Product Rule of Three Terms
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DERIVATIONS and PROJECTIONS on JORDAN TRIPLES an Introduction to Nonassociative Algebra, Continuous Cohomology, and Quantum Functional Analysis
DERIVATIONS AND PROJECTIONS ON JORDAN TRIPLES An introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis Bernard Russo July 29, 2014 This paper is an elaborated version of the material presented by the author in a three hour minicourse at V International Course of Mathematical Analysis in Andalusia, at Almeria, Spain September 12-16, 2011. The author wishes to thank the scientific committee for the opportunity to present the course and to the organizing committee for their hospitality. The author also personally thanks Antonio Peralta for his collegiality and encouragement. The minicourse on which this paper is based had its genesis in a series of talks the author had given to undergraduates at Fullerton College in California. I thank my former student Dana Clahane for his initiative in running the remarkable undergraduate research program at Fullerton College of which the seminar series is a part. With their knowledge only of the product rule for differentiation as a starting point, these enthusiastic students were introduced to some aspects of the esoteric subject of non associative algebra, including triple systems as well as algebras. Slides of these talks and of the minicourse lectures, as well as other related material, can be found at the author's website (www.math.uci.edu/∼brusso). Conversely, these undergraduate talks were motivated by the author's past and recent joint works on derivations of Jordan triples ([116],[117],[200]), which are among the many results discussed here. Part I (Derivations) is devoted to an exposition of the properties of derivations on various algebras and triple systems in finite and infinite dimensions, the primary questions addressed being whether the derivation is automatically continuous and to what extent it is an inner derivation. -
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. -
Notes on Calculus II Integral Calculus Miguel A. Lerma
Notes on Calculus II Integral Calculus Miguel A. Lerma November 22, 2002 Contents Introduction 5 Chapter 1. Integrals 6 1.1. Areas and Distances. The Definite Integral 6 1.2. The Evaluation Theorem 11 1.3. The Fundamental Theorem of Calculus 14 1.4. The Substitution Rule 16 1.5. Integration by Parts 21 1.6. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. Partial Fractions 32 1.8. Integration using Tables and CAS 39 1.9. Numerical Integration 41 1.10. Improper Integrals 46 Chapter 2. Applications of Integration 50 2.1. More about Areas 50 2.2. Volumes 52 2.3. Arc Length, Parametric Curves 57 2.4. Average Value of a Function (Mean Value Theorem) 61 2.5. Applications to Physics and Engineering 63 2.6. Probability 69 Chapter 3. Differential Equations 74 3.1. Differential Equations and Separable Equations 74 3.2. Directional Fields and Euler’s Method 78 3.3. Exponential Growth and Decay 80 Chapter 4. Infinite Sequences and Series 83 4.1. Sequences 83 4.2. Series 88 4.3. The Integral and Comparison Tests 92 4.4. Other Convergence Tests 96 4.5. Power Series 98 4.6. Representation of Functions as Power Series 100 4.7. Taylor and MacLaurin Series 103 3 CONTENTS 4 4.8. Applications of Taylor Polynomials 109 Appendix A. Hyperbolic Functions 113 A.1. Hyperbolic Functions 113 Appendix B. Various Formulas 118 B.1. Summation Formulas 118 Appendix C. Table of Integrals 119 Introduction These notes are intended to be a summary of the main ideas in course MATH 214-2: Integral Calculus. -
Lecture 9: Partial Derivatives
Math S21a: Multivariable calculus Oliver Knill, Summer 2016 Lecture 9: Partial derivatives ∂ If f(x,y) is a function of two variables, then ∂x f(x,y) is defined as the derivative of the function g(x) = f(x,y) with respect to x, where y is considered a constant. It is called the partial derivative of f with respect to x. The partial derivative with respect to y is defined in the same way. ∂ We use the short hand notation fx(x,y) = ∂x f(x,y). For iterated derivatives, the notation is ∂ ∂ similar: for example fxy = ∂x ∂y f. The meaning of fx(x0,y0) is the slope of the graph sliced at (x0,y0) in the x direction. The second derivative fxx is a measure of concavity in that direction. The meaning of fxy is the rate of change of the slope if you change the slicing. The notation for partial derivatives ∂xf,∂yf was introduced by Carl Gustav Jacobi. Before, Josef Lagrange had used the term ”partial differences”. Partial derivatives fx and fy measure the rate of change of the function in the x or y directions. For functions of more variables, the partial derivatives are defined in a similar way. 4 2 2 4 3 2 2 2 1 For f(x,y)= x 6x y + y , we have fx(x,y)=4x 12xy ,fxx = 12x 12y ,fy(x,y)= − − − 12x2y +4y3,f = 12x2 +12y2 and see that f + f = 0. A function which satisfies this − yy − xx yy equation is also called harmonic. The equation fxx + fyy = 0 is an example of a partial differential equation: it is an equation for an unknown function f(x,y) which involves partial derivatives with respect to more than one variables. -
Anti-Chain Rule Name
Anti-Chain Rule Name: The most fundamental rule for computing derivatives in calculus is the Chain Rule, from which all other differentiation rules are derived. In words, the Chain Rule states that, to differentiate f(g(x)), differentiate the outside function f and multiply by the derivative of the inside function g, so that d f(g(x)) = f 0(g(x)) · g0(x): dx The Chain Rule makes computation of nearly all derivatives fairly easy. It is tempting to think that there should be an Anti-Chain Rule that makes antidiffer- entiation as easy as the Chain Rule made differentiation. Such a rule would just reverse the process, so instead of differentiating f and multiplying by g0, we would antidifferentiate f and divide by g0. In symbols, Z F (g(x)) f(g(x)) dx = + C: g0(x) Unfortunately, the Anti-Chain Rule does not exist because the formula above is not Z always true. For example, consider (x2 + 1)2 dx. We can compute this integral (correctly) by expanding and applying the power rule: Z Z (x2 + 1)2 dx = x4 + 2x2 + 1 dx x5 2x3 = + + x + C 5 3 However, we could also think about our integrand above as f(g(x)) with f(x) = x2 and g(x) = x2 + 1. Then we have F (x) = x3=3 and g0(x) = 2x, so the Anti-Chain Rule would predict that Z F (g(x)) (x2 + 1)2 dx = + C g0(x) (x2 + 1)3 = + C 3 · 2x x6 + 3x4 + 3x2 + 1 = + C 6x x5 x3 x 1 = + + + + C 6 2 2 6x Since these answers are different, we conclude that the Anti-Chain Rule has failed us. -
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. -
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. -
Laplace Transform
Chapter 7 Laplace Transform The Laplace transform can be used to solve differential equations. Be- sides being a different and efficient alternative to variation of parame- ters and undetermined coefficients, the Laplace method is particularly advantageous for input terms that are piecewise-defined, periodic or im- pulsive. The direct Laplace transform or the Laplace integral of a function f(t) defined for 0 ≤ t< 1 is the ordinary calculus integration problem 1 f(t)e−stdt; Z0 succinctly denoted L(f(t)) in science and engineering literature. The L{notation recognizes that integration always proceeds over t = 0 to t = 1 and that the integral involves an integrator e−stdt instead of the usual dt. These minor differences distinguish Laplace integrals from the ordinary integrals found on the inside covers of calculus texts. 7.1 Introduction to the Laplace Method The foundation of Laplace theory is Lerch's cancellation law 1 −st 1 −st 0 y(t)e dt = 0 f(t)e dt implies y(t)= f(t); (1) R R or L(y(t)= L(f(t)) implies y(t)= f(t): In differential equation applications, y(t) is the sought-after unknown while f(t) is an explicit expression taken from integral tables. Below, we illustrate Laplace's method by solving the initial value prob- lem y0 = −1; y(0) = 0: The method obtains a relation L(y(t)) = L(−t), whence Lerch's cancel- lation law implies the solution is y(t)= −t. The Laplace method is advertised as a table lookup method, in which the solution y(t) to a differential equation is found by looking up the answer in a special integral table. -
9.2. Undoing the Chain Rule
< Previous Section Home Next Section > 9.2. Undoing the Chain Rule This chapter focuses on finding accumulation functions in closed form when we know its rate of change function. We’ve seen that 1) an accumulation function in closed form is advantageous for quickly and easily generating many values of the accumulating quantity, and 2) the key to finding accumulation functions in closed form is the Fundamental Theorem of Calculus. The FTC says that an accumulation function is the antiderivative of the given rate of change function (because rate of change is the derivative of accumulation). In Chapter 6, we developed many derivative rules for quickly finding closed form rate of change functions from closed form accumulation functions. We also made a connection between the form of a rate of change function and the form of the accumulation function from which it was derived. Rather than inventing a whole new set of techniques for finding antiderivatives, our mindset as much as possible will be to use our derivatives rules in reverse to find antiderivatives. We worked hard to develop the derivative rules, so let’s keep using them to find antiderivatives! Thus, whenever you have a rate of change function f and are charged with finding its antiderivative, you should frame the task with the question “What function has f as its derivative?” For simple rate of change functions, this is easy, as long as you know your derivative rules well enough to apply them in reverse. For example, given a rate of change function …. … 2x, what function has 2x as its derivative? i.e. -
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. -
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 ∇ .