Chapter 16: Vector Calculus
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Glossary Physics (I-Introduction)
1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay. -
Line, Surface and Volume Integrals
Line, Surface and Volume Integrals 1 Line integrals Z Z Z Á dr; a ¢ dr; a £ dr (1) C C C (Á is a scalar ¯eld and a is a vector ¯eld) We divide the path C joining the points A and B into N small line elements ¢rp, p = 1;:::;N. If (xp; yp; zp) is any point on the line element ¢rp, then the second type of line integral in Eq. (1) is de¯ned as Z XN a ¢ dr = lim a(xp; yp; zp) ¢ rp N!1 C p=1 where it is assumed that all j¢rpj ! 0 as N ! 1. 2 Evaluating line integrals The ¯rst type of line integral in Eq. (1) can be written as Z Z Z Á dr = i Á(x; y; z) dx + j Á(x; y; z) dy C CZ C +k Á(x; y; z) dz C The three integrals on the RHS are ordinary scalar integrals. The second and third line integrals in Eq. (1) can also be reduced to a set of scalar integrals by writing the vector ¯eld a in terms of its Cartesian components as a = axi + ayj + azk. Thus, Z Z a ¢ dr = (axi + ayj + azk) ¢ (dxi + dyj + dzk) C ZC = (axdx + aydy + azdz) ZC Z Z = axdx + aydy + azdz C C C 3 Some useful properties about line integrals: 1. Reversing the path of integration changes the sign of the integral. That is, Z B Z A a ¢ dr = ¡ a ¢ dr A B 2. If the path of integration is subdivided into smaller segments, then the sum of the separate line integrals along each segment is equal to the line integral along the whole path. -
1 Curl and Divergence
Sections 15.5-15.8: Divergence, Curl, Surface Integrals, Stokes' and Divergence Theorems Reeve Garrett 1 Curl and Divergence Definition 1.1 Let F = hf; g; hi be a differentiable vector field defined on a region D of R3. Then, the divergence of F on D is @ @ @ div F := r · F = ; ; · hf; g; hi = f + g + h ; @x @y @z x y z @ @ @ where r = h @x ; @y ; @z i is the del operator. If div F = 0, we say that F is source free. Note that these definitions (divergence and source free) completely agrees with their 2D analogues in 15.4. Theorem 1.2 Suppose that F is a radial vector field, i.e. if r = hx; y; zi, then for some real number p, r hx;y;zi 3−p F = jrjp = (x2+y2+z2)p=2 , then div F = jrjp . Theorem 1.3 Let F = hf; g; hi be a differentiable vector field defined on a region D of R3. Then, the curl of F on D is curl F := r × F = hhy − gz; fz − hx; gx − fyi: If curl F = 0, then we say F is irrotational. Note that gx − fy is the 2D curl as defined in section 15.4. Therefore, if we fix a point (a; b; c), [gx − fy](a; b; c), measures the rotation of F at the point (a; b; c) in the plane z = c. Similarly, [hy − gz](a; b; c) measures the rotation of F in the plane x = a at (a; b; c), and [fz − hx](a; b; c) measures the rotation of F in the plane y = b at the point (a; b; c). -
The Matrix Calculus You Need for Deep Learning
The Matrix Calculus You Need For Deep Learning Terence Parr and Jeremy Howard July 3, 2018 (We teach in University of San Francisco's MS in Data Science program and have other nefarious projects underway. You might know Terence as the creator of the ANTLR parser generator. For more material, see Jeremy's fast.ai courses and University of San Francisco's Data Institute in- person version of the deep learning course.) HTML version (The PDF and HTML were generated from markup using bookish) Abstract This paper is an attempt to explain all the matrix calculus you need in order to understand the training of deep neural networks. We assume no math knowledge beyond what you learned in calculus 1, and provide links to help you refresh the necessary math where needed. Note that you do not need to understand this material before you start learning to train and use deep learning in practice; rather, this material is for those who are already familiar with the basics of neural networks, and wish to deepen their understanding of the underlying math. Don't worry if you get stuck at some point along the way|just go back and reread the previous section, and try writing down and working through some examples. And if you're still stuck, we're happy to answer your questions in the Theory category at forums.fast.ai. Note: There is a reference section at the end of the paper summarizing all the key matrix calculus rules and terminology discussed here. arXiv:1802.01528v3 [cs.LG] 2 Jul 2018 1 Contents 1 Introduction 3 2 Review: Scalar derivative rules4 3 Introduction to vector calculus and partial derivatives5 4 Matrix calculus 6 4.1 Generalization of the Jacobian . -
Vector Calculus and Multiple Integrals Rob Fender, HT 2018
Vector Calculus and Multiple Integrals Rob Fender, HT 2018 COURSE SYNOPSIS, RECOMMENDED BOOKS Course syllabus (on which exams are based): Double integrals and their evaluation by repeated integration in Cartesian, plane polar and other specified coordinate systems. Jacobians. Line, surface and volume integrals, evaluation by change of variables (Cartesian, plane polar, spherical polar coordinates and cylindrical coordinates only unless the transformation to be used is specified). Integrals around closed curves and exact differentials. Scalar and vector fields. The operations of grad, div and curl and understanding and use of identities involving these. The statements of the theorems of Gauss and Stokes with simple applications. Conservative fields. Recommended Books: Mathematical Methods for Physics and Engineering (Riley, Hobson and Bence) This book is lazily referred to as “Riley” throughout these notes (sorry, Drs H and B) You will all have this book, and it covers all of the maths of this course. However it is rather terse at times and you will benefit from looking at one or both of these: Introduction to Electrodynamics (Griffiths) You will buy this next year if you haven’t already, and the chapter on vector calculus is very clear Div grad curl and all that (Schey) A nice discussion of the subject, although topics are ordered differently to most courses NB: the latest version of this book uses the opposite convention to polar coordinates to this course (and indeed most of physics), but older versions can often be found in libraries 1 Week One A review of vectors, rotation of coordinate systems, vector vs scalar fields, integrals in more than one variable, first steps in vector differentiation, the Frenet-Serret coordinate system Lecture 1 Vectors A vector has direction and magnitude and is written in these notes in bold e.g. -
Chapter 8 Change of Variables, Parametrizations, Surface Integrals
Chapter 8 Change of Variables, Parametrizations, Surface Integrals x0. The transformation formula In evaluating any integral, if the integral depends on an auxiliary function of the variables involved, it is often a good idea to change variables and try to simplify the integral. The formula which allows one to pass from the original integral to the new one is called the transformation formula (or change of variables formula). It should be noted that certain conditions need to be met before one can achieve this, and we begin by reviewing the one variable situation. Let D be an open interval, say (a; b); in R , and let ' : D! R be a 1-1 , C1 mapping (function) such that '0 6= 0 on D: Put D¤ = '(D): By the hypothesis on '; it's either increasing or decreasing everywhere on D: In the former case D¤ = ('(a);'(b)); and in the latter case, D¤ = ('(b);'(a)): Now suppose we have to evaluate the integral Zb I = f('(u))'0(u) du; a for a nice function f: Now put x = '(u); so that dx = '0(u) du: This change of variable allows us to express the integral as Z'(b) Z I = f(x) dx = sgn('0) f(x) dx; '(a) D¤ where sgn('0) denotes the sign of '0 on D: We then get the transformation formula Z Z f('(u))j'0(u)j du = f(x) dx D D¤ This generalizes to higher dimensions as follows: Theorem Let D be a bounded open set in Rn;' : D! Rn a C1, 1-1 mapping whose Jacobian determinant det(D') is everywhere non-vanishing on D; D¤ = '(D); and f an integrable function on D¤: Then we have the transformation formula Z Z Z Z ¢ ¢ ¢ f('(u))j det D'(u)j du1::: dun = ¢ ¢ ¢ f(x) dx1::: dxn: D D¤ 1 Of course, when n = 1; det D'(u) is simply '0(u); and we recover the old formula. -
Surface Regularized Geometry Estimation from a Single Image
SURGE: Surface Regularized Geometry Estimation from a Single Image Peng Wang1 Xiaohui Shen2 Bryan Russell2 Scott Cohen2 Brian Price2 Alan Yuille3 1University of California, Los Angeles 2Adobe Research 3Johns Hopkins University Abstract This paper introduces an approach to regularize 2.5D surface normal and depth predictions at each pixel given a single input image. The approach infers and reasons about the underlying 3D planar surfaces depicted in the image to snap predicted normals and depths to inferred planar surfaces, all while maintaining fine detail within objects. Our approach comprises two components: (i) a four- stream convolutional neural network (CNN) where depths, surface normals, and likelihoods of planar region and planar boundary are predicted at each pixel, followed by (ii) a dense conditional random field (DCRF) that integrates the four predictions such that the normals and depths are compatible with each other and regularized by the planar region and planar boundary information. The DCRF is formulated such that gradients can be passed to the surface normal and depth CNNs via backpropagation. In addition, we propose new planar-wise metrics to evaluate geometry consistency within planar surfaces, which are more tightly related to dependent 3D editing applications. We show that our regularization yields a 30% relative improvement in planar consistency on the NYU v2 dataset [24]. 1 Introduction Recent efforts to estimate the 2.5D layout of a depicted scene from a single image, such as per-pixel depths and surface normals, have yielded high-quality outputs respecting both the global scene layout and fine object detail [2, 6, 7, 29]. Upon closer inspection, however, the predicted depths and normals may fail to be consistent with the underlying surface geometry. -
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. -
CS 468 (Spring 2013) — Discrete Differential Geometry 1 the Unit Normal Vector of a Surface. 2 Surface Area
CS 468 (Spring 2013) | Discrete Differential Geometry Lecture 7 Student Notes: The Second Fundamental Form Lecturer: Adrian Butscher; Scribe: Soohark Chung 1 The unit normal vector of a surface. Figure 1: Normal vector of level set. • The normal line is a geometric feature. The normal direction is not (think of a non-orientable surfaces such as a mobius strip). If possible, you want to pick normal directions that are consistent globally. For example, for a manifold, you can pick normal directions pointing out. • Locally, you can find a normal direction using tangent vectors (though you can't extend this globally). • Normal vector of a parametrized surface: E1 × E2 If TpS = spanfE1;E2g then N := kE1 × E2k This is just the cross product of two tangent vectors normalized by it's length. Because you take the cross product of two vectors, it is orthogonal to the tangent plane. You can see that your choice of tangent vectors and their order determines the direction of the normal vector. • Normal vector of a level set: > [DFp] N := ? TpS kDFpk This is because the gradient at any point is perpendicular to the level set. This makes sense intuitively if you remember that the gradient is the "direction of the greatest increase" and that the value of the level set function stays constant along the surface. Of course, you also have to normalize it to be unit length. 2 Surface Area. • We want to be able to take the integral of the surface. One approach to the problem may be to inegrate a parametrized surface in the parameter domain. -
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. -
Multivariable and Vector Calculus
Multivariable and Vector Calculus Lecture Notes for MATH 0200 (Spring 2015) Frederick Tsz-Ho Fong Department of Mathematics Brown University Contents 1 Three-Dimensional Space ....................................5 1.1 Rectangular Coordinates in R3 5 1.2 Dot Product7 1.3 Cross Product9 1.4 Lines and Planes 11 1.5 Parametric Curves 13 2 Partial Differentiations ....................................... 19 2.1 Functions of Several Variables 19 2.2 Partial Derivatives 22 2.3 Chain Rule 26 2.4 Directional Derivatives 30 2.5 Tangent Planes 34 2.6 Local Extrema 36 2.7 Lagrange’s Multiplier 41 2.8 Optimizations 46 3 Multiple Integrations ........................................ 49 3.1 Double Integrals in Rectangular Coordinates 49 3.2 Fubini’s Theorem for General Regions 53 3.3 Double Integrals in Polar Coordinates 57 3.4 Triple Integrals in Rectangular Coordinates 62 3.5 Triple Integrals in Cylindrical Coordinates 67 3.6 Triple Integrals in Spherical Coordinates 70 4 Vector Calculus ............................................ 75 4.1 Vector Fields on R2 and R3 75 4.2 Line Integrals of Vector Fields 83 4.3 Conservative Vector Fields 88 4.4 Green’s Theorem 98 4.5 Parametric Surfaces 105 4.6 Stokes’ Theorem 120 4.7 Divergence Theorem 127 5 Topics in Physics and Engineering .......................... 133 5.1 Coulomb’s Law 133 5.2 Introduction to Maxwell’s Equations 137 5.3 Heat Diffusion 141 5.4 Dirac Delta Functions 144 1 — Three-Dimensional Space 1.1 Rectangular Coordinates in R3 Throughout the course, we will use an ordered triple (x, y, z) to represent a point in the three dimensional space. -
Vector Fields
Vector Calculus Independent Study Unit 5: Vector Fields A vector field is a function which associates a vector to every point in space. Vector fields are everywhere in nature, from the wind (which has a velocity vector at every point) to gravity (which, in the simplest interpretation, would exert a vector force at on a mass at every point) to the gradient of any scalar field (for example, the gradient of the temperature field assigns to each point a vector which says which direction to travel if you want to get hotter fastest). In this section, you will learn the following techniques and topics: • How to graph a vector field by picking lots of points, evaluating the field at those points, and then drawing the resulting vector with its tail at the point. • A flow line for a velocity vector field is a path ~σ(t) that satisfies ~σ0(t)=F~(~σ(t)) For example, a tiny speck of dust in the wind follows a flow line. If you have an acceleration vector field, a flow line path satisfies ~σ00(t)=F~(~σ(t)) [For example, a tiny comet being acted on by gravity.] • Any vector field F~ which is equal to ∇f for some f is called a con- servative vector field, and f its potential. The terminology comes from physics; by the fundamental theorem of calculus for work in- tegrals, the work done by moving from one point to another in a conservative vector field doesn’t depend on the path and is simply the difference in potential at the two points.