1.14 Tensor Calculus I: Tensor Fields
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A Brief but Important Note on the Product Rule
A brief but important note on the product rule Peter Merrotsy The University of Western Australia [email protected] The product rule in the Australian Curriculum he product rule refers to the derivative of the product of two functions Texpressed in terms of the functions and their derivatives. This result first naturally appears in the subject Mathematical Methods in the senior secondary Australian Curriculum (Australian Curriculum, Assessment and Reporting Authority [ACARA], n.d.b). In the curriculum content, it is mentioned by name only (Unit 3, Topic 1, ACMMM104). Elsewhere (Glossary, p. 6), detail is given in the form: If h(x) = f(x) g(x), then h'(x) = f(x) g'(x) + f '(x) g(x) (1) or, in Leibniz notation, d dv du ()u v = u + v (2) dx dx dx presupposing, of course, that both u and v are functions of x. In the Australian Capital Territory (ACT Board of Senior Secondary Studies, 2014, pp. 28, 43), Tasmania (Tasmanian Qualifications Authority, 2014, p. 21) and Western Australia (School Curriculum and Standards Authority, 2014, WACE, Mathematical Methods Year 12, pp. 9, 22), these statements of the Australian Senior Mathematics Journal vol. 30 no. 2 product rule have been adopted without further commentary. Elsewhere, there are varying attempts at elaboration. The SACE Board of South Australia (2015, p. 29) refers simply to “differentiating functions of the form , using the product rule.” In Queensland (Queensland Studies Authority, 2008, p. 14), a slight shift in notation is suggested by referring to rules for differentiation, including: d []f (t)g(t) (product rule) (3) dt At first, the Victorian Curriculum and Assessment Authority (2015, p. -
1.2 Topological Tensor Calculus
PH211 Physical Mathematics Fall 2019 1.2 Topological tensor calculus 1.2.1 Tensor fields Finite displacements in Euclidean space can be represented by arrows and have a natural vector space structure, but finite displacements in more general curved spaces, such as on the surface of a sphere, do not. However, an infinitesimal neighborhood of a point in a smooth curved space1 looks like an infinitesimal neighborhood of Euclidean space, and infinitesimal displacements dx~ retain the vector space structure of displacements in Euclidean space. An infinitesimal neighborhood of a point can be infinitely rescaled to generate a finite vector space, called the tangent space, at the point. A vector lives in the tangent space of a point. Note that vectors do not stretch from one point to vector tangent space at p p space Figure 1.2.1: A vector in the tangent space of a point. another, and vectors at different points live in different tangent spaces and so cannot be added. For example, rescaling the infinitesimal displacement dx~ by dividing it by the in- finitesimal scalar dt gives the velocity dx~ ~v = (1.2.1) dt which is a vector. Similarly, we can picture the covector rφ as the infinitesimal contours of φ in a neighborhood of a point, infinitely rescaled to generate a finite covector in the point's cotangent space. More generally, infinitely rescaling the neighborhood of a point generates the tensor space and its algebra at the point. The tensor space contains the tangent and cotangent spaces as a vector subspaces. A tensor field is something that takes tensor values at every point in a space. -
A Brief Tour of Vector Calculus
A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0. -
A Quotient Rule Integration by Parts Formula Jennifer Switkes ([email protected]), California State Polytechnic Univer- Sity, Pomona, CA 91768
A Quotient Rule Integration by Parts Formula Jennifer Switkes ([email protected]), California State Polytechnic Univer- sity, Pomona, CA 91768 In a recent calculus course, I introduced the technique of Integration by Parts as an integration rule corresponding to the Product Rule for differentiation. I showed my students the standard derivation of the Integration by Parts formula as presented in [1]: By the Product Rule, if f (x) and g(x) are differentiable functions, then d f (x)g(x) = f (x)g(x) + g(x) f (x). dx Integrating on both sides of this equation, f (x)g(x) + g(x) f (x) dx = f (x)g(x), which may be rearranged to obtain f (x)g(x) dx = f (x)g(x) − g(x) f (x) dx. Letting U = f (x) and V = g(x) and observing that dU = f (x) dx and dV = g(x) dx, we obtain the familiar Integration by Parts formula UdV= UV − VdU. (1) My student Victor asked if we could do a similar thing with the Quotient Rule. While the other students thought this was a crazy idea, I was intrigued. Below, I derive a Quotient Rule Integration by Parts formula, apply the resulting integration formula to an example, and discuss reasons why this formula does not appear in calculus texts. By the Quotient Rule, if f (x) and g(x) are differentiable functions, then ( ) ( ) ( ) − ( ) ( ) d f x = g x f x f x g x . dx g(x) [g(x)]2 Integrating both sides of this equation, we get f (x) g(x) f (x) − f (x)g(x) = dx. -
Curl, Divergence and Laplacian
Curl, Divergence and Laplacian What to know: 1. The definition of curl and it two properties, that is, theorem 1, and be able to predict qualitatively how the curl of a vector field behaves from a picture. 2. The definition of divergence and it two properties, that is, if div F~ 6= 0 then F~ can't be written as the curl of another field, and be able to tell a vector field of clearly nonzero,positive or negative divergence from the picture. 3. Know the definition of the Laplace operator 4. Know what kind of objects those operator take as input and what they give as output. The curl operator Let's look at two plots of vector fields: Figure 1: The vector field Figure 2: The vector field h−y; x; 0i: h1; 1; 0i We can observe that the second one looks like it is rotating around the z axis. We'd like to be able to predict this kind of behavior without having to look at a picture. We also promised to find a criterion that checks whether a vector field is conservative in R3. Both of those goals are accomplished using a tool called the curl operator, even though neither of those two properties is exactly obvious from the definition we'll give. Definition 1. Let F~ = hP; Q; Ri be a vector field in R3, where P , Q and R are continuously differentiable. We define the curl operator: @R @Q @P @R @Q @P curl F~ = − ~i + − ~j + − ~k: (1) @y @z @z @x @x @y Remarks: 1. -
Matrices and Tensors
APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM.. -
A Some Basic Rules of Tensor Calculus
A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e. -
General Relativity Fall 2019 Lecture 13: Geodesic Deviation; Einstein field Equations
General Relativity Fall 2019 Lecture 13: Geodesic deviation; Einstein field equations Yacine Ali-Ha¨ımoud October 11th, 2019 GEODESIC DEVIATION The principle of equivalence states that one cannot distinguish a uniform gravitational field from being in an accelerated frame. However, tidal fields, i.e. gradients of gravitational fields, are indeed measurable. Here we will show that the Riemann tensor encodes tidal fields. Consider a fiducial free-falling observer, thus moving along a geodesic G. We set up Fermi normal coordinates in µ the vicinity of this geodesic, i.e. coordinates in which gµν = ηµν jG and ΓνσjG = 0. Events along the geodesic have coordinates (x0; xi) = (t; 0), where we denote by t the proper time of the fiducial observer. Now consider another free-falling observer, close enough from the fiducial observer that we can describe its position with the Fermi normal coordinates. We denote by τ the proper time of that second observer. In the Fermi normal coordinates, the spatial components of the geodesic equation for the second observer can be written as d2xi d dxi d2xi dxi d2t dxi dxµ dxν = (dt/dτ)−1 (dt/dτ)−1 = (dt/dτ)−2 − (dt/dτ)−3 = − Γi − Γ0 : (1) dt2 dτ dτ dτ 2 dτ dτ 2 µν µν dt dt dt The Christoffel symbols have to be evaluated along the geodesic of the second observer. If the second observer is close µ µ λ λ µ enough to the fiducial geodesic, we may Taylor-expand Γνσ around G, where they vanish: Γνσ(x ) ≈ x @λΓνσjG + 2 µ 0 µ O(x ). -
Tensor Calculus and Differential Geometry
Course Notes Tensor Calculus and Differential Geometry 2WAH0 Luc Florack March 10, 2021 Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8]. Contents Preface iii Notation 1 1 Prerequisites from Linear Algebra 3 2 Tensor Calculus 7 2.1 Vector Spaces and Bases . .7 2.2 Dual Vector Spaces and Dual Bases . .8 2.3 The Kronecker Tensor . 10 2.4 Inner Products . 11 2.5 Reciprocal Bases . 14 2.6 Bases, Dual Bases, Reciprocal Bases: Mutual Relations . 16 2.7 Examples of Vectors and Covectors . 17 2.8 Tensors . 18 2.8.1 Tensors in all Generality . 18 2.8.2 Tensors Subject to Symmetries . 22 2.8.3 Symmetry and Antisymmetry Preserving Product Operators . 24 2.8.4 Vector Spaces with an Oriented Volume . 31 2.8.5 Tensors on an Inner Product Space . 34 2.8.6 Tensor Transformations . 36 2.8.6.1 “Absolute Tensors” . 37 CONTENTS i 2.8.6.2 “Relative Tensors” . 38 2.8.6.3 “Pseudo Tensors” . 41 2.8.7 Contractions . 43 2.9 The Hodge Star Operator . 43 3 Differential Geometry 47 3.1 Euclidean Space: Cartesian and Curvilinear Coordinates . 47 3.2 Differentiable Manifolds . 48 3.3 Tangent Vectors . 49 3.4 Tangent and Cotangent Bundle . 50 3.5 Exterior Derivative . 51 3.6 Affine Connection . 52 3.7 Lie Derivative . 55 3.8 Torsion . 55 3.9 Levi-Civita Connection . 56 3.10 Geodesics . 57 3.11 Curvature . 58 3.12 Push-Forward and Pull-Back . 59 3.13 Examples . 60 3.13.1 Polar Coordinates in the Euclidean Plane . -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Matrix Calculus
Appendix D Matrix Calculus From too much study, and from extreme passion, cometh madnesse. Isaac Newton [205, §5] − D.1 Gradient, Directional derivative, Taylor series D.1.1 Gradients Gradient of a differentiable real function f(x) : RK R with respect to its vector argument is defined uniquely in terms of partial derivatives→ ∂f(x) ∂x1 ∂f(x) , ∂x2 RK f(x) . (2053) ∇ . ∈ . ∂f(x) ∂xK while the second-order gradient of the twice differentiable real function with respect to its vector argument is traditionally called the Hessian; 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 ∂x1 ∂x1∂x2 ··· ∂x1∂xK 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 2 K f(x) , ∂x2∂x1 ∂x2 ··· ∂x2∂xK S (2054) ∇ . ∈ . .. 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 ∂xK ∂x1 ∂xK ∂x2 ∂x ··· K interpreted ∂f(x) ∂f(x) 2 ∂ ∂ 2 ∂ f(x) ∂x1 ∂x2 ∂ f(x) = = = (2055) ∂x1∂x2 ³∂x2 ´ ³∂x1 ´ ∂x2∂x1 Dattorro, Convex Optimization Euclidean Distance Geometry, Mεβoo, 2005, v2020.02.29. 599 600 APPENDIX D. MATRIX CALCULUS The gradient of vector-valued function v(x) : R RN on real domain is a row vector → v(x) , ∂v1(x) ∂v2(x) ∂vN (x) RN (2056) ∇ ∂x ∂x ··· ∂x ∈ h i while the second-order gradient is 2 2 2 2 , ∂ v1(x) ∂ v2(x) ∂ vN (x) RN v(x) 2 2 2 (2057) ∇ ∂x ∂x ··· ∂x ∈ h i Gradient of vector-valued function h(x) : RK RN on vector domain is → ∂h1(x) ∂h2(x) ∂hN (x) ∂x1 ∂x1 ··· ∂x1 ∂h1(x) ∂h2(x) ∂hN (x) h(x) , ∂x2 ∂x2 ··· ∂x2 ∇ . -
Infinitesimal Calculus
Infinitesimal Calculus Δy ΔxΔy and “cannot stand” Δx • Derivative of the sum/difference of two functions (x + Δx) ± (y + Δy) = (x + y) + Δx + Δy ∴ we have a change of Δx + Δy. • Derivative of the product of two functions (x + Δx)(y + Δy) = xy + Δxy + xΔy + ΔxΔy ∴ we have a change of Δxy + xΔy. • Derivative of the product of three functions (x + Δx)(y + Δy)(z + Δz) = xyz + Δxyz + xΔyz + xyΔz + xΔyΔz + ΔxΔyz + xΔyΔz + ΔxΔyΔ ∴ we have a change of Δxyz + xΔyz + xyΔz. • Derivative of the quotient of three functions x Let u = . Then by the product rule above, yu = x yields y uΔy + yΔu = Δx. Substituting for u its value, we have xΔy Δxy − xΔy + yΔu = Δx. Finding the value of Δu , we have y y2 • Derivative of a power function (and the “chain rule”) Let y = x m . ∴ y = x ⋅ x ⋅ x ⋅...⋅ x (m times). By a generalization of the product rule, Δy = (xm−1Δx)(x m−1Δx)(x m−1Δx)...⋅ (xm −1Δx) m times. ∴ we have Δy = mx m−1Δx. • Derivative of the logarithmic function Let y = xn , n being constant. Then log y = nlog x. Differentiating y = xn , we have dy dy dy y y dy = nxn−1dx, or n = = = , since xn−1 = . Again, whatever n−1 y dx x dx dx x x x the differentials of log x and log y are, we have d(log y) = n ⋅ d(log x), or d(log y) n = . Placing these values of n equal to each other, we obtain d(log x) dy d(log y) y dy = .