Double dot product mathematica The Wikipedia page Isai linked to basically says it all, but I think it is worth unpacking some of the definitions given there here with a bit more motivation. So my question is: How do we expand (using tensor properties) a double dot product of the basis vectors to a simpler one? [tex](e_ie_je_ke_l):(e_me_n The dot product of two vectors a and b can be interpreted as the product of two lengths: the length of a orthogonally projected onto b, and the length of b itself. Dot を2個のテンソル および に適用した結果は，やはりテンソルで となる． Dot を 階のテンソルと 階のテンソルに適用すると， 階のテンソルが与えられる． »; Dot は， SparseArray オブジェクトおよび構造化配列オブジェクトに使うこ The tensor product is another way to multiply vectors, in addition to the dot and cross products. A double-dot product between two tensors becomes a single-dot product in the flattened matrix representation, i. Unicode has a code point from 2200 to 22FF for mathematical operators. Products. The dot product of two real vectors is the sum of the componentwise products of the vectors. I want to multiply them with Matlab and I know in Matlab it becomes: A : B = trace (A*B) but Dot[A,B] (*which makes no sense*) A*B (*makes absolutely no sense, but I am desperate at this point*) Tr[A,Transpose[B]] (*But I think this only works for rank 2 tensors*) Edit: Here is a bit more context. Thus Mathematica does the least surprising thing, which is to assume Dot[a,b]==Dot[b,a], and not Dot[a,b]==Conjugate[Dot[b,a]]. 9, p. VectorAngle — angle between two vectors. ; Product [f, {i, i min, i max}] can be entered as f. In addition, there are also many other mathematical symbols part of Unicode system like integrals, Ok I have seen the tensor double dot scalar product defined two ways and it all boils down to how the multiplication is defined. System Modeler; gives the dot product of the two 3-vectors v 1, v 2 in the default coordinate Inner[f, list1, list2, g] is a generalization of Dot in which f plays the role of multiplication and g of addition. I know when multiplying two tensor with double dot product (:) that means inner product, the order of result will be decrease two times. 5: The Dot and Cross Mathematica computes the dot product operation between two vectors when we place a period in between them: {a, b}. the dot product of the 1. Learning Objectives. Is there anyway to get mathematica, e. B will also grow larger as Dot — scalar dot product. In Euclidean geometry, the dot product The dot operator symbol is used in math to represent multiplication and, in the context of linear algebra, as the dot product operator. It follows immediately that X·Y=0 if X is perpendicular to Y. " The dot product (or inner product) of a tensor T and a vector a produces a vector b = T . 4k bronze badges. It is to automatically sum any index appearing twice from 1 to 3. The result of the tensor product of aand bis not a scalar, like the dot product, nor a (pseudo)-vector like the cross-product. In spite of its name, Mathematica does not use a dot (. The asterisk command can be applied only when two matrices have the same dimensions; in this case the output is the matrix containing corresponding products of corresponding entry. Simplifying symbolic expressions involving dot products. 274k 34 34 gold badges 600 600 silver badges 1. and you seem to want to actually make the matrices: that is implemented as KroneckerProduct in Mathematica. 1. Dot product for lists. The double dot product is an important concept of mathematical algebra. I'm trying to find the double dot product of the projection tensor P and a matrix which are denoted by the following: I = Array[KroneckerDelta, {3,3}]; J = The "double inner product" and "double dot product" are referring to the same thing- a double contraction over the last two indices of the first tensor and the first two indices of the what is for the case: Double contraction of two 4th tensors? For arbitrary rank tensors with any number of contractions between them, you can use Flatten and then Dot to Unicode: 00A8. In all the textbooks (for example New's Introduction to Nonlinear Optics, Eq. e. Cross [v 1, v 2, ] gives the dual (Hodge star) of the wedge product of the v Usually operator has name in continuum mechacnis like 'dot product', 'double dot product' and so on. The double dot product produces a scalar, the double cross product, a dyadic, and the mixed products, a vector. Making statements based on opinion; back them up with references or personal experience. In this post, I will show that this choice has some important implications. $$\partial_{x_\mu} x \cdot x = 2 x_\mu \tag{1}$$ so the result that I would like to have is actually 2 x These express~ns~a~b~g~eralized. All of the entries in these tensors are “machine precision”, which roughly translates to a C++ double. Find the dot product of A and B, treating the rows as vectors. The dot product operation can be performed in one of two ways. UnitVector — unit vector along a coordinate direction. The Wolfram Language uses state-of-the-art algorithms to work with both dense and sparse matrices, and incorporates a number of powerful original algorithms, especially for high-precision and symbolic matrices. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics. The tensor product of vectors aand bis denoted a bin mathematics but simply abwith no special product symbol in mechanics. In general, Cross [v 1, v 2, , v n-1] is a totally antisymmetric product which takes vectors of length n and yields a vector of length n that is orthogonal to all of the v i. dot product, to do a one by one multiplication or multiplicative mapping. Dimensions to contract in A and B, specified as vectors. Because the product is generally denoted with a dot between the vectors, it is also called the dot product. OverDot [ expr , 2 ] can be entered using and \[DoubleDot] , while OverDot [ expr , 3 ] can be entered using and \[TripleDot] . M1*M2*M3*M4*M5**Mn. Hand calculation of dot products involves only simple multiplication and addition. =d^2x/dt^2. Examples A is second order tensor and B is fourth order tensor. A pair of overdots placed over a symbol, as in x^. Wolfram Notebook Assistant + LLM Kit. System Modeler; Wolfram Player; So I came across this expression: $$ \nabla\mathbf{A}:\nabla\mathbf{A}=\partial_iA_j\partial_iA_j. g. I would like to create a function to matrix multiply (dot product) n matrices together, where n is large, for example . ) to represent this function. For example, let $\vec{u} = [u_1,u_2], \vec{v}=[v_1,v_2]$, then $ \nabla\vec{u}:\nabla\vec{v}=\nabla u_x $\begingroup$ Thanks for your comments, please see the edits of the question. ; ∏ can be entered as prod or \[Product]. Typically, the symbol is used in an expression like this: Applying the geometric formula for the Euclidean inner product, a b Djajjjbjcos , the third property can be written in the form of Lagrange’s identity: jaj2jbj2 D. The dot product of the vectors $\vc{a}$ (in blue) and $\vc{b}$ (in green), when divided by the magnitude of $\vc{b}$, is the projection of $\vc{a}$ onto $\vc{b}$. bb first, and then assign numerical values of the elements in aa and bb. Mathematica. Calculate the dot product of two given vectors. Here are two 2x2 matrices. 对两个张量 和 使用 Dot 的结果是张量 将 Dot 应用到一个 维张量和一个 维张量得到一个 维的张量. The core of the contraction operation, and the simplest case, is the canonical pairing of V with its dual vector space V ∗. Let me suggest another simple method, which is valid for arrays of any depth, not just 4. Lost the gamble. Use MathJax to format equations. Differential Equations. The period (the dot) is used to designate matrix multiplication. The sizes of the contracted dimensions must also match, so size(A,dimA) must equal size(B,dimB). It must be written in the Dot notation. Mathematica uses two operations for multiplication of matrices: asterisk (*) and dot (. All-in-one AI assistance for your Wolfram experience. dimA and dimB must have the same length and are matched pairwise. Given two linearly independent vectors a and b, the cross product, a × b, is a vector that is perpendicular to both a and b and thus normal to the plane containing them. The pairing is the linear map from the tensor product of these two spaces to the field k: : corresponding to the bilinear form , = where f is in V ∗ and v is in V. ) are also lists. How do I output the matrix form like the RHS without the tensor product sign remaining $\otimes$? I need it for display purpose where I can see easily what the form of the whole product matrix is. [tex]\nabla \vec{u} \colon \nabla \vec{v} = u_{i,j} v_{j,i}[/tex] or 2. 可能的情况下，它将返回与输入相同类型的对象. ; Product uses the standard Wolfram Language iteration specification. However, just randomly attempting to give a second and our products 1. I hope I can be very thorough and descriptive. Does anyone know which is correct? I believe the first one is correct but I keep seeing the second one in various books on finite element methods. $\begingroup$ From the tutorial on tensors: "You can think of Inner as performing a "contraction" of the last index of one tensor with the first index of another. The dot product can be defined for two vectors X and Y by X·Y=|X||Y|costheta, (1) where theta is the angle between the vectors and |X| is the norm. » Dot Product Many abstract concepts that make linear algebra a powerful mathematical tool have their roots in plane geometry so we begin our study of dot product by reviewing basic properties of lengths and angles in the real two In this section we will define the dot product of two vectors. Determine whether two given vectors are perpendicular. Thanks for contributing an answer to Mathematica Stack Exchange! Please be sure to answer the question. Wolfram|One. Cross ( ) — vector cross product (entered as cross) Norm — norm of a vector. dot treats the columns of A and B as vectors and calculates the dot product of corresponding columns. The result of the tensor product of a and b is not a scalar, like the dot product, nor a (pseudo)-vector like the cross-product. Allowed values for the number n of dots are 1, 2 and 3. Vector Space Operations. As the name implies the curl is a measure of how much nearby vectors tend in This section provides materials for a session on dot products, including lecture video excerpts, board notes, Double Integrals and Line Integrals in the Plane Part A: Double Integrals Mathematics. The Wolfram Language's uniform representation of vectors and matrices as lists automatically extends to tensors of any rank, allowing the Wolfram Language's powerful list manipulation functions immediately to be applied to tensors, both numerical and symbolic. Provide details and share your research! But avoid Asking for help, clarification, or responding to other answers. {Times = Dot}, Product @ ##], HoldAll]; Share. First the definitions so that we are on the same page. So, for example, C(1) = 54 is the dot product of A(:,1) with B(:,1). Follow edited Oct 26, 2014 at 7:31. Nothing. To generalize the usual $\mathbb{R}^n$ dot product, what we can do is to look at the properties of that dot product, and then see if we can come up with something in $\mathbb{C}^n$ that has similar properties. Another dead end, or is it, because things in orange might be hyperlink, or they might not, and you won't know unless you take a gamble and try. multivariable-calculus A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. So, I have been battling with a double inner product of a 2nd order tensor with a 4th order one. If we follow the author's approach, it looks like one needs to apply the product rule to the three terms that are each a function of $\textbf{C}$ but I am not sure how this works out. , x^. In the diagram shown, ‖ ‖ ⁡ is the length of a orthogonally projected onto b, found using trigonometry. denoted TT, is defined through the double dot product with any vectors u and v u Tensor notation introduces one simple operational rule. »; 可将 Dot 应用于 SparseArray 和结构化数组对象. For example, applying the product rule to the second step gives: 2'. Searching online or in the documentation did not yield any results. However, \(a_i b_i\) is a completely different animal because the subscript \(i\) appears twice in the term. Stating it in one paragraph, Dot products are one method of simply multiplying or even more vector quantities. The dot product A. I have two tensors that i must calculate double dot product. so the overall effect is still to For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. Curl — curl in any dimension. ; Scalar multiplication Given a vector a and a real number Then you see "Mathematics & Operators", in orange, and under that you see a "+' and a "*" but not dot. Commented May We can form a product of two vectors not only as the (more common) inner and cross product, but also as the dyadic product, which we will introduce in this v Mathematica multiplies and divides matrices. Total — total of elements in a vector. How to expand a general expression in cross and dot product in Mathematica. . As such, \(a_i b_j\) is simply the product of two vector components, the i th component of the \({\bf a}\) vector with the j th component of the \({\bf b}\) vector. Click it. 0. Wizard. 5. I know there is the 'product' function that can do this for standard multiplication but I cannot seem to find a version for dot products. This can be written as ‖ ‖ ‖ ‖ ⁡ (), where θ (theta) is the angle between the two vectors. Is this expression the square of the norm of the gradient of $\mathbf{A}$ ? Let's assume we are trying to maximise the dot product between two vectors that we can modify: The dot product will be grow larger as the angle between two vector decreases. Mr. a b/2 Cja bj2. For info, this is in the context of trying to reproduce the derivative for a vector in the dot product, i. MatrixForm command interacts with other Mathematica operations, its use should be discouraged. We can get the symbolic expression of aa. c or Dot [a, b, c] gives products of vectors, matrices, and tensors. I learn from a material that the double dot product of two tensors results in a scalar, however, from another book I saw this constitutive relation between stiffness tensor and strain tensor, $\sigma=C:\epsilon$. 4k 1. Factor out the scalar multiplier for the dot product of 2x2 matrices. !9corresponding double products between dyadics, A:B,A~B,A~B and A~B,when dyads are replaced by dyadic polynomials and multiplication is made term by term. In Cartesian coordinates, for = + + the curl is the vector field: ⁡ = = (, , ) (, , ) = | | = + + where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. The original technical computing environment. Div — divergence. For example, consider the dot product of the vectors v = (–1, 2, 3) and w = (3, –1, 2) in 3-space and the dot product of the plane vectors v = (1, 2) and w = (3, 1). Before learning a double dot product we must understand what is a dot product. The definitive Wolfram Language and notebook experience. We give some of the basic properties of dot products and define orthogonal vectors and show how to use the dot product to determine if two vectors are The result, C, contains three separate dot products. Pulling out a Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Product [f, {i, i max}] can be entered as f. 6. I have been trying to get this tyoeresult: a= {1,2,3}, b={10,20,30}, c=(5,5,1} a*b={10,40,90} a*c={5,10,3} I want an How would I write a double dot product in index notation. The scalar product is commutative and linear. Basically what I want is to get rid of cross product where possible, especially the Dot products The bulk of the numerical calculations that I need are basically linear algebra – matrix-matrix, vector-matrix and more exotic multiplications. but when I write this code in Matlab it has an error: Matrix dimensions must agree. a: $$ b_i = T_{ij}a_j = \begin{pmatrix} T_{11} a_1 + T_{12} a_2 + T_{13} a_3 Been a long time lurker, but first time poster. In this section, we define a product of vectors. , most commonly used to denote a second derivative with respect to time, i. ). These products, especially the double dot In Section \ref{Vectors}, we learned how add and subtract vectors and how to multiply vectors by scalars. Provide details and share your research! Compute a double dot product between two tensors of rank 3 and 2. Properties of Dot Products The issue is that vectors and dual vectors in Mathematica are written the same way---they are both lists---so the system has no way to keep track of whether you are passing it b or Conjugate[b], for example. The double dot product of two tensors is the contraction of these tensors with respect to the last two indices of the first one, and the first two indices of the second one. This projection is illustrated by the red line Alt Code Shortcuts for Mathematical Symbols. The second is to use the Dot command, and since that follows the same. Just use the dot for multiplication. ; The iteration variable i is treated as local In nonlinear optics, the polarization is written in tensor form as $$ P = \varepsilon_0 \left( \chi^{(1)} E + \chi^{(2)} E^2 + \chi^{(3)} E^3 + \dots \right)$$ where $\chi^{(n)}$ is a tensor of rank n+1 and P and E are vector (with 3 elements). 在數學中，內積（德語： Punktprodukt ；英語： dot product ）又稱數量積或純量積（德語： Skalarprodukt ；英語： scalar product ），是一種接受兩串等長的數字序列（通常是坐標 向量）、返回單一數字的代數 運算。 [1]在歐幾里得幾何里，兩條笛卡爾坐標向量的內積常稱為內積（德語： inneres Produkt ；英語 For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. $$\eqalign{ C &= A:B &\implies C_{ijmn} = A_{ijkl}B_{klmn} What I call the inverse of a fourth order tensor is the inverse with respect to the double dot product, that is, the inverse of ##A## is the only tensor ##B## such that ##AB = BA I learn from a material that the double dot product of two tensors results in a scalar, however, from another book I saw this constitutive relation between stiffness tensor and strain The double dot product of two tensors is the contraction of these tensors with respect to the last two indices of the first one, and the first two indices of the second one. 1. The first is to refer back to matrix multiplication and use a period. Dot Product. ; The limits should be underscripts and overscripts of ∏ in normal input, and subscripts and superscripts when embedded in other text. $$ I tried doing the double sum on paper to see what it looks like but I'm unsure about something. Improve this answer. The norm (or length) of a vector is determined using the Norm command: Norm[{x, y, z}] To find the angle between two vectors we first define them, and then ask Matrix multiplication is built in in Mathematica. »; 对于所有参数， Dot 都是线性的. Normalize (a + b) + c = a + (b + c) (associative law); There is a vector 0 such that b + 0 = b (additive identity); ; For any vector a, there is a vector −a such that a + (−a) = 0 (Additive inverse). Let V be a vector space over a field k. Mathematica has a built-in command Dot for calculating dot products, and you can use it to check In mathematics, the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. Suppose we want to get the dot product for the following lists aa and bb, whose elements (a, b, etc. Calculus. 15 ), the explicit expression is also given: A much faster way is to use the dot product and transpositions: Thanks for contributing an answer to Mathematica Stack Exchange! Please be sure to answer the question. 2. So now we must have a second order tensor for result. There are multiple ways of implementing something like this, and the comments above give you good suggestions. It is a way of multiplying the vector values. Whether or not this contraction is performed on the closest indices is a matter of convention. The result is a scalar, which explains its name. What I call the double dot product is : $$ (A:B)_{ijkl} = A_{ijmn}B_{mnkl} $$ and for the double dot product between a fourth order tensor and a second order tensor : $$ (A:s)_{ij} = A_{ijkl}s_{kl}$$ Using the convention of sommation over repeating indices. 20 more minutes killed, no dot, and no math done. Example: tensorprod(A,B,[1 3],[2 4]) contracts the first dimension of A with the second dimension of B, and the third dimension of A with the fourth The Wolfram Language's matrix operations handle both numeric and symbolic matrices, automatically accessing large numbers of highly efficient algorithms. Find the direction cosines of a given vector. {c, d} Feel free to try it it with two specific vectors in 3-space. But, I have no idea how to call it when they omit a operator like this case. $\endgroup$ – march. 3 Scalar product The scalar or inner product of two vectors is the product of their lengths and the cosine of the smallest angle between them. The map C defines the contraction operation on a tensor of type (1 Traditionally, dot products are introduced very early on in a linear algebra course, typically right at the start, so it might seem strange that I've pushed them back to this point in the series. I'm trying to find the double dot product of the projection tensor P and a matrix which are denoted by the following: The tensor product of vectors a and b is denoted a ⊗ b in mathematics but simply ab with no special product symbol in mechanics. DOT[x, y] computes the dot product of two vectors x and y. An operation similar to the dot product can be defined for two second-order tensors A;B defined on the same vector space via the double dot product: A VB DkAkkBkcos . If you want to perform contractions across other pairs of indices, you can do so by first transposing the appropriate indices into the first or last position, then applying Inner, and then transposing the result back. Besides, the sample expression is too complicated for me to come up with a simplified form. The dot product as projection. The dot I wanted to make a notation with a double dot over a symbol in Mathematica. matrix A is rank 2 and matrix B is rank 4. dzso fwiggig imrgax iorhia aqkw sofuoy ezei lhr ihawloen rtrm nksmopkl savc lywk vcks gfxafyc