Lecture 9: Partial Derivatives
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Differentiation Rules (Differential Calculus)
Differentiation Rules (Differential Calculus) 1. Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x. 2. Alternate Notations for (D f )(x) d d f (x) d f 0 (1) For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), dx , dx (x), f (x), f (x). The “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy 0 whereas f (x) is the value of it at x. If y = f (x), then Dxy, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f ,“D f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy 0 Historical note: Newton used y,˙ while Leibniz used dx . About a century later Lagrange introduced y and Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below. -
Policy Gradient
Lecture 7: Policy Gradient Lecture 7: Policy Gradient David Silver Lecture 7: Policy Gradient Outline 1 Introduction 2 Finite Difference Policy Gradient 3 Monte-Carlo Policy Gradient 4 Actor-Critic Policy Gradient Lecture 7: Policy Gradient Introduction Policy-Based Reinforcement Learning In the last lecture we approximated the value or action-value function using parameters θ, V (s) V π(s) θ ≈ Q (s; a) Qπ(s; a) θ ≈ A policy was generated directly from the value function e.g. using -greedy In this lecture we will directly parametrise the policy πθ(s; a) = P [a s; θ] j We will focus again on model-free reinforcement learning Lecture 7: Policy Gradient Introduction Value-Based and Policy-Based RL Value Based Learnt Value Function Implicit policy Value Function Policy (e.g. -greedy) Policy Based Value-Based Actor Policy-Based No Value Function Critic Learnt Policy Actor-Critic Learnt Value Function Learnt Policy Lecture 7: Policy Gradient Introduction Advantages of Policy-Based RL Advantages: Better convergence properties Effective in high-dimensional or continuous action spaces Can learn stochastic policies Disadvantages: Typically converge to a local rather than global optimum Evaluating a policy is typically inefficient and high variance Lecture 7: Policy Gradient Introduction Rock-Paper-Scissors Example Example: Rock-Paper-Scissors Two-player game of rock-paper-scissors Scissors beats paper Rock beats scissors Paper beats rock Consider policies for iterated rock-paper-scissors A deterministic policy is easily exploited A uniform random policy -
Introduction to Shape Optimization
Introduction to Shape optimization Noureddine Igbida1 1Institut de recherche XLIM, UMR-CNRS 6172, Facult´edes Sciences et Techniques, Universit´ede Limoges 123, Avenue Albert Thomas 87060 Limoges, France. Email : [email protected] Preliminaries on PDE 1 Contents 1. PDE ........................................ 2 2. Some differentiation and differential operators . 3 2.1 Gradient . 3 2.2 Divergence . 4 2.3 Curl . 5 2 2.4 Remarks . 6 2.5 Laplacian . 7 2.6 Coordinate expressions of the Laplacian . 10 3. Boundary value problem . 12 4. Notion of solution . 16 1. PDE A mathematical model is a description of a system using mathematical language. The process of developing a mathematical model is termed mathematical modelling (also written model- ing). Mathematical models are used in many area, as in the natural sciences (such as physics, biology, earth science, meteorology), engineering disciplines (e.g. computer science, artificial in- telligence), in the social sciences (such as economics, psychology, sociology and political science); Introduction to Shape optimization N. Igbida physicists, engineers, statisticians, operations research analysts and economists. Among this mathematical language we have PDE. These are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic. In this section, we give some basic example of elliptic partial differential equation (PDE) of second order : standard Laplacian and Laplacian with variable coefficients. -
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 -
CHAPTER 3: Derivatives
CHAPTER 3: Derivatives 3.1: Derivatives, Tangent Lines, and Rates of Change 3.2: Derivative Functions and Differentiability 3.3: Techniques of Differentiation 3.4: Derivatives of Trigonometric Functions 3.5: Differentials and Linearization of Functions 3.6: Chain Rule 3.7: Implicit Differentiation 3.8: Related Rates • Derivatives represent slopes of tangent lines and rates of change (such as velocity). • In this chapter, we will define derivatives and derivative functions using limits. • We will develop short cut techniques for finding derivatives. • Tangent lines correspond to local linear approximations of functions. • Implicit differentiation is a technique used in applied related rates problems. (Section 3.1: Derivatives, Tangent Lines, and Rates of Change) 3.1.1 SECTION 3.1: DERIVATIVES, TANGENT LINES, AND RATES OF CHANGE LEARNING OBJECTIVES • Relate difference quotients to slopes of secant lines and average rates of change. • Know, understand, and apply the Limit Definition of the Derivative at a Point. • Relate derivatives to slopes of tangent lines and instantaneous rates of change. • Relate opposite reciprocals of derivatives to slopes of normal lines. PART A: SECANT LINES • For now, assume that f is a polynomial function of x. (We will relax this assumption in Part B.) Assume that a is a constant. • Temporarily fix an arbitrary real value of x. (By “arbitrary,” we mean that any real value will do). Later, instead of thinking of x as a fixed (or single) value, we will think of it as a “moving” or “varying” variable that can take on different values. The secant line to the graph of f on the interval []a, x , where a < x , is the line that passes through the points a, fa and x, fx. -
Vector Calculus and Differential Forms with Applications To
Vector Calculus and Differential Forms with Applications to Electromagnetism Sean Roberson May 7, 2015 PREFACE This paper is written as a final project for a course in vector analysis, taught at Texas A&M University - San Antonio in the spring of 2015 as an independent study course. Students in mathematics, physics, engineering, and the sciences usually go through a sequence of three calculus courses before go- ing on to differential equations, real analysis, and linear algebra. In the third course, traditionally reserved for multivariable calculus, stu- dents usually learn how to differentiate functions of several variable and integrate over general domains in space. Very rarely, as was my case, will professors have time to cover the important integral theo- rems using vector functions: Green’s Theorem, Stokes’ Theorem, etc. In some universities, such as UCSD and Cornell, honors students are able to take an accelerated calculus sequence using the text Vector Cal- culus, Linear Algebra, and Differential Forms by John Hamal Hubbard and Barbara Burke Hubbard. Here, students learn multivariable cal- culus using linear algebra and real analysis, and then they generalize familiar integral theorems using the language of differential forms. This paper was written over the course of one semester, where the majority of the book was covered. Some details, such as orientation of manifolds, topology, and the foundation of the integral were skipped to save length. The paper should still be readable by a student with at least three semesters of calculus, one course in linear algebra, and one course in real analysis - all at the undergraduate level. -
Antiderivatives 307
4100 AWL/Thomas_ch04p244-324 8/20/04 9:02 AM Page 307 4.8 Antiderivatives 307 4.8 Antiderivatives We have studied how to find the derivative of a function. However, many problems re- quire that we recover a function from its known derivative (from its known rate of change). For instance, we may know the velocity function of an object falling from an initial height and need to know its height at any time over some period. More generally, we want to find a function F from its derivative ƒ. If such a function F exists, it is called an anti- derivative of ƒ. Finding Antiderivatives DEFINITION Antiderivative A function F is an antiderivative of ƒ on an interval I if F¿sxd = ƒsxd for all x in I. The process of recovering a function F(x) from its derivative ƒ(x) is called antidiffer- entiation. We use capital letters such as F to represent an antiderivative of a function ƒ, G to represent an antiderivative of g, and so forth. EXAMPLE 1 Finding Antiderivatives Find an antiderivative for each of the following functions. (a) ƒsxd = 2x (b) gsxd = cos x (c) hsxd = 2x + cos x Solution (a) Fsxd = x2 (b) Gsxd = sin x (c) Hsxd = x2 + sin x Each answer can be checked by differentiating. The derivative of Fsxd = x2 is 2x. The derivative of Gsxd = sin x is cos x and the derivative of Hsxd = x2 + sin x is 2x + cos x. The function Fsxd = x2 is not the only function whose derivative is 2x. The function x2 + 1 has the same derivative. -
Numerical Differentiation
Jim Lambers MAT 460/560 Fall Semester 2009-10 Lecture 23 Notes These notes correspond to Section 4.1 in the text. Numerical Differentiation We now discuss the other fundamental problem from calculus that frequently arises in scientific applications, the problem of computing the derivative of a given function f(x). Finite Difference Approximations 0 Recall that the derivative of f(x) at a point x0, denoted f (x0), is defined by 0 f(x0 + h) − f(x0) f (x0) = lim : h!0 h 0 This definition suggests a method for approximating f (x0). If we choose h to be a small positive constant, then f(x + h) − f(x ) f 0(x ) ≈ 0 0 : 0 h This approximation is called the forward difference formula. 00 To estimate the accuracy of this approximation, we note that if f (x) exists on [x0; x0 + h], 0 00 2 then, by Taylor's Theorem, f(x0 +h) = f(x0)+f (x0)h+f ()h =2; where 2 [x0; x0 +h]: Solving 0 for f (x0), we obtain f(x + h) − f(x ) f 00() f 0(x ) = 0 0 + h; 0 h 2 so the error in the forward difference formula is O(h). We say that this formula is first-order accurate. 0 The forward-difference formula is called a finite difference approximation to f (x0), because it approximates f 0(x) using values of f(x) at points that have a small, but finite, distance between them, as opposed to the definition of the derivative, that takes a limit and therefore computes the derivative using an “infinitely small" value of h. -
Review of Differentiation and Integration
Schreyer Fall 2018 Review of Differentiation and Integration for Ordinary Differential Equations In this course you will be expected to be able to differentiate and integrate quickly and accurately. Many students take this course after having taken their previous course many years ago, at another institution where certain topics may have been omitted, or just feel uncomfortable with particular techniques. Because understanding this material is so important to being successful in this course, we have put together this brief review packet. In this packet you will find sample questions and a brief discussion of each topic. If you find the material in this pamphlet is not sufficient for you, it may be necessary for you to use additional resources, such as a calculus textbook or online materials. Because this is considered prerequisite material, it is ultimately your responsibility to learn it. The topics to be covered include Differentiation and Integration. 1 Differentiation Exercises: 1. Find the derivative of y = x3 sin(x). ln(x) 2. Find the derivative of y = cos(x) . 3. Find the derivative of y = ln(sin(e2x)). Discussion: It is expected that you know, without looking at a table, the following differentiation rules: d [(kx)n] = kn(kx)n−1 (1) dx d h i ekx = kekx (2) dx d 1 [ln(kx)] = (3) dx x d [sin(kx)] = k cos kx (4) dx d [cos(kx)] = −k sin x (5) dx d [uv] = u0v + uv0 (6) dx 1 d u u0v − uv0 = (7) dx v v2 d [u(v(x))] = u0(v)v0(x): (8) dx We put in the constant k into (1) - (5) because a very common mistake to make is something d e2x like: e2x = (when the correct answer is 2e2x). -
Just the Definitions, Theorem Statements, Proofs On
MATH 3333{INTERMEDIATE ANALYSIS{BLECHER NOTES 55 Abbreviated notes version for Test 3: just the definitions, theorem statements, proofs on the list. I may possibly have made a mistake, so check it. Also, this is not intended as a REPLACEMENT for your classnotes; the classnotes have lots of other things that you may need for your understanding, like worked examples. 6. The derivative 6.1. Differentiation rules. Definition: Let f :(a; b) ! R and c 2 (a; b). If the limit lim f(x)−f(c) exists and is finite, then we say that f is differentiable at x!c x−c 0 df c, and we write this limit as f (c) or dx (c). This is the derivative of f at c, and also obviously equals lim f(c+h)−f(c) by setting x = c + h or h = x − c. If f is h!0 h differentiable at every point in (a; b), then we say that f is differentiable on (a; b). Theorem 6.1. If f :(a; b) ! R is differentiable at at a point c 2 (a; b), then f is continuous at c. Proof. If f is differentiable at c then f(x) − f(c) f(x) = (x − c) + f(c) ! f 0(c)0 + f(c) = f(c); x − c as x ! c. So f is continuous at c. Theorem 6.2. (Calculus I differentiation laws) If f; g :(a; b) ! R is differentiable at a point c 2 (a; b), then (1) f(x) + g(x) is differentiable at c and (f + g)0(c) = f 0(c) + g0(c). -
MATH 162: Calculus II Differentiation
MATH 162: Calculus II Framework for Mon., Jan. 29 Review of Differentiation and Integration Differentiation Definition of derivative f 0(x): f(x + h) − f(x) f(y) − f(x) lim or lim . h→0 h y→x y − x Differentiation rules: 1. Sum/Difference rule: If f, g are differentiable at x0, then 0 0 0 (f ± g) (x0) = f (x0) ± g (x0). 2. Product rule: If f, g are differentiable at x0, then 0 0 0 (fg) (x0) = f (x0)g(x0) + f(x0)g (x0). 3. Quotient rule: If f, g are differentiable at x0, and g(x0) 6= 0, then 0 0 0 f f (x0)g(x0) − f(x0)g (x0) (x0) = 2 . g [g(x0)] 4. Chain rule: If g is differentiable at x0, and f is differentiable at g(x0), then 0 0 0 (f ◦ g) (x0) = f (g(x0))g (x0). This rule may also be expressed as dy dy du = . dx x=x0 du u=u(x0) dx x=x0 Implicit differentiation is a consequence of the chain rule. For instance, if y is really dependent upon x (i.e., y = y(x)), and if u = y3, then d du du dy d (y3) = = = (y3)y0(x) = 3y2y0. dx dx dy dx dy Practice: Find d x d √ d , (x2 y), and [y cos(xy)]. dx y dx dx MATH 162—Framework for Mon., Jan. 29 Review of Differentiation and Integration Integration The definite integral • the area problem • Riemann sums • definition Fundamental Theorem of Calculus: R x I: Suppose f is continuous on [a, b]. -
Using Surface Integrals for Checking the Archimedes' Law of Buoyancy
Using surface integrals for checking the Archimedes’ law of buoyancy F M S Lima Institute of Physics, University of Brasilia, P.O. Box 04455, 70919-970, Brasilia-DF, Brazil E-mail: [email protected] Abstract. A mathematical derivation of the force exerted by an inhomogeneous (i.e., compressible) fluid on the surface of an arbitrarily-shaped body immersed in it is not found in literature, which may be attributed to our trust on Archimedes’ law of buoyancy. However, this law, also known as Archimedes’ principle (AP), does not yield the force observed when the body is in contact to the container walls, as is more evident in the case of a block immersed in a liquid and in contact to the bottom, in which a downward force that increases with depth is observed. In this work, by taking into account the surface integral of the pressure force exerted by a fluid over the surface of a body, the general validity of AP is checked. For a body fully surrounded by a fluid, homogeneous or not, a gradient version of the divergence theorem applies, yielding a volume integral that simplifies to an upward force which agrees to the force predicted by AP, as long as the fluid density is a continuous function of depth. For the bottom case, this approach yields a downward force that increases with depth, which contrasts to AP but is in agreement to experiments. It also yields a formula for this force which shows that it increases with the area of contact. PACS numbers: 01.30.mp, 01.55.+b, 47.85.Dh Submitted to: Eur.