect to a variable x include: When taking higher, n th order derivatives, the operator may be written: The derivative of a function f of an argument x is sometimes given as either of the following: The D notation's use and creation

374-453: Is credited to Oliver Heaviside , who considered differential operators of the form in his study of differential equations . One of the most frequently seen differential operators is the Laplacian operator , defined by Another differential operator is the Θ operator, or theta operator , defined by This is sometimes also called the homogeneity operator , because its eigenfunctions are

SECTION 10

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408-444: Is defined by ⟨ f , g ⟩ = ∫ a b f ( x ) ¯ g ( x ) d x , {\displaystyle \langle f,g\rangle =\int _{a}^{b}{\overline {f(x)}}\,g(x)\,dx,} where the line over f ( x ) denotes the complex conjugate of f ( x ). If one moreover adds the condition that f or g vanishes as x →

442-429: Is justified (i.e., independent of order of differentiation) because of the symmetry of second derivatives . The polynomial p obtained by replacing partials ∂ ∂ x i {\displaystyle {\frac {\partial }{\partial x_{i}}}} by variables ξ i {\displaystyle \xi _{i}} in P is called the total symbol of P ; i.e.,

476-755: Is not the case, i.e., if α ≤ β {\textstyle \alpha \leq \beta } as multi-indices, then d α i d x i α i x i β i = β i ! ( β i − α i ) ! x i β i − α i {\displaystyle {\frac {d^{\alpha _{i}}}{dx_{i}^{\alpha _{i}}}}x_{i}^{\beta _{i}}={\frac {\beta _{i}!}{(\beta _{i}-\alpha _{i})!}}x_{i}^{\beta _{i}-\alpha _{i}}} for each i {\displaystyle i} and

510-479: The Fourier transform as follows. Let ƒ be a Schwartz function . Then by the inverse Fourier transform, This exhibits P as a Fourier multiplier . A more general class of functions p ( x ,ξ) which satisfy at most polynomial growth conditions in ξ under which this integral is well-behaved comprises the pseudo-differential operators . The conceptual step of writing a differential operator as something free-standing

544-663: The Schwarzian derivative . Given a nonnegative integer m , an order- m {\displaystyle m} linear differential operator is a map P {\displaystyle P} from a function space F 1 {\displaystyle {\mathcal {F}}_{1}} on R n {\displaystyle \mathbb {R} ^{n}} to another function space F 2 {\displaystyle {\mathcal {F}}_{2}} that can be written as: P = ∑ | α | ≤ m

578-416: The eigenspaces of Θ are the spaces of homogeneous functions . ( Euler's homogeneous function theorem ) In writing, following common mathematical convention, the argument of a differential operator is usually placed on the right side of the operator itself. Sometimes an alternative notation is used: The result of applying the operator to the function on the left side of the operator and on the right side of

612-556: The monomials in z : Θ ( z k ) = k z k , k = 0 , 1 , 2 , … {\displaystyle \Theta (z^{k})=kz^{k},\quad k=0,1,2,\dots } In n variables the homogeneity operator is given by Θ = ∑ k = 1 n x k ∂ ∂ x k . {\displaystyle \Theta =\sum _{k=1}^{n}x_{k}{\frac {\partial }{\partial x_{k}}}.} As in one variable,

646-2075: The power rule for the ordinary derivative ; if α and β are in { 0 , 1 , 2 , … } {\textstyle \{0,1,2,\ldots \}} , then Suppose α = ( α 1 , … , α n ) {\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})} , β = ( β 1 , … , β n ) {\displaystyle \beta =(\beta _{1},\ldots ,\beta _{n})} , and x = ( x 1 , … , x n ) {\displaystyle x=(x_{1},\ldots ,x_{n})} . Then we have that ∂ α x β = ∂ | α | ∂ x 1 α 1 ⋯ ∂ x n α n x 1 β 1 ⋯ x n β n = ∂ α 1 ∂ x 1 α 1 x 1 β 1 ⋯ ∂ α n ∂ x n α n x n β n . {\displaystyle {\begin{aligned}\partial ^{\alpha }x^{\beta }&={\frac {\partial ^{\vert \alpha \vert }}{\partial x_{1}^{\alpha _{1}}\cdots \partial x_{n}^{\alpha _{n}}}}x_{1}^{\beta _{1}}\cdots x_{n}^{\beta _{n}}\\&={\frac {\partial ^{\alpha _{1}}}{\partial x_{1}^{\alpha _{1}}}}x_{1}^{\beta _{1}}\cdots {\frac {\partial ^{\alpha _{n}}}{\partial x_{n}^{\alpha _{n}}}}x_{n}^{\beta _{n}}.\end{aligned}}} For each i {\textstyle i} in { 1 , … , n } {\textstyle \{1,\ldots ,n\}} ,

680-419: The X.Org Server DDX (chemistry) , a collective name for DDT and its breakdown products DDE and DDD See also [ edit ] 3DDX , a rhythm video game DXD (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title DDX . If an internal link led you here, you may wish to change the link to point directly to

714-420: The adjoint of P is defined in L (Ω) by duality in the analogous manner: for all smooth L functions f , g . Since smooth functions are dense in L , this defines the adjoint on a dense subset of L : P is a densely defined operator . The Sturm–Liouville operator is a well-known example of a formal self-adjoint operator. This second-order linear differential operator L can be written in

SECTION 20

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748-405: The cotangent bundle by the coordinate differentials d x , which determine fiber coordinates ξ i . In terms of a basis of frames e μ , f ν of E and F , respectively, the differential operator P decomposes into components on each section u of E . Here P νμ is the scalar differential operator defined by With this trivialization, the principal symbol can now be written In

782-556: The cotangent space over a fixed point x of X , the symbol σ P {\displaystyle \sigma _{P}} defines a homogeneous polynomial of degree k in T x ∗ X {\displaystyle T_{x}^{*}X} with values in Hom ⁡ ( E x , F x ) {\displaystyle \operatorname {Hom} (E_{x},F_{x})} . A differential operator P and its symbol appear naturally in connection with

816-431: The definition of the scalar product. It is therefore sometimes chosen as a definition of the adjoint operator. When T ∗ {\displaystyle T^{*}} is defined according to this formula, it is called the formal adjoint of T . A (formally) self-adjoint operator is an operator equal to its own (formal) adjoint. If Ω is a domain in R , and P a differential operator on Ω, then

850-537: The extension of many formulae from elementary calculus to the corresponding multi-variable case. Below are some examples. In all the following, x , y , h ∈ C n {\displaystyle x,y,h\in \mathbb {C} ^{n}} (or R n {\displaystyle \mathbb {R} ^{n}} ), α , ν ∈ N 0 n {\displaystyle \alpha ,\nu \in \mathbb {N} _{0}^{n}} , and f , g ,

884-455: The form This property can be proven using the formal adjoint definition above. This operator is central to Sturm–Liouville theory where the eigenfunctions (analogues to eigenvectors ) of this operator are considered. Differentiation is linear , i.e. where f and g are functions, and a is a constant. Any polynomial in D with function coefficients is also a differential operator. We may also compose differential operators by

918-984: The function x i β i {\displaystyle x_{i}^{\beta _{i}}} only depends on x i {\displaystyle x_{i}} . In the above, each partial differentiation ∂ / ∂ x i {\displaystyle \partial /\partial x_{i}} therefore reduces to the corresponding ordinary differentiation d / d x i {\displaystyle d/dx_{i}} . Hence, from equation ( 1 ), it follows that ∂ α x β {\displaystyle \partial ^{\alpha }x^{\beta }} vanishes if α i > β i {\textstyle \alpha _{i}>\beta _{i}} for at least one i {\textstyle i} in { 1 , … , n } {\textstyle \{1,\ldots ,n\}} . If this

952-408: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=DDX&oldid=1029375502 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Differential operator In mathematics , a differential operator is an operator defined as

986-401: The notation ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is used for the scalar product or inner product . This definition therefore depends on the definition of the scalar product (or inner product). In the functional space of square-integrable functions on a real interval ( a , b ) , the scalar product

1020-415: The operator, and the difference obtained when applying the differential operator to the functions on both sides, are denoted by arrows as follows: Such a bidirectional-arrow notation is frequently used for describing the probability current of quantum mechanics. Given a linear differential operator T {\displaystyle T} T u = ∑ k = 0 n

1054-606: The rule Some care is then required: firstly any function coefficients in the operator D 2 must be differentiable as many times as the application of D 1 requires. To get a ring of such operators we must assume derivatives of all orders of the coefficients used. Secondly, this ring will not be commutative : an operator gD isn't the same in general as Dg . For example we have the relation basic in quantum mechanics : The subring of operators that are polynomials in D with constant coefficients is, by contrast, commutative. It can be characterised another way: it consists of

DDX - Misplaced Pages Continue

1088-599: The symbol, namely, is called the principal symbol of P . While the total symbol is not intrinsically defined, the principal symbol is intrinsically defined (i.e., it is a function on the cotangent bundle). More generally, let E and F be vector bundles over a manifold X . Then the linear operator is a differential operator of order k {\displaystyle k} if, in local coordinates on X , we have where, for each multi-index α, P α ( x ) : E → F {\displaystyle P^{\alpha }(x):E\to F}

1122-640: The total symbol of P above is: p ( x , ξ ) = ∑ | α | ≤ m a α ( x ) ξ α {\displaystyle p(x,\xi )=\sum _{|\alpha |\leq m}a_{\alpha }(x)\xi ^{\alpha }} where ξ α = ξ 1 α 1 ⋯ ξ n α n . {\displaystyle \xi ^{\alpha }=\xi _{1}^{\alpha _{1}}\cdots \xi _{n}^{\alpha _{n}}.} The highest homogeneous component of

1156-841: The translation-invariant operators. Multi-index An n -dimensional multi-index is an n {\textstyle n} - tuple of non-negative integers (i.e. an element of the n {\textstyle n} - dimensional set of natural numbers , denoted N 0 n {\displaystyle \mathbb {N} _{0}^{n}} ). For multi-indices α , β ∈ N 0 n {\displaystyle \alpha ,\beta \in \mathbb {N} _{0}^{n}} and x = ( x 1 , x 2 , … , x n ) ∈ R n {\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})\in \mathbb {R} ^{n}} , one defines: The multi-index notation allows

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