Primer on the Basics of Tensor Analysis and the Laplacian in Generalized Coordinates (original) (raw)
Tensors: A guide for undergraduate students
American Journal of Physics, 2013
A guide on tensors is proposed for undergraduate students in physics or engineering that ties directly to vector calculus in orthonormal coordinate systems. We show that once orthonormality is relaxed, a dual basis, together with the contravariant and covariant components, naturally emerges. Manipulating these components requires some skill that can be acquired more easily and quickly once a new notation is adopted. This notation distinguishes multi-component quantities in different coordinate systems by a differentiating sign on the index labelling the component rather than on the label of the quantity itself. This tiny stratagem, together with simple rules openly stated at the beginning of this guide, allows an almost automatic, easy-to-pursue procedure for what is otherwise a cumbersome algebra. By the end of the paper, the reader will be skillful enough to tackle many applications involving tensors of any rank in any coordinate system, without indexmanipulation obstacles standing in the way. V
Introduction to Tensor Calculus for General Relativity
It is su cient to develop the needed di erential geometry as a straightforward extension of linear algebra and vector calculus. However, it is important to keep in mind the geometrical interpretation of physical quantities. For this reason, we will not shy from using abstract concepts like p o i n ts, curves and vectors, and we will distinguish between a vector A and its components A . U n l i k e some other authors (e.g., Weinberg 1972), we will introduce geometrical objects in a coordinate-free manner, only later introducing coordinates for the purpose of simplifying calculations. This approach requires that we distinguish vectors from the related objects called one-forms. Once the di erences and similarities between vectors, one-forms and tensors are clear, we will adopt a uni ed notation that makes computations easy. We begin with vectors. A vector is a quantity with a magnitude and a direction. This primitive concept, familiar from undergraduate physics and mathematics, applies equally in general relativity. An example of a vector is dx, the di erence vector between two in nitesimally close points of spacetime. Vectors form a linear algebra (i.e., a vector ~space). If A is a vector and a is a real number (scalar) then aA is a vector with the same direction (or the opposite direction, if a < 0) whose length is multiplied by jaj. If ~~~Ã and B are vectors then so is A + B. These results are as valid for vectors in a curved four-dimensional spacetime as they are for vectors in three-dimensional Euclidean space. Note that we h a ve i n troduced vectors without mentioning coordinates or coordinate transformations. Scalars and vectors are invariant under coordinate transformations ṽector components are not. The whole point of writing the laws of physics (e.g., F = mã) using scalars and vectors is that these laws do not depend on the coordinate system imposed by the physicist. From equation (37), because df is a scalar and dx is a vector component, @f =@x must be the component of a one-form, not a vector. The notation @ , with its covariant (subscript) index, reinforces our view that the partial derivative is the component o f a õne-form and not a vector. We denote the gradient one-form by r. L i k e all one-forms, the gradient m a y be decomposed into a sum over basis one-forms e ~ . Using equation (37) and equation ( ) as the requirement f o r a d u a l b a s i s , w e conclude that the gradient is r ~ e ~ @ in a coordinate basis : (38) Note that we m ust write the basis one-form to the left of the partial derivative operator, for the basis one-form itself may depend on position! We will return to this point i n Section 4 when we discuss the covariant derivative. In the present case, it is clear from equation ( ) that we m ust let the derivative act only on the function f . W e can now rewrite equation ( ) in the coordinate-free manner df = h ~ rf dxi : (39) If we w ant the directional derivative o f f along any particular direction, we simply replace dx by a v ector pointing in the desired direction (e.g., the tangent v ector to some curve). Also, if we let f X equal one of the coordinates, using equation ( ) the gradient gives us the corresponding basis one-form: ~ rx = e ~ in a coordinate basis : (40) 14 The third use of coordinates is that they can be used to describe the distance between two points of spacetime. However, coordinates alone are not enough. We also need the metric tensor. We write the squared distance between two spacetime points as ds 2 = jdx j 2 g(dx dx) = dx dx : (41) 4 Di erentiation and Integration In this section we discuss di erentiation and integration in curved spacetime. These might seem like a delicate subjects but, given the tensor algebra that we h a ve d e v eloped, tensor calculus is straightforward. Consider rst the gradient of a scalar eld f X . We h a ve already shown in Section 2 that the gradient operator r is a one-form (an object that is invariant under coordinate transformations) and that, in a coordinate basis, its components are simply the partial derivatives with respect to the coordinates: where @ (@=@x ). We h a ve i n troduced a second notation, r , called the covariant derivative with respect to x . By de nition, the covariant d e r i v ative behaves like t h e component of a one-form. But, from equation ( ), this is also true of the partial derivative operator @ . W h y h a ve w e in troduced a new symbo l? Before answering this question, let us rst note that the gradient, because it behaves like a tensor of rank (0 1) (a one-form), changes the rank of a tensor eld from (m n) to (m n + 1). (This is obviously true for the gradient of a scalar eld, with m = n = 0.) That is, application of the gradient i s l i k e taking the tensor product with a one-form. The di erence is that the components are not the product of the components, because r is not a number. Nevertheless, the resulting object must be a tensor of rank (m n + 1) e.g., its components must transform like components of a (m n + 1) tensor. The gradient of a scalar eld f is a (0 1) tensor with components (@ f ). coordinates ( z). The line element (cf. eq. 50) now becomes ds 2 = d 2 + 2 d 2 + dz 2 . Because ẽ and ẽ are independent o f z and ẽz is itself constant, no new non-vanishing Christo el symbols appear. Now consider a related but di erent manifold: a cylinder. A cylinder is simply a surface of constant in our three-dimensional Euclidean space. This two-dimensional space is mapped by coordinates ( z ), with basis vectors ẽ and ẽz . What are the gradients of these basis vectors? They vanish! But, how can that be? From equation ( ), @ ẽ = ; ẽ . H a ve w e forgotten about the ẽ direction? This example illustrates an important lesson. We cannot project tensors into basis vectors that do not exist in our manifold, whether it is a two-dimensional cylinder or a four-dimensional spacetime. A cylinder exists as a two-dimensional mathematical surface whether or not we c hoose to embed it in a three-dimensional Euclidean space. If it happens that we c a n e m bed our manifold into a simpler higher-dimensional space, we do so only as a matter of calculational convenience. If the result of a calculation is a v ector normal to our manifold, we m ust discard this result because this direction does not exist in our manifold. If this conclusion is troubling, consider a cylinder as seen by a t wo-dimensional ant crawling on its surface. If the ant goes around in circles about the z-axis it is moving in the ẽ direction. The ant w ould say that its direction is not changing as it moves along the circle. We conclude that the Christo el symbols indeed all vanish for a cylinder described by coordinates ( z ). Later we will return to the question of how t o e v aluate the Christo el symbols in general. First we i n vestigate the gradient of one-forms and of general tensor elds.
Springer eBooks, 2019
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A Some Basic Rules of Tensor Calculus
2006
The direct notation operates with scalars, vectors and tens ors as physical objects defined in the three dimensional space. A vector (first rank te sor)a is considered as a directed line segment rather than a triple of numbers (co ordinates). A second rank tensorA is any finite sum of ordered vector pairs A = a ⊗ b + . . . + c ⊗ d. The scalars, vectors and tensors are handled as invariant (inde pe nt from the choice of the coordinate system) objects. This is the reason for the us e of the direct notation in the modern literature of mechanics and rheology, e.g. [29 , 3 , 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of v ectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write
Textbook on Tensor Calculus and Differential Geometry
2012
The Metric Tensor 78 2.1.1 Fundamental Contravariant Tensor 83 2.1.2 Length of a Curve 92 2.2 Associated Tensors 94 2.2.1 Magnitude of a Vector 97 2.2.2 Angle Between Two Vectors 100 2.2.3 Orthogonality of Two Vectors 101 2.3 Some Loci 104 2.3.1 Coordinate Curve 105 2.3.2 Hypersurface 107 2.3.3 Orthogonal Ennuple 111 2.4 Affine Coordinates 113 2.5 Curvilinear Coordinates 114 2.5.1 Coordinate Surfaces 115 2.5.2 Coordinate Curves 116 2.5.3 Line Element 120 2.5.4 Length of a Vector 120 2.5.5 Angle between Two Vectors 122 2.5.6 Reciprocal Base System 124 2.5.7 Partial Derivative 129 2.6 Exercises 133 4. Riemannian Geometry .
Universitext, 2005
Part I Basic Tensor Algebra Tensor Spaces 3 X Contents 3.5 Einstein's contraction of the tensor product 3.6 Matrix representation of tensors 3.6.1 First-order tensors 3.6.2 Second-order tensors 3.7 Exercises 4 Change-of-basis in Tensor Spaces 65 4.1 Introduction 65 4.2 Change of basis in a third-order tensor product space ....... 65 4.3 Matrix representation of a change-of-basis in tensor spaces ... 4.4 General criteria for tensor character 4.5 Extension to homogeneous tensors 4.6 Matrix operation rules for tensor expressions 74 4.6.1 Second-order tensors (matrices) 4.6.2 Third-order tensors 77 4.6.3 Fourth-order tensors 78 4.7 Change-of-basis invariant tensors: Isotropic tensors 4.8 Main isotropic tensors 4.8.1 The null tensor 4.8.2 Zero-order tensor (scalar invariant) 4.8.3 Kronecker's delta 4.9 Exercises