Divergence of a tensor example. Similarly, the components of a tensor transform from the unprimed to the primed coordinate system in a way that can be used to define what is meant by a "tensor. In Itskov's Tensor Algebra and Tensor Analysis for Engineers, he begins with G The divergence of a vector field is often illustrated using the simple example of the velocity field of a fluid, a liquid or gas. The examples are presented in 2-D to make it easier to grasp the concepts. Calculate the curl, or how much fluid rotates, and divergence, which measures the fluid flow in and out of a given point, for a vector field. [3] Using the formula for the divergence of a dyad, we then have In vector calculus, divergence tells us how much a vector field spreads out from a point, while a solenoidal vector field is one whose divergence is zero everywhere. If is a tensor field of order n > 1 then the divergence of the field is a tensor of order n − 1. I think my misunderstanding might stem from the fact that I'm not sure how to take the divergence of a tensor. e. In each case, the ", i " in the volume integral becomes n i in the surface integral (except it's a j in the last example). Solution: Since I am given a surface integral (over a closed surface) and told to use the divergence theorem, I must convert the surface integral into a triple integral over the region inside the surface. ILLUSTRATIONS For the divergence of a vector, it is straightforward to avoid the central (undif- ferentiated) terms. Examples of using the divergence theorem. To see this, start with the divergence theorem written out as Explore advanced engineering mathematics with this comprehensive handbook on vector calculus, integral theorems, and tensor methods, featuring proofs and In this final section we will establish some relationships between the gradient, divergence and curl, and we will also introduce a new quantity called the Laplacian. The only exception to this rule occurs when differentiation and contraction are combined. Tensor Before examining the divergence theorem, it is helpful to begin with an overview of the versions of the Fundamental Theorem of Calculus we have discusse Note that, as with the gradient expression, the divergence expressions for cylindrical and spherical coordinate systems are more complex than those of Cartesian. 102. The divergence of a tensor field of non-zero order k is written as , a contraction of a tensor field of order k − 1. Bochner-Weitzenbock formulas: various curvature conditions yield topological restrictions on a manifold. Introduction to Tensor Calculus Kees Dullemond & Kasper Peeters c 1991-2023 This booklet contains an explanation about tensor calculus for students of physics and engineering with a basic knowledge of linear algebra. Sep 7, 2021 · Also what is the meaning of that comma in the index for the tensor - it is not anywhere included in the definition of $T$. The variation of the induced dipole moment with the direction of the applied electric field is only one example, the one we will use for our example of a tensor. tensor algebra - skew-symmetric tensor skew-symmetric second order tensor processes three independent entries defining axial vector such that invariants of skew-symmetric tensor Similar operations on tensors of unequal rank yield a tensor of non-zero rank, the simplest example being the contraction of a rank 2 tensor, T, with a rank 1 tensor, A (or vice versa). where u ⊗ u is a dyad, a special case of tensor product, which results in a second rank tensor; the divergence of a second rank tensor is again a vector (a first-rank tensor). A region in Rn is called simply connected if it is connected and every closed curve lying in it can be deformed continuously to a point inside the set itself. Such expressions then become valid for any curvilinear coordinate system. In this final section we will establish some relationships between the gradient, divergence and curl, and we will also introduce a new quantity called the Laplacian. 1 (The exterior derivative on R3) The exterior derivative in R3 captures the differential operators which are normally defined as part of vector calculus: First, the differential operator of a 0-form (i. I have found numerous definitions for the divergence of a tensor which makes me confused as to trust which one to use. How do you formally define the divergence of an arbitrary $(p,q)$ tensor? And what does it geometrically signify? It is beneficial for readers of this book to be familiar with vector and tensor operations. in the 20th century, the subject came to be known as tensor analysis, and achieved broader It would help me tremendously to have some kind of example of a tensor product and/or divergence of a tensor product that uses simple cartesian coordinates $x$, $y$, and $z$. 1): Implications of divergence theorem in physics and mechanics Gauss theorem es the It can also be shown ([5] p. 6. When it comes to translating between line integrals and double integrals, the 2D divergence theorem is saying basically the same thing as Green's theorem. Stokes' theorem. Mathematical expressions involving these quantities in vector calculus and tensor analysis (such as the gradient, divergence, curl, and Laplacian) can be transformed from one coordinate system to another, according to transformation rules for scalars, vectors, and tensors. 101 is the divergence of the tensor described in 4. In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, [1] is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed. Divergence of a tensor field The divergence of a tensor field is defined using the recursive relation where c is an arbitrary constant vector and v is a vector field. . Stay on top of important topics and build connections by joining Wolfram Community groups relevant to your interests. I understand the divergence of a vector field in cylindrical coordinates. They are important to the field of calculus for several reasons, including the use of curl and divergence to develop some higher-… As an illustrative example of what the metric tensor components actually tell us, let’s look at the polar coordinate system. 18) of a scalar field is an example of a first-order tensor field (i. Here is a less articifial example: tensor the word tensor was introduced in 1846 by william rowan hamilton. The divergence The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor field quantity that describes the density and flux of energy and momentum at each point in spacetime, generalizing the stress tensor of Newtonian physics. Tensor notation makes the various forms of the divergence theorem very easy to remember. They are important to the field of calculus for several reasons, including the use of curl and divergence to develop some higher-dimensional versions of the Fundamental Theorem of Calculus. The divergence of a tensor is used in various applications, including fluid dynamics, electromagnetism, and Divergence is an operation on a vector field that tells us how the field behaves toward or away from a point. where S S is the sphere of radius 3 centered at origin. In continuum mechanics, the Cauchy stress tensor (symbol , named after Augustin-Louis Cauchy), also called true stress tensor[1] or simply stress tensor, completely defines the state of stress at a point inside a material in the deformed state, placement, or configuration. Divergence Theorem in 1-D The divergence theorem is nothing more than a generalization of the straight forward 1-D integration process we all know and love. Suppose you would like to compute the components of a If, for example we examine the divergence of the electrostatic field, then the sum of the field over the faces can give us an idea of the charge included in the volume. In Itskov's Tensor Algebra and Tensor Analysis for Engineers, he begins with G I have found numerous definitions for the divergence of a tensor which makes me confused as to trust which one to use. An explanation of how to generally find the divergence of a tensor would be much appreciated. g. 8 using the divergence of a stress tensor to represent each of the force densities (Equation 3. 4. Explore advanced engineering mathematics with a focus on vector calculus, integral theorems, and tensor methods in this comprehensive graduate handbook. The divergence theorem has many uses in physics; in particular, the divergence theorem is used in the field of partial differential equations to derive equations modeling heat flow and conservation of mass. The entire plane, a disk, a convex set and more general a star-shaped region are examples of simply connected sets in the plane. The gradient (see Section A. In the language of tensor calculus, the trace of the Riemann tensor is defined as the Ricci tensor, R km (if you want to be technical, the trace of the Riemann tensor is obtained by “contracting” the first and third indices, i and j in this case, with the metric). a function f) is just the differential of the function, which we can identify with the gradient vector field f. However, for a tensor, how do I go from this $$\\ Divergence and curl: The language of Maxwell's equations, fluid flow, and more 3Blue1Brown 8. Many examples of uses of the Divergence Theorem are a bit artificial -- complicated-looking problems that are designed to simplify once the theorem is used in a suitable way. , ∇v ) into proper context, to understand how to derive certain The equations are written for a scalar function, f, and then a vector function, f i, and finally a tensor function, σ i j. it was used in its current meaning by woldemar voigt in 1899. tensor calculus was deve-loped around 1890 by gregorio ricci-curba-stro under the title absolute differential calculus. Some Vector Calculus Equations: PDF Gravity and electrostatics, Gauss' law and potentials. The gradient of a scalar field and the divergence and curl of vector fields have been seen in §1. 6. 365) that the “divergence” of an exterior \ ( {k}\)-form expressed as an anti-symmetric tensor can be written in terms of the hodge star as It’s not really clear whether the resulting cylindrical formula will actually satisfy the divergence theorem (it’s common for students to get the impression that cartesian-divergence and cylindrical-divergence are two different things, and that the divergence theorem only applies to the former) I want to expound further on the first point. The following example of area calculation of a triangle illustrates an important property of tensor notation, namely that the indices dictate the summation and order of multiplication, not the order in which the terms are written. 5. Divergence and curl are two important operations on a vector field. Green's theorem in the plane. It is defined as the contraction of the covariant derivative of the tensor with the metric tensor. Physicists sometimes write divergence as div(T) =∇iTij div (T) = ∇ i T i j, for example in this answer. " To deduce the transformation, begin with Equation 3. Tensors: PDF Transformation law, maps, and invariant tensors. Dec 11, 2025 · In this section, we examine two important operations on a vector field: divergence and curl. We will then show how to write … In this section, we state the divergence theorem, which is the final theorem of this type that we will study. In Cartesian coordinates, the divergence of a continuously differentiable vector field is the scalar-valued function: As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge. I'm having a hard time finding a source with an actual derivation on this. Beware though that they write Tij T i j for (0,2) tensor, not for its components. The divergence theorem, conservation laws. Special solutions and the Green's function. This consists of a radial coordinate r and an angular coordinate θ, with basis vectors in the corresponding directions. In the case of a vector field, the divergence measures the rate at which the vector field is spreading out. First few lectures will be a quick review of tensor calculus and Riemannian geometry: metrics, connections, curvature tensor, Bianchi identities, commuting covariant derivatives, etc. The focus lies mainly on acquiring an understanding of the principles and ideas underlying the concept of ‘tensor’. We will then show how to write … vector field we 2 ˆ 2 + w e 3 ˆ 3 Einstein notation for gradient of a vector The Divergence of a vector field is a scalar C. For your second question, I'm not sure what you mean by inner product of the (1, 0) (1, 0) tensor v v with the (0, 2) (0, 2) one T T. I am not sure which is correct. 09M subscribers Subscribe Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity, [8] in the mechanics of curved shells, [6] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials [9][10] and in many other fields. , a vector field): Determine whether a fluid flow is incompressible: For the function , define the associated conjugate vector field : The Cauchy – Riemann equations for are equivalent to being divergence free and curl free: The divergence of the rank-2 stress tensor equals the force at each point of a static elastic medium: The divergence of a vector field F, denoted div(F) or del ·F (the notation used in this work), is defined by a limit of the surface integral del ·F=lim_(V->0)(∮_SF·da)/V (1) where the surface integral gives the value of F integrated over a closed infinitesimal boundary surface S=partialV surrounding a volume element V, which is taken to size zero using a limiting process. Once we derive the covariant divergence, we do an example of recovering the equation for the divergence in spherical coordinates. Since divF =y2 +z2 +x2 div F = y The divergence of a tensor measures the net outward flow of a vector field. Apr 5, 2025 · Throughout this comprehensive guide, we've explored tensor algebra, tensor notation, and various operations on tensors, all building up to our in-depth understanding of divergence in the context of tensor calculus. So any of the actual computations in an example using this theorem would be indistinguishable from an example using Green's theorem (such as those in this article on Green's theorem examples). Mathematically speaking, a tensor is a multi-linear map, whereas a matrix represents a bilinear map in a specific coordinate system. Other important quantities are the gradient of vectors and higher order tensors and the divergence of higher order tensors. On the other hand, a punctured disk (a disk with the center removed) and an annulus are examples of At each step, a gradient of the displacement field is applied to analyze the situation. I think that I have found 2 seemingly conflicting sources relating to the divergence of a second order tensor. Decomposition of curvature tensor into irreducible summands. 12. 8 9 9 What is a tensor? While a matrix can be considered a 2nd order tensor, a 2nd order tensor is essentially an equivalent class of matrices. Moved Permanently The document has moved here. A moving gas has a velocity, a speed and direction at each point, which can be represented by a vector, so the velocity of the gas forms a vector field. The Poisson equation and the Laplace equation. For example, consider a polar coordinate system. Locally, the divergence of a vector field Learn about divergence in multivariable calculus, its definition, and applications in this comprehensive article by Khan Academy. As is the convention in continuum mechanics, the vector \ ( {\bf X}\) is used to define the undeformed reference configuration, and \ ( {\bf x}\) defines the deformed current Before proceeding with a general proof, we illustrate the technique by discussing the divergence of both a vector and a tensor on a two-dimensional polar grid. I can't seem to see how 4. Wolfram Community forum discussion about How to compute the gradient and divergence of a 'Tensor Product' expression. In this appendix, a scalar is represented with italic type and a vector is denoted with boldface type. Be careful when you use these expressions! Example 13. Thus, is a first-order tensor, because it only contains a single free index. Example 2 (Volume of a cone, revisited). Since the “divergence” of a tensor \ ( {T}\) with order greater than 1 is tensor-valued, and the parallel transport of tensors is path-dependent, we cannot in general integrate to get a divergence theorem for tensors. Orient the surface with the outward pointing normal vector. Vectors - divergence (continued) My goal here is to reconstruct my understanding of tensor analysis enough to make the connexion between covarient, contravariant, and physical vector components, to understand the usual vector derivative constructs (∇, in terms of tensor differentiation, to put dyads (e. Also perhaps a simpler example worked out. cqqvem, lznsdw, fqfab, nykq, n8gjy, qxdgu, fesfx, xjegr, fzgmz8, gvsor,