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# Desmos Double Integral Calculator

They are mostly standard functions written as you might expect. Type in any integral to get the solution, steps and graph This website uses cookies to ensure you get the best experience. series for negative integral exponents. It is suitable for all types of users, whether they are looking to carry out basic or advanced mathematics; Graphing Calculator has it all. Online Scientific Calculator. Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. How to Calculate Variance. I find this simple idea gets students talking to each other more, and becoming much more selective over their choice of questions. S is the sensitivity factor. Final integral value is the sum of integral for each partial intervals. Functions like sin and sqrt work, as do special constants like pi and e. Cornering ABS makes use of the Bosch IMU (Inertial Measurement Unit) platform to optimise front and rear braking power even in critical situations with the bike at considerable lean angles. And the integral of the pseudo-gravitationally time dilated tick-rate over the turnaround-time is pre-defined in the scenarion, as the acceleration of the traveller's frame shall change his velocity in the stationary twin's frame from +v to -v in a certain approximate distance. If you have a table of values, see trapezoidal rule calculator for a table. 5: 5978: 81: dy/dx integral calculator. Examples: sinc exp dome. The answer is 7. If there’s a button for it on a traditional calculator, there’s probably a way to type it now. A particular solution to a differential equation corresponding to a specific value of the equation's free parameters. So, do I need to find functions for f(x) or is there something else?. You can also get a better visual and understanding of the function and area under the curve using our graphing tool. Enter your expressions in the same order as you would write them on paper. Therefore, each table represents a ratio. Description. To save time in simplifying straight forward equations and questions, it’s best that we make use of equation solvers available online. It is suitable for all types of users, whether they are looking to carry out basic or advanced mathematics; Graphing Calculator has it all. Integrand function is replaced by the Lagrange polynomials of different degree, integration of which yields the numerical integration formulas with different degree of accuracy. Standard Integrals 116 54. Download The. Calculations at a cardioid (heart-shaped curve), an epicycloid with one arc. scrolling and zooming the displayed range, built-in function calculator, numerical integration, solving differential equations numerically. The EXP function is a built-in function in Excel that is categorized as a Math/Trig Function. The x-intercept is the point at which the graph crosses the x-axis. GraphyCalc is a 3d graphing calculator, allowing you to interactively explore 3d functions. Metric units worksheet. Integral Calculator. A mechanical device that computes area integrals is the planimeter, which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element linkage effects Green's theorem, converting the quadratic polar integral to a linear integral. Use Desmos to investigate the beautiful world of integral calculus. An antiderivative is a function that reverses what the derivative does. 0 m and the time is 1. It starts with getting something as simple as sin(1/x) or x/(x^2 + 5x + 1) to plot reasonably well, and gets harder from there soon (edit: if you want to complicate things, pick a function. Definition of a vertical asymptote: The line x = x 0 is a "vertical asymptote" of f(x) if and only if f(x) approaches + or - as x approaches x 0 from the left or from the right. Simply enter in as long of a string of numbers to average that you like into the box and separate the numbers by a comma and then press calculate to get the average of all of the numbers. You may see all of our other area calculations by going to our Area Calculator. A cardioid is formed by a circle of the diameter a, which adjacently rolls around another circle of the same size. Taylor's. Functions 3D Plotter is an application to drawing functions of several variables and surface in the space R3 and to calculate indefinite integrals or definite integrals. Double facts worksheets. This is a formula calculator. This is a tentative schedule that may be changed based on the needs of the students, the needs of the instructor, or any apocalyptic events that may occur during the semester. Indefinite and definite integrals, answers, alternate forms. The initial velocity is 14 m/s the final velocity is 0. Triple Integral Calculator. Description. Type in any integral to get the solution, free steps and graph This website uses cookies to ensure you get the best experience. Note, that integral expression may seems a little different in inline and display math mode - in inline mode the integral symbol and the limits are compressed. CE Calculators > Moment of Inertia Calculator for plane sections (rectangular, circular, pentagon, hexagon, Channel, Angle, T and H sections Moment of inertia or second moment of area is important for determining the strength of beams and columns of a structural system. You are going to get an approximate value of 20. The calculator will approximate the integral using the trapezoidal rule, with steps shown. • Graph in polar coordinates. One function has many antiderivatives, but they all take the form of a function plus an arbitrary constant. Antiderivatives are a key part of indefinite integrals. Then fill in the lower bound, upper bound, and the integrand and you’ll be on your way! Evaluate a definite integral in an instant, or plot an integral with varying bounds. Radical Equation:. By using this website, you agree to our Cookie Policy. Keyword Research: People who searched integral calculator dy also searched. Resources Learn Desmos Accessibility Assessments Partnerships API Docs. Units: 4 Prerequisite: Completion of MATH 16A or 30 with grade of "C" or better Advisory: Completion of MATH 27 with grade of "C" or better Hours: 72 lecture Differentiation and integration of trigonometric functions, functions of several variables, partial derivatives, double integrals, introduction to differential equations, sequences and. Available as a mobile and desktop website as well as native iOS and Android apps. Properties of the Integral 116 55. No Download or Signup. Why are they equal? Find the direction of the vectors u = < 2 , 5 > and v = < - 2 , - 5 >. Download The. Now try to divide 1 by an even bigger number. com/calculator). Teachers can create a class with student logins - and monitor student progress. Antiderivatives are the opposite of derivatives. In some cases, inverse trigonometric functions are valuable. Surface Integrals of Vector Fields. The Algorithm The bisection method is an algorithm, and we will explain it in terms of its steps. Slope Field Calculator. "It's not like a graphing calculator app is going to be that difficult to program" It is surprisingly hard to get something that produces reliable plots. Suppose that $$f(x,y)$$ assumes both positive and negatives values on the rectangle $$R\text{,}$$ as shown on the left of Figure 11. By using this website, you agree to our Cookie Policy. Go to: Double Integral — Triple Integral Tool to calculate the integral of a function. Description: Given a closed interval [a,b] on which f changes sign, we divide the interval in half and note that f must change sign on either the right or the left half (or be zero at the midpoint of [a,b]. Karl's Calculus Tutor : Related Rates. So I just finished my CALC 2 class with the highest marks in the class and I wanted to put together a “guide” on how to study for calculus. com/calculator). The x-intercept is the point at which the graph crosses the x-axis. Visual Calculus is a powerful tool to compute and graph limit, derivative, integral, 3D vector, partial derivative function, double integral, triple integral, series, ODE etc. The computation of an definite integral over an interval consist in measuring the area under the curve of the function to integrate. Double Integrals Help Links: S. CE Calculators > Moment of Inertia Calculator for plane sections (rectangular, circular, pentagon, hexagon, Channel, Angle, T and H sections Moment of inertia or second moment of area is important for determining the strength of beams and columns of a structural system. Learn more. Free definite integral calculator - solve definite integrals with all the steps. Hints help you try the next step on your own. Complementary and supplementary worksheet. Easy to use and 100% Free! We also have several other calculators. 1/ 4 – 1/ the integral 1 to 4, 4x of ln of 3x dx. If you have a table of values, see Simpson's rule calculator for a table. In other words interpolation is the technique to estimate the value of a mathematical function, for any intermediate value of the independent variable. The Integral Calculator supports definite and indefinite integrals (antiderivatives) as well as integrating functions with many variables. Integrand function is replaced by the Lagrange polynomials of different degree, integration of which yields the numerical integration formulas with different degree of accuracy. Download The. Why are they equal? Find the direction of the vectors u = < 2 , 5 > and v = < - 2 , - 5 >. [11] For negative values of x , the terms integral part or integer part of x are sometimes instead taken to be the value of the ceiling function, i. Student Loan Calculator – Was made by the College Board to make it easier for college students to stay on top for their student loan payments. The cardioid has the diameter 2a on its symmetry axis. If desired to convert a 2D cylindrical coordinate, then the user just enters values into the r and φ form fields and leaves the 3rd field, the z field, blank. s = rθ, when θ is measured in radians “Life is full of circles. ” — Nora Roberts. Simply enter in as long of a string of numbers to average that you like into the box and separate the numbers by a comma and then press calculate to get the average of all of the numbers. Then, fill in the lower bound, upper bound, and the integrand and you'll be on your way! You can find the integral on the Desmos keyboard by clicking on FUNCTIONSand then MISC. Slope Field Calculator. Look below to see them all. Plot the polar coordinates graph keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in addition you can see which keywords most interested customers on the this website. Of course, any monotonic function with constant unequal horizontal asymptotes is a Heaviside step function under appropriate scaling and possible reflection. Here's what I don't get/don't know what supposed to do: I tried graphing this on desmos and found it REALLY hard to find functions that would work and I'm guessing that the tangents would have to be close to (0,0). Comparing rates worksheet. A typical application of the logistic equation is a common model of population growth (see also population dynamics), originally due to Pierre-François Verhulst in 1838, where the rate of reproduction is proportional to both the existing population and the amount of available resources, all else being equal. Problems are also posted on Twitter using the hashtag #mathpotd. If you have the calculator set to radian mode, you get tan − 1 ( 10 3 ) ≈ 1. Type in any integral to get the solution, free steps and graph This website uses cookies to ensure you get the best experience. Information; Calendar; Calendar. Antiderivatives are a key part of indefinite integrals. , the volume of a geometrical figure) or even a triple integral (e. ) The first is called the cardioid (“heart-like”) and the second is the deltoid (“triangle-like”). Mines are rare with only 6 of them in the game but they can generate big scores. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. , the total mass of a metal rod). One function has many antiderivatives, but they all take the form of a function plus an arbitrary constant. Description: Given a closed interval [a,b] on which f changes sign, we divide the interval in half and note that f must change sign on either the right or the left half (or be zero at the midpoint of [a,b]. x n into equal line segments. Type in any integral to get the solution, steps and graph This website uses cookies to ensure you get the best experience. Why are they equal? Find the direction of the vectors u = < 2 , 5 > and v = < - 2 , - 5 >. Integral expression can be added using the \int_{lower}^{upper} command. Free volume of solid of revolution calculator - find volume of solid of revolution step-by-step This website uses cookies to ensure you get the best experience. , the volume of a geometrical figure) or even a triple integral (e. From Zurab's integral representation for the Apéry's constant to almost impossible integrals I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). Compute volumes under surfaces, surface area and other types of two-dimensional integrals using Wolfram|Alpha's double integral calculator. Enter the length of all three sides. Cornering ABS makes use of the Bosch IMU (Inertial Measurement Unit) platform to optimise front and rear braking power even in critical situations with the bike at considerable lean angles. Equations reducible to quadratic form calculator. The seventh and final line does the double integral required. I’m not exactly sure to be honest. No Download or Signup. The deﬁnite integral as a function of its integration bounds 117 56. Free area under between curves calculator - find area between functions step-by-step This website uses cookies to ensure you get the best experience. Technology’s Role. (I can also tell that this must be a geometric series because of the form given for each term: as the index increases, each term will be multiplied by an additional factor of –2. By using this website, you agree to our Cookie Policy. Only few simple trigonometric equations can be solved without any use of calculator but not at all. Please pick the appropriate calculator from below to begin. com has ranked N/A in N/A and 4,015,265 on the world. Geometry worksheets. Visual Calculus. Stage 7: Double-Blind Group Peer Review Students use a feedback form to work as group to provide feedback to anonymous papers. It can also be used for calculating the length of curves and other approximations too. Desmos mostly just saved me from writing out the horrible quadratic solution but it does also give a reassuring check that I made no slips. Type in any integral to get the solution, free steps and graph This website uses cookies to ensure you get the best experience. Integrated review can be used in corequisite courses, or simply to help students who enter without a full understanding of prerequisite skills and concepts. Functions like sin and sqrt work, as do special constants like pi and e. The definite integral as a function of its integration bounds (VII. Solve this integral of a sharply peaked function without a calculator. Integration is often introduced as the reverse process to differentiation, and has wide applications, for example in finding areas under curves and volumes of solids. Features: 1) Scientific Calculator. Customary units worksheet. Variance is a measure of how spread out a data set is. At the moment, there are two ways we can interpret the value of the double integral. Double Integrals Help Links: S. Nonetheless, it isn't obvious from taking a look at the function how the derivative arises. Antiderivatives are the opposite of derivatives. FREQUENCY DOMAIN AND FOURIER TRANSFORMS So, x(t) being a sinusoid means that the air pressure on our ears varies pe- riodically about some ambient pressure in a manner indicated by the sinusoid. The last thing that we want to do is to use the product rule and chain rule multiple times. Our printable absolute value worksheets meticulously designed for 6th grade and 7th grade students include exercises like finding the absolute value of positive and negative integers, performing simple addition, subtraction, multiplication and division involving the absolute value of real numbers and more. Double facts worksheets. Applications of the Integral. Then graph f(x). The answer is 7. Why are they equal? Find the direction of the vectors u = < 2 , 5 > and v = < - 2 , - 5 >. The user controls the number of partitions, the upper and lower limits, and the method used to estimate the integral. The sixth line substitutes the components from the parametrization into the real-valued function we want to integrate. Type in any integral to get the solution, free steps and graph This website uses cookies to ensure you get the best experience. While the core ideas of calculus (derivatives and integrals) are not hard to understand, calculus is a demanding subject because it requires a broad and thorough background of algebra and functions. Symbolab though offers extensive features despite it being free. Free math lessons and math homework help from basic math to algebra, geometry and beyond. So, first we must have to introduce the trigonometric functions to explore them thoroughly. Radical Equation:. 7) Substitution in Integrals (VII. Comparing rates worksheet. In other words interpolation is the technique to estimate the value of a mathematical function, for any intermediate value of the independent variable. These questions have been designed to help you gain deep understanding of the concept of limits which is of major importance in understanding calculus concepts such as the derivative and integrals of a function. Free Algebra Solver and Algebra Calculator showing step by step solutions. Why is the difference between the two directions equal to 180°?. Integral Graph. In mathematics, the Riemann sum is defined as the approximation of an integral by a finite sum. Complementary and supplementary word problems worksheet. Then graph f(x). It can do basic stuff like most calculator apps. Then press '=' to display the result. You are going to get an approximate value of 20. Many complex integrals can be reduced to expressions involving the beta function. Evaluating a piecewise function adds an extra step to the whole proceedings. Homework Equations I'm doing this in relation to statistical thermodynamics - I think I need to use Sterling's Approximation or a gamma function. Demos Graphing Calculator is a good mobile substitute for the more popular graphing calculators. 2 CHAPTER 4. You may see all of our other area calculations by going to our Area Calculator. Double integral have 2 variables. ADVERTISEMENTS: The following points highlight the two main types of population growth curves. Visual Calculus. A mechanical device that computes area integrals is the planimeter, which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element linkage effects Green's theorem, converting the quadratic polar integral to a linear integral. Now try to divide 1 by an even bigger number. Price: Free. Provided by Alexa ranking, integral-calculator. It can do basic stuff like most calculator apps. (I can also tell that this must be a geometric series because of the form given for each term: as the index increases, each term will be multiplied by an additional factor of –2. These questions have been designed to help you gain deep understanding of the concept of limits which is of major importance in understanding calculus concepts such as the derivative and integrals of a function. A set of questions on the concepts of the limit of a function in calculus are presented along with their answers. You are going to get an approximate value of 20. I liked the midpoint and distance. The initial velocity is 14 m/s the final velocity is 0. Simple to grasp and easy to use, but powerful features are available when you need them. Enter the length of all three sides. It can also do graphs, tables, stats, and more. From Zurab's integral representation for the Apéry's constant to almost impossible integrals I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). An antiderivative is a function that reverses what the derivative does. 2 The simple building block We start with a simple "engine" for the development of negative exponents, namely, (1 x) 1. Enter in graphing or “y =” mode. After working with the Desmos calculator for a while, I wanted to give the geometry tool a try. The Dirac delta is not strictly a function, because any real function that is equal to zero everywhere but at a single point must have a total integral equal to zero, but for many purposes this definition can be manipulated as a function. Have a Quizizz game code? Join a game here. This is the first of the three versions of cos 2. When you calculate a definite integral such as $\int_a^b f(x)dx$, what kind of object should you end up with? A function or something simpler? How do this contrast with the indefinite integral $\int f(x) dx$? Right sum. Indeed, since $$and$$ are not "robust" commands (in the LaTeX sense of the word "robust"), it's perilous to use them in the arguments of "moving" commands; no such difficulties arise with $. 3 Interpretation of Double Riemann Sums and Double integrals. Our tools power Products Classroom Activities Graphing Calculator Scientific Calculator Four Function Calculator Matrix Calculator Test. On your calculator, the base 10 logarithm is noted by log, and the base e logarithm is noted by ln. Visual Calculus is a powerful tool to compute and graph limit, derivative, integral, 3D vector, partial derivative function, double integral, triple integral, series, ODE etc. Instead of ^ you can also type a single quote ' or a double quote ". It can do basic stuff like most calculator apps. If there’s a button for it on a traditional calculator, there’s probably a way to type it now. Using Newton-Cotes formulas, the integration interval is divided by points x 1,x 2,x 3. So this is a geometric series with common ratio r = –2. series for negative integral exponents. 3) The indefinite integral (VII. Go to: Double Integral — Triple Integral Tool to calculate the integral of a function. FREQUENCY DOMAIN AND FOURIER TRANSFORMS So, x(t) being a sinusoid means that the air pressure on our ears varies pe- riodically about some ambient pressure in a manner indicated by the sinusoid. It helps to gain experience by displaying the full working process of solving the problem and exercises. Free indefinite integral calculator - solve indefinite integrals with all the steps. scrolling and zooming the displayed range, built-in function calculator, numerical integration, solving differential equations numerically. The intercepts of a graph are points at which the graph crosses the axes. Log InorSign Up. Use this tool to find the approximate area from a curve to the x axis. So, 7 is. So, first we must have to introduce the trigonometric functions to explore them thoroughly. I decided to try a pattern that I saw on the Pattern In Islamic Art website. 6) Properties of the Integral (VII. DA: 24 PA: 18 MOZ Rank: 10. Keyword Research: People who searched integral calculator dy also searched. We have a whole lot of quality reference tutorials on subjects starting from intermediate algebra syllabus to equation. Type in any integral to get the solution, steps and graph This website uses cookies to ensure you get the best experience. If you have a table of values, see Simpson's rule calculator for a table. It was named after the German mathematician Riemann in 19 th century. The geometry tool requires much less algebraic manipulation. One full period of this graph is shown in red above. A beautiful, free online scientific calculator with advanced features for evaluating percentages, fractions, exponential functions, logarithms, trigonometry, statistics, and more. Read Integral Approximations to learn more. Note also that you may let f be a continuous function on [a. Homework Statement ∫ e1000((sinx)/x) dx [0 to 1000 : bound of integration]. Radical Equation:. 7) Substitution in Integrals (VII. The seventh and final line does the double integral required. s = rθ, when θ is measured in radians “Life is full of circles. series for negative integral exponents. Best Graphing Calculator Online We have the most sophisticated and comprehensive TI 84 type graphing calculator online. Technology’s Role. The intercepts of a graph are points at which the graph crosses the axes. Triple Integral Calculator. The recurrence relation of the beta function is given by. For example, y=2x{1 and v = < - 4 , 6 >. Tectonics, 24, TC4013. In both cases, the rolling circle is given a radius of 1, and in both cases the centers of the two circles remain at a distance of 2. You can also check your answers! Interactive graphs/plots help visualize and better understand the functions. If desired to convert a 2D cylindrical coordinate, then the user just enters values into the r and φ form fields and leaves the 3rd field, the z field, blank. Stage 7: Double-Blind Group Peer Review Students use a feedback form to work as group to provide feedback to anonymous papers. Then press '=' to display the result. 2 The simple building block We start with a simple "engine" for the development of negative exponents, namely, (1 x) 1. Subsection 11. By using this website, you agree to our Cookie Policy. The Integral Calculator supports definite and indefinite integrals (antiderivatives) as well as integrating functions with many variables. Funcions 3D plotter calculates the analytic and numerical integral and too calculates partial derivatives with respect to x and y for 2 variabled functions. Type in any integral to get the solution, steps and graph. We have a whole lot of quality reference tutorials on subjects starting from intermediate algebra syllabus to equation. So, 7 is. Desmos Graphing Calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Graphing Calculator has a formidable reputation of being the top alternative to heavy complex hardware calculators. adding a domain and range is allowed. Parker earned a BA in mathematics with a double minor in Computer Science and Women's Studies from Elmira College, and MA in Educational Psychology from Marist College. Complementary and supplementary word problems worksheet. Omni Calculator solves 1297 problems anywhere from finance and business to health. 2 The simple building block We start with a simple "engine" for the development of negative exponents, namely, (1 x) 1. One function has many antiderivatives, but they all take the form of a function plus an arbitrary constant. Let me show you the graph of this function:. Note: use your eyes and common sense when using this! Some curves don't work well, for example tan(x), 1/x near 0, and functions with sharp changes give bad results. Note, that integral expression may seems a little different in inline and display math mode - in inline mode the integral symbol and the limits are compressed. Calculator supports derivatives up to 10th order as well as complex functions. The integral calculator gives chance to count integrals of functions online free. InteGreat! allows the user to visually explore the idea of integration through approximating the integral value with partitions. The level of difficulty will vary, but most problems should not require any specialized knowledge beyond calculus. In the classroom over 14 years, Ms. Questions and Answers on Limits in Calculus. Only few simple trigonometric equations can be solved without any use of calculator but not at all. While it would be possible to test different numbers using pen and paper using the formulas that I plugged into WolframAlpha, it would be considerably more time-consuming and frustrating. Here's a quick video tutorial on using integrals in the Desmos Graphing Calculator (https://www. How to Calculate Variance. The trace of one point on the rolling circle produces this shape. Cornering ABS makes use of the Bosch IMU (Inertial Measurement Unit) platform to optimise front and rear braking power even in critical situations with the bike at considerable lean angles. Standard Integrals 116 54. Free area under between curves calculator - find area between functions step-by-step This website uses cookies to ensure you get the best experience. Radical Equation:. Free definite integral calculator - solve definite integrals with all the steps. This website uses cookies to ensure you get the best experience. It can also do graphs, tables, stats, and more. The calculator will approximate the integral using the trapezoidal rule, with steps shown. 3) The indefinite integral (VII. 0 m and the time is 1. scrolling and zooming the displayed range, built-in function calculator, numerical integration, solving differential equations numerically. Definition of a vertical asymptote: The line x = x 0 is a "vertical asymptote" of f(x) if and only if f(x) approaches + or - as x approaches x 0 from the left or from the right. com is the right site to pay a visit to!. Let’s create some discrete plots using Matlab function 'stem'. Probability Calculator in GeoGebra: 1D Statistics: The Normal Distribution: Distributions in Desmos: functions > Dist: Probability Calculator: Hypothesis testing: Sampling distributions: Kinematics: Kinematics exam question: Force and motion: Resolving forces exam question: Moments: Balancing act: Projectiles: Trajectory of a Projectile. It’s so fast and easy you won’t want to do the math again!. Type in any integral to get the solution, steps and graph This website uses cookies to ensure you get the best experience. The numbers of simple integral graphs on , 2, nodes are 0, 2, 3, 6, 10, 20, 33, 71,. Funcions 3D plotter calculates the analytic and numerical integral and too calculates partial derivatives with respect to x and y for 2 variabled functions. These topics, concepts,. Thanks so much for sharing. A cardioid is formed by a circle of the diameter a, which adjacently rolls around another circle of the same size. Cornering ABS makes use of the Bosch IMU (Inertial Measurement Unit) platform to optimise front and rear braking power even in critical situations with the bike at considerable lean angles. It can do basic stuff like most calculator apps. Desmos - Free Math Resources https://teacher. Integrated Review is now included in all Sullivan 11th Edition MyLab Math courses, including Precalculus, Algebra & Trigonometry, and Trigonometry. Read Integral Approximations to learn more. Description. The recurrence relation of the beta function is given by. So, do I need to find functions for f(x) or is there something else?. Our tools power assessments and major curricula. Explore math with our beautiful, free online graphing calculator. Then press '=' to display the result. Online reference and tools - calculators, converters, math, electricity, web design. A cardioid is formed by a circle of the diameter a, which adjacently rolls around another circle of the same size. Includes all the functions and options you might need. com reaches roughly 771 users per day and delivers about 23,142 users each month. Here's what I don't get/don't know what supposed to do: I tried graphing this on desmos and found it REALLY hard to find functions that would work and I'm guessing that the tangents would have to be close to (0,0). Proper and Improper Integrals and the example with the desmos graphing calculator: Week 8: IV. Instead of using a calculator, use Microsoft Excel to do. Note, that integral expression may seems a little different in inline and display math mode - in inline mode the integral symbol and the limits are compressed. The answer is 7. Keyword CPC PCC Volume Score; find dy/dx integral calculator: 0. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. If you have a table of values, see Simpson's rule calculator for a table. How to Convert Repeating Decimals to Fractions. integral of C of (2xy) dx + (xy^(2)) dy I got the double integral of 2xy-2x dydx but I am having a hard time coming up with the bounds of integration. The fundamental idea in calculus is to make calculations on functions as a variable “gets close to” or approaches a certain value. At this point, the y-coordinate is zero. For more about how to use the Integral Calculator, go to "Help" or take a look at the examples. J – Shaped Curve 2. You can convert a logarithm with a base that is not 10 or e to an equivalent logarithm with base 10 or e. Evaluating a piecewise function adds an extra step to the whole proceedings. A beautiful, free online scientific calculator with advanced features for evaluating percentages, fractions, exponential functions, logarithms, trigonometry, statistics, and more. In other words interpolation is the technique to estimate the value of a mathematical function, for any intermediate value of the independent variable. Geometry worksheets. Integrand function is replaced by the Lagrange polynomials of different degree, integration of which yields the numerical integration formulas with different degree of accuracy. Area and perimeter worksheets. Nerdy Details: The formula you supply calculates the height z from the variables x and y between -1 and 1. It not only provides online calculators (graphing, scientific, and four function), but it has many activities students can do. The most common application of Riemann sum is considered in finding the areas of lines or graphs. Integrated review can be used in corequisite courses, or simply to help students who enter without a full understanding of prerequisite skills and concepts. 1ME Cornering ABS ECU, an integral part of the Ducati Safety Pack (DSP). This website uses cookies to ensure you get the best experience. For help refer to the Instruction manual and to the Online Scientific Calculator blog entries. Only few simple trigonometric equations can be solved without any use of calculator but not at all. Double Integral Calculator Added Apr 29, 2011 by scottynumbers in Mathematics Computes the value of a double integral; allows for function endpoints and changes to order of integration. Keyword CPC PCC Volume Score; find dy/dx integral calculator: 0. Products Classroom Activities Graphing Calculator Scientific Calculator Four Function Calculator Matrix Calculator Test Practice Geometry Tool. Initially developed to serve as a software sketchbook and to teach fundamentals of computer programming within a visual context, Processing also has evolved into a tool for generating finished professional work. So, 7 is. GraphyCalc is a 3d graphing calculator, allowing you to interactively explore 3d functions. Areas between graphs. The EXP function is a built-in function in Excel that is categorized as a Math/Trig Function. Learn more. You can also check your answers! Interactive graphs/plots help visualize and better understand the functions. For example if you have integral of (1/(2. Read Integral Approximations to learn more. This is a formula calculator. Simpson's rule Calculator Simpson's rule is a technique to calculate the approximation of definite curve and is used to find area beneath or above the parabola. Number 8, let us see what number 8 has for us. The Algorithm The bisection method is an algorithm, and we will explain it in terms of its steps. 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# Fun With Maxwell Distribution

The Maxwell distribution describes the velocities of the molecules of an ideal gas at equilibrium. The probability for finding a molecule with velocities between $$\vec{v}$$ and $$\vec{v}+d\vec{v}$$ is given by $dP=\left(\frac{m}{2\pi k T}\right)^{3/2} e^{-\frac{m v^{2}}{2 k T}} dv_{x} d v_{y} d v_{z} \quad \text{(Maxwell Distribution)}$ where $$m$$ is the mass of the molecules, $$k$$ is the Boltzmann constant and T is the temperature of the gas. If the mean speed of the molecules of a gas is $$<v>=300~\text{m/s}$$, what is the mean of the inverse of the speed $$<1/v>$$ in seconds per meter?

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# Existence of pure quantum gravity

#### atyy

Is pure quantum gravity known to exist?

I had thought it exists in 3D, but Strominger writes http://arxiv.org/abs/0906.1313 "Determining Z for pure 3D quantum Einstein gravity - if it exists - is an important open problem"

Eg. Does the Turaev-Viro model not describe 3D QG?

Related Beyond the Standard Model News on Phys.org

#### marcus

Gold Member
Dearly Missed
I'd like to hear your summary, if you have looked those sources over and have some general impressions.

It is striking how much work still is being devoted to 3D pure QG.
As I recall (odd bit of trivia) 3D QG was the topic of Witten's talk at the Strings 2007 conference in Madrid. At the end, someone from the audience asked him what the connetion was with the string program and he told them (I don't remember exactly what it was.)

That same week Loops 2007 was going on at Morelia in Mexico and Rovelli gave a kind of "where do we go from here" roadmap talk at the end, where he mentioned that since so much had been done on 3D he would urge people not to work on it unless they had some really good reason in mind--some direct bearing on the harder problem of 4D. It impressed me because he was actually telling the young researchers there to "work on the hard problem! Don't keep doodling around with 3D just because it is easier to get publishable results."

Every effort to lead involves risk. You only know later how it turns out. It may be that there is some essential insight still to be gotten from 3D QG! Perhaps Strominger will surprise us with some valuable discovery there. Or maybe not. I'd be glad to hear what you envision as possibilities.

Then the more interesting question: what do you conclude about 4D QG? Does a theory exist? What state does it seem to be in? Are you thinking of some perturbative approach? I mean where you have a series including the terms allowed by symmetry. Or some TQFT approach where you start with a BF formulation?

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#### atyy

Criteria:

1) It should be a consistent quantum theory
2) It should have the right semiclassical/classical limit

and maybe

3) It should have the BTZ black hole and give us its entropy

#### Sardano

Criteria:

1) It should be a consistent quantum theory
2) It should have the right semiclassical/classical limit

and maybe

3) It should have the BTZ black hole and give us its entropy

Several months ago, I attended some Enrique Alvarez lectures on Quantum Gravity, and the first thing he did was to show us that it was not possible to obtain a Quantum Gravity theory ignoring all the others interaction. That you should take all of them into account in order to have a well defined Quantum Gravity theory, which would be an unified theory of the four known interactions.

In light of this comments, some guy asked that then LQG should be wrong!... Enrique Alvarez with all due respect said that he thinks LQG is not correct but that all the possibilites should be investigated... there was big laughs on the audience.

The argument was based on QFT, the theories that are being experimentally cheked in accelerators all over the world in hundred of ways (GR has only a few experimental tests). I will look for the reference of this lectures and I will post it, in case someone is interested.

#### atyy

Several months ago, I attended some Enrique Alvarez lectures on Quantum Gravity, and the first thing he did was to show us that it was not possible to obtain a Quantum Gravity theory ignoring all the others interaction. That you should take all of them into account in order to have a well defined Quantum Gravity theory, which would be an unified theory of the four known interactions.
How about the 3D case? Strominger's http://arxiv.org/abs/0906.1313 (fourth problem) indicate that he's unsure, particularly in the AdS case, where there's the BTZ black hole.

I had long thought the 3D case was covered by Witten's Chern-Simons theory, or by the Turaev-Viro theory. But Strominger refers to Witten's http://arxiv.org/abs/0706.3359 which casts doubt on the former, and I'm not sure Turaev-Viro is known to give the right semiclassical limit.

I will look for the reference of this lectures and I will post it, in case someone is interested.
That's be cool if you can find it, thanks.

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#### Finbar

Several months ago, I attended some Enrique Alvarez lectures on Quantum Gravity, and the first thing he did was to show us that it was not possible to obtain a Quantum Gravity theory ignoring all the others interaction. That you should take all of them into account in order to have a well defined Quantum Gravity theory, which would be an unified theory of the four known interactions.
What he say's here http://arxiv.org/pdf/1011.0543 is

"Then all couplings in the effective Lagrangian become of order unity, and there is no decoupling limit in which gravitation can be considered by itself in isolation from all other interactions. This then seems the minimum prize one has to pay for being interested in quantum gravity; all couplings in the derivative expansion become important simultaneously. No significant differences appear when supergravity is considered.
In conclusion, it does not seem likely that much progress can be made by some- how quantizing Einstein-Hilbert’s Lagrangian in isolation. To study quantum gravity means to study all other interactions as well."

But these interactions can still be purely gravitational ones. He is not saying that we have to couple the theory to matter to get a consistent theory just that we shouldn't only consider the EH Lagrangian.

So i think maybe you misunderstood him? We don't need a unify QG with the other forces to get a consistent theory.

#### fzero

Homework Helper
Gold Member
But these interactions can still be purely gravitational ones. He is not saying that we have to couple the theory to matter to get a consistent theory just that we shouldn't only consider the EH Lagrangian.

So i think maybe you misunderstood him? We don't need a unify QG with the other forces to get a consistent theory.
No, what he's saying is that if there is matter and nongravitational interactions at scales $$E \ll M_P$$, then at $$E \sim M_P$$ these interactions remain and, furthermore, their couplings are of order unity. There is therefore no regime where pure gravity exists in the presence of other interactions. There are further arguments in http://arxiv.org/abs/hep-th/0601001 that gravity is always the weakest force in such a scenario.

This is an important concept when we ask about how to quantize gravity in a real-world scenario. Whether or not there are quantum theories of pure gravity as toy models remains an interesting question, but perhaps not one that is completely relevant to describing our universe.

#### atyy

So I guess the situation is

-Turaev-Viro model describes 3D Riemannian gravity with positive cosmological constant
http://arxiv.org/abs/gr-qc/9905087
http://arxiv.org/abs/gr-qc/0311066

-However with positive cosmological constant, there are no known observables for someone inside the universe, so Witten examines the case of negative cosmological constant.
-Witten's Chern-Simons proposal for 3D (Lorentzian?) gravity with negative cosmological constant is probably wrong.
-So he makes a new proposal for pure 3D AdS quantum gravity to be described by 2D boundary CFT
http://arxiv.org/abs/0706.3359
http://math.ucr.edu/home/baez/week254.html

#### atyy

So I guess the situation is

-Turaev-Viro model describes 3D Riemannian gravity with positive cosmological constant
http://arxiv.org/abs/gr-qc/9905087
http://arxiv.org/abs/gr-qc/0311066

-However with positive cosmological constant, there are no known observables for someone inside the universe, so Witten examines the case of negative cosmological constant.
-Witten's Chern-Simons proposal for 3D (Lorentzian?) gravity with negative cosmological constant is probably wrong.
-So he makes a new proposal for pure 3D AdS quantum gravity to be described by 2D boundary CFT
http://arxiv.org/abs/0706.3359
http://math.ucr.edu/home/baez/week254.html
Actually, I'm not so sure about the Turaev-Viro model and 3D gravity. Here are a few more papers with interesting discussions of the relationship.

http://arxiv.org/abs/hep-th/9110057
"Therefore it would be better to consider the Regge action appeared in our model as the CS action, rather than the Einstein action. The strange i factor may indicate the subtlety in the correspondence between the Euclidean gravity and a CS theory."

http://arxiv.org/abs/hep-th/9804185v1
"Thus, if there indeed exist a phase of the quantum BF theory in which non-degenerate E fields dominate, then one can gain some control over the spacetime volume in (2+1) quantum gravity using the results from BF theory. If, on the other hand, the path integral of BF theory is always dominated by highly degenerate solutions, as may well be the case, then the two theories have very little to do with each other, and the spacetime volume of BF theory is not the same as the volume in gravity. At the present stage of our understanding of quantum gravity it is hard to tell which of this two possibilities is realized."

http://arxiv.org/abs/1012.4784
"In three dimensions, the Turaev-Viro invariant is related to Euclidean gravity with a positive cosmological constant. This relation can be inferred from two perspectives. The first one is the investigation of the semi-classical limit of the model via the asymptotics of the amplitude for the three-simplexes (given by a quantum 6j symbol). ... The asymptotics of the quantum 6j symbol has been computed in [43] and yields the cosine of the Regge action with a cosmological term.

An alternative way to relate the Turaev-Viro model to three-dimensional gravity with positive cosmological constant is obtained from the correspondence between three-dimensional gravity, BF theory and Chern-Simons gauge theory. ... The fact that the Turaev-Viro sum of a manifold M is equal to the modulus square of the Reshetikhin-Turaev invariant of M, relates the Turaev-Viro state sum to BF theory with a positive cosmological constant"

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#### Finbar

No, what he's saying is that if there is matter and nongravitational interactions at scales $$E \ll M_P$$, then at $$E \sim M_P$$ these interactions remain and, furthermore, their couplings are of order unity. There is therefore no regime where pure gravity exists in the presence of other interactions. There are further arguments in http://arxiv.org/abs/hep-th/0601001 that gravity is always the weakest force in such a scenario.

This is an important concept when we ask about how to quantize gravity in a real-world scenario. Whether or not there are quantum theories of pure gravity as toy models remains an interesting question, but perhaps not one that is completely relevant to describing our universe.
Yes i agree that to make contact with reality we must couple gravity to matter. But there is nothing he says that suggests we have to unify the forces to obtain a consistent theory which is what you implied. There's a big difference between unification and coupling.

#### suprised

If string theory is any right, then it seems that pure gravity is not consistent at higher loop orders. In order to have a modular invariant partition function, and thus a unitary theory, extra degrees of freedom (compactified extra dimensions for example) appear to be necessary.

This may be a good testing ground between strings and LQG: can the loopy people construct a consistent quantum theory of pure gravity? It would be interesting to see whether a constraint on matter appears in their framework or not, eventually.

#### atyy

If string theory is any right, then it seems that pure gravity is not consistent at higher loop orders. In order to have a modular invariant partition function, and thus a unitary theory, extra degrees of freedom (compactified extra dimensions for example) appear to be necessary.

This may be a good testing ground between strings and LQG: can the loopy people construct a consistent quantum theory of pure gravity? It would be interesting to see whether a constraint on matter appears in their framework or not, eventually.
I think there are quite a few LQG people who are thinking in the direction that pure gravity cannot work. On the more conservative end, there is LQC which seems to always incorporate matter, and on the less conservative end, there is the idea explored Freidel, Livine, Oriti, Ryan and others that unification should take place. Also, Freidel and Krasnov have tried to make links to AdS/CFT. This is all very piecemeal, but it indicates diversity in the LQG community (loss of direction;).

OTOH, even though the handwavy arguments indicate that pure QG won't get the BH right, Polchinski's text still acknoweldeges Asymptotic Safety as not having been formally ruled out. I think the most interesting result in this direction is that of Causal Dynamical Triangulations, which seems to have a ground state that is like a de Sitter universe.

#### marcus

Gold Member
Dearly Missed
...
This may be a good testing ground between strings and LQG: can the loopy people construct a consistent quantum theory of pure gravity? ...
I'm not sure what you mean. The current review of the status of the program discusses the inclusion of matter---I assume that has always been the goal. Some progress on including matter has been made recently as reported at the end of http://arxiv.org/abs/1102.3660 .

Currently quite a lot of effort in that direction, some attempts Atyy mentioned plus the recent Rovelli et al "Spinfoam Fermions". Will it be successful? No one can say.

The most recent broad audience review of the Loop program is http://arxiv.org/abs/1012.4707 . That is easy to read and would also discuss progress on including matter.

By including matter I don't mean incorporating a detailed version of the standard model, as I expect everyone understands. Rather more modestly, one wants to see how fermion degrees of freedom should inhabit the geometry as represented by the network or cell-complex (i.e. foam).

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#### atyy

I'm not sure what you mean. The current review of the status of the program discusses the inclusion of matter---I assume that has always been the goal. Some progress on including matter has been made recently as reported at the end of http://arxiv.org/abs/1102.3660 .

Currently quite a lot of effort in that direction, some attempts Atyy mentioned plus the recent Rovelli et al "Spinfoam Fermions". Will it be successful? No one can say.
Well, the question is does it require matter to be consistent? A very long line of LQG thought, right through and embodied by the Zakopane lectures is that even though matter can be incorporated (and ultimately must be to make an experimental prediction), it is not needed.

#### marcus

Gold Member
Dearly Missed
That's very interesting Atyy, but the actual direction of current researh involves the inclusion of matter, both in the full theory (Rovelli et al Spinfoam Fermions paper) and most actively in the application to cosmology. I don't know about "very long line of thought". Pragmatically what matters is what people do, not what they you or I think about it. Or am I mistaken about this?

#### atyy

That's very interesting Atyy, but the actual direction of current researh involves the inclusion of matter, both in the full theory (Rovelli et al Spinfoam Fermions paper) and most actively in the application to cosmology. I don't know about "very long line of thought". Pragmatically what matters is what people do, not what they you or I think about it. Or am I mistaken about this?
I think you are mistaken. For example, up to the point where, as you like to say, Rovelli says "This is the theory", there is no matter (as interpreted by him), indicating that he believes there can be a consistent theory without matter.

(My suspicion is he is near a consistent theory, but one that does not contain pure gravity - but CDT rebuts this argument, so I am curious).

#### marcus

Gold Member
Dearly Missed
...(My suspicion is he is near a consistent theory, but one that does not contain pure gravity - but CDT rebuts this argument, so I am curious).
We may never know, since by the end of the papers you mentioned he is already talking about how to include matter. In the selfsame theory (and the progress towards it that has recently occurred.)

Theories are made to be tested: that will happen and decide. The rest is academic.

You pointed out to me that the bridge between the full theory and CMB cosmology is still not complete. It has only reached a "halfway" stage that still employs a manifold but has more degrees of freedom (including some anisotropy/inhomogeneity) than the old LQC did.

I'd guess that completing that bridge is the most urgent item on the agenda (because it leads to a test of the full theory by CMB observations, not just a test of the Loop cosmology application.)

#### atyy

No, it's clear. Otherwise the semiclassical limit having Regge-like terms is wrong.

#### marcus

Gold Member
Dearly Missed
No, it's clear. Otherwise the semiclassical limit having Regge-like terms is wrong.
I don't understand Atyy. I guess you need to use more words with me and explain a bit more. What is clear? Don't mind if you think you are repeating the obvious, just spell it out.
Can you point me to an equation in 1102.3660 which would be wrong if such and such were the case? I know that there is a discussion of semiclassical limit and that the Regge action appears there.

Could you be talking about eqn 101 on page 18? I wonder what would make that eqn wrong? But basically what is "it" that you say is clear? Sorry for any slow lumberingness.

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#### atyy

It's clear that Rovelli intends to quantize pure gravity. In his words "Loop quantum gravity is resolutely a theory of quantum gravity that does not address the unification problem." http://arxiv.org/abs/1012.4707 p7

#### marcus

Gold Member
Dearly Missed
It's clear that Rovelli intends to quantize pure gravity. In his words "Loop quantum gravity is resolutely a theory of quantum gravity that does not address the unification problem." http://arxiv.org/abs/1012.4707 p7
I see part of what you are saying. According to you a theory which includes matter can still be "pure gravity".
All it has to do is allow matter to ride in the geometry and take part in gravity---but not encorporate the standard model generation, particle interactions etc. Then it is still "pure".

It can have all the matter it wants as long as it does not "address the unification problem", as Rovelli puts it. And in the same paper he discusses including fermion fields into the spinfoam picture. Resolutely, one might say

#### Sardano

Some of you are saying that LQG includes "matter fields". Ok, but for me this is a joke. Is this "matter content" of the lagrangian the Satandard Model or any kind of matter content that includes the Standard Model?. If the answer is yes: then okey, LQG still have a chance of being correct. Is the answer is no, then LQG has nothing to say and Enriques argument still apllies. I have heard that LQC "resolves" the Big Bang into a Big Bounce. Please, dont make me laugh, because the matter content of such theories is ridiculous, like putting just a scalar.

#### mitchell porter

Three threads on 3D gravity, I don't know where I should post!

In http://physics.stackexchange.com/questions/6393/can-a-cft-2-which-cant-be-factorized-into-chiral-and-antichiral-parts-and-or-h" [Broken], Moshe Rozali says of Witten's "Three-dimensional gravity revisited": "I was wondering also what is the impact of this paper on the community of people playing with gauge theory formulations of pure gravity (in 3 and 4 dimensions). This should really give them a pause, since 3dim pure gravity is the most common toy model for them." The thread itself is about how a technical conjecture of Witten's in the paper is wrong, but I believe Rozali is referring to the introductory remarks, in which Witten says his own earlier work on 3D gravity as Chern-Simons theory was wrong.

Personally, I am torn on whether to engage with this issue. I can see a way forward: We compare the "Turaev-Viro model" - about which I know nothing, but which is serving in these threads as the exemplar of pure 3D gravity (is it "canonically quantized gravity"?) - with Witten's pure 3D gravity in AdS. We can exhibit the mathematical guts of both "theories", and try to compare them. This can be done - at least, I believe I understand Witten's paper - it just requires a bit of hard work. I'm reluctant to take the lead, but I could be induced into a dialogue if someone (atyy?) wants to talk about how Turaev-Viro works or has questions about Witten's paper.

Last edited by a moderator:

#### atyy

Three threads on 3D gravity, I don't know where I should post!

In http://physics.stackexchange.com/questions/6393/can-a-cft-2-which-cant-be-factorized-into-chiral-and-antichiral-parts-and-or-h" [Broken], Moshe Rozali says of Witten's "Three-dimensional gravity revisited": "I was wondering also what is the impact of this paper on the community of people playing with gauge theory formulations of pure gravity (in 3 and 4 dimensions). This should really give them a pause, since 3dim pure gravity is the most common toy model for them." The thread itself is about how a technical conjecture of Witten's in the paper is wrong, but I believe Rozali is referring to the introductory remarks, in which Witten says his own earlier work on 3D gravity as Chern-Simons theory was wrong.

Personally, I am torn on whether to engage with this issue. I can see a way forward: We compare the "Turaev-Viro model" - about which I know nothing, but which is serving in these threads as the exemplar of pure 3D gravity (is it "canonically quantized gravity"?) - with Witten's pure 3D gravity in AdS. We can exhibit the mathematical guts of both "theories", and try to compare them. This can be done - at least, I believe I understand Witten's paper - it just requires a bit of hard work. I'm reluctant to take the lead, but I could be induced into a dialogue if someone (atyy?) wants to talk about how Turaev-Viro works or has questions about Witten's paper.
I don't think we can compare the Turaev-Viro model and Witten's paper, or at least if we compare them, they are about different physical systems.

My understanding of Witten's paper is that it's about 3D quantum gravity with negative cosmological constant. He essentially retracts his earlier proposal that this is described by a Chern-Simons quantum theory, and makes a new proposal that it is described by a certain CFT.

The Turaev-Viro model is a spin-foam model. It is related to the Ponzano-Regge model which is a spin-foam model that uses normal groups, and is supposed to describe 3D Euclidean quantum gravity. The Ponzano-Regge model is divergent. The Turaev-Viro model (which apparently wasn't originally intended for gravity) replaces Ponzano-Regge's normal groups with quantum groups, and becomes finite. The quantum group seems to lead to a positive cosmological constant.

So I think the TV model has a Euclidean bulk with positive cc, but I presume Witten has a Lorentzian bulk with negative cc.

Some of the things I don't understand about the TV model are:
1) What is a good semiclassical limit for pure 3D quantum gravity?
2) Does the TV model have that limit?

Mizoguchi and Tada http://arxiv.org/abs/hep-th/9110057 examined that limit and concluded maybe it wasn't really gravity: "Therefore it would be better to consider the Regge action appeared in our model as the CS action, rather than the Einstein action. The strange i factor may indicate the subtlety in the correspondence between the Euclidean gravity and a CS theory."

Looking at the thread you linked to, it seems Witten's conjecture is wrong. So there is no known definition of pure 3D quantum gravity in AdS. And from Mizoguchi and Tada, the TV model also doesn't describe 3D quantum gravity with positive cc. It's funny how alluring (but misleading?) the similarity with Chern-Simons.(The TV model seems to have a much tighter link to the Levin-Wen models than to 3D gravity.)

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### #ActualCornstalks

Posted 15 January 2013 - 11:30 AM

SiCrane mentions some very good points. I'm going to dive a little deeper into the internals of memory management and try to explain why the above is such a big problem.

There are various strategies for managing allocated memory. The allocator is what actually allocates the memory and frees it when you call new/delete. One very common allocation strategy is to request memory from the operating system when more is needed, but when its freed, don't immediately hand it back to the operating system. Keep it around. Why? Well, chances are the programmer is going to allocate more memory some time soon. If you give it all back to the OS right now, you'll just have to request more form the OS soon anyway (and these can all be pretty slow operations), so if the allocator hangs onto it, when you request more memory the allocator has some already available to give you. Yay, increase in speed! (there are other practical reasons why, but this is a big one) (and of course, if the allocator has a ton of free memory it's holding onto, it might decide to hand some of it back to the OS, but still keep a "reserve" of free memory hanging around).

Now, as you allocate memory and free it, your memory becomes fragmented. If you looked at it as one continuous block of memory, you'd have free chunks and allocated chunks mixed all over. How does the allocator keep track of what's allocated and what's free? Remember, this is the allocator! It can't just ask some other allocator for memory and keep a nice std::vector<ptr> of free and allocated chunks of memory. It's a pain in the butt to do, but here's a common way it's done:

Linked lists. In memory. When you allocate memory, the allocator does something like this:

void* malloc(size_t bytes) // same idea for new/new[] (horribly simplified, though)
{
void* ptr = RequestBytesFromOperatingSystem(bytes + HEADER_SIZE);

return (char*)ptr + HEADER_SIZE; // skip the header and give return the usable block to the user
}

The allocator allocates an extra few bytes to keep a header for each block of memory. This header usually says how large the block is, if the block is free or allocated and in use, if the next block (at ptr + bytes + HEADER_SIZE) is free or allocated and in use, and if the previous block is free or allocated and in use. If the block is free, the allocator might use some of the space in the block to store pointers to the previous and next free blocks (maintaining a linked list).

So now let's take this idea for memory blocks, and what SiCrane said above, and document them to figure out how a block of memory might look to the allocator

/*
Potential structure of a memory block allocated by new (that is, this block is allocated and in use):
Upper 29 bits: size of this block in memory
Bit 2: 1 if previous block is free, 0 if previous block is allocated
Bit 1: 1 if next block is free, 0 if next block is allocated
Bit 0: 1 if this block is free, 0 if this block is allocated
Pointer into segment table (4 bytes) (also acts as padding to maintain 8-byte alignment, like many CPUs need)
Instead of wasting the padding, we'll store which segment into our segment table this block should be
inserted into when its freed (so we don't have compute it when we free it, yay speed!)
User data (N bytes)
This is the memory the programmer is actually able to use

Potential structure of a memory block allocated by new [] (that is, this block is allocated and in use):
Upper 29 bits: size of this block in memory
Bit 2: 1 if previous block is free, 0 if previous block is allocated
Bit 1: 1 if next block is free, 0 if next block is allocated
Bit 0: 1 if this block is free, 0 if this block is allocated
Number of elements (4 bytes) (also acts as padding to maintain 8-byte alignment, like many CPUs need)
Instead of wasting the padding, we'll use it to store the number of elements in this array
User data (N bytes)
This is the memory the programmer is actually able to use

Potential structure of a free block of memory (after it's been freed by delete or delete []):
Upper 29 bits: size of this block in memory
Bit 2: 1 if previous block is free, 0 if previous block is allocated
Bit 1: 1 if next block is free, 0 if next block is allocated
Bit 0: 1 if this block is free, 0 if this block is allocated
Pointer to the next free block (4 bytes)
This way we can maintain a linked list of free blocks
Block data (N bytes)
Junk
*/

There are various allocation strategies and memory block formats, but the above encapsulates some common practices and ideas shared by many allocators.

Now, when the allocator tries to free a block of memory, it's going to try to use the block's header (the first 4 bytes of the block), and the following 4 bytes (what those bytes represent depends on whether or not you called new or new [], and depending on whether you call delete or delete [] it will change how those bytes are interpreted). If you call the delete on memory you used new [], the allocator will think the 4 bytes after the header are a pointer into the segment table (which they're not; those 4 bytes represent the number of elements in the array), and boom, you've successfully corrupted your memory (or hopefully crashed). If you call delete [] on memory you used new, the allocator will think the 4 bytes after the header are the number of elements in the array (which they're not; those 4 bytes represent a pointer into the segment table), and boom, you've successfully corrupted your memory (or hopefully crashed).

If different block formats and allocation strategies are used, it's possible to completely destroy the allocator's linked list of free/allocated blocks by calling the wrong delete, and now you've got memory leaks, you're overwriting data and double allocating the same blocks of memory for different uses, etc. Bad, bad, bad mojo.

Mixing delete with new [] or delete [] with new is not safe! Yes, it's possible the allocator implements new as new [1] (which would be a valid way to do it), and delete as delete [], but banking on that idea is like playing Russian roulette (with a gun that has more than one round in it).

Writing a correct, proper, fast, and efficient allocator is very hard work and involves lots of tricks. The last thing you need as an allocator is someone screwing up your tricks and hacks by freeing memory wrong.

### #2Cornstalks

Posted 15 January 2013 - 11:28 AM

SiCrane mentions some very good points. I'm going to dive a little deeper into the internals of memory management and try to explain why the above is such a big problem.

There are various strategies for managing allocated memory. The allocator is what actually allocates the memory and frees it when you call new/delete. One very common allocation strategy is to request memory from the operating system when more is needed, but when its freed, don't immediately hand it back to the operating system. Keep it around. Why? Well, chances are the programmer is going to allocate more memory some time soon. If you give it all back to the OS right now, you'll just have to request more form the OS soon anyway (and these can all be pretty slow operations), so if the allocator hangs onto it, when you request more memory the allocator has some already available to give you. Yay, increase in speed! (there are other practical reasons why, but this is a big one)

Now, as you allocate memory and free it, your memory becomes fragmented. If you looked at it as one continuous block of memory, you'd have free chunks and allocated chunks mixed all over. How does the allocator keep track of what's allocated and what's free? Remember, this is the allocator! It can't just ask some other allocator for memory and keep a nice std::vector<ptr> of free and allocated chunks of memory. It's a pain in the butt to do, but here's a common way it's done:

Linked lists. In memory. When you allocate memory, the allocator does something like this:

void* malloc(size_t bytes) // same idea for new/new[] (horribly simplified, though)
{
void* ptr = RequestBytesFromOperatingSystem(bytes + HEADER_SIZE);

return (char*)ptr + HEADER_SIZE; // skip the header and give return the usable block to the user
}

The allocator allocates an extra few bytes to keep a header for each block of memory. This header usually says how large the block is, if the block is free or allocated and in use, if the next block (at ptr + bytes + HEADER_SIZE) is free or allocated and in use, and if the previous block is free or allocated and in use. If the block is free, the allocator might use some of the space in the block to store pointers to the previous and next free blocks (maintaining a linked list).

So now let's take this idea for memory blocks, and what SiCrane said above, and document them to figure out how a block of memory might look to the allocator

/*
Potential structure of a memory block allocated by new (that is, this block is allocated and in use):
Upper 29 bits: size of this block in memory
Bit 2: 1 if previous block is free, 0 if previous block is allocated
Bit 1: 1 if next block is free, 0 if next block is allocated
Bit 0: 1 if this block is free, 0 if this block is allocated
Pointer into segment table (4 bytes) (also acts as padding to maintain 8-byte alignment, like many CPUs need)
Instead of wasting the padding, we'll store which segment into our segment table this block should be
inserted into when its freed (so we don't have compute it when we free it, yay speed!)
User data (N bytes)
This is the memory the programmer is actually able to use

Potential structure of a memory block allocated by new [] (that is, this block is allocated and in use):
Upper 29 bits: size of this block in memory
Bit 2: 1 if previous block is free, 0 if previous block is allocated
Bit 1: 1 if next block is free, 0 if next block is allocated
Bit 0: 1 if this block is free, 0 if this block is allocated
Number of elements (4 bytes) (also acts as padding to maintain 8-byte alignment, like many CPUs need)
Instead of wasting the padding, we'll use it to store the number of elements in this array
User data (N bytes)
This is the memory the programmer is actually able to use

Potential structure of a free block of memory (after it's been freed by delete or delete []):
Upper 29 bits: size of this block in memory
Bit 2: 1 if previous block is free, 0 if previous block is allocated
Bit 1: 1 if next block is free, 0 if next block is allocated
Bit 0: 1 if this block is free, 0 if this block is allocated
Pointer to the next free block (4 bytes)
This way we can maintain a linked list of free blocks
Block data (N bytes)
Junk
*/

There are various allocation strategies and memory block formats, but the above encapsulates some common practices and ideas shared by many allocators.

Now, when the allocator tries to free a block of memory, it's going to try to use the block's header (the first 4 bytes of the block), and the following 4 bytes (what those bytes represent depends on whether or not you called new or new [], and depending on whether you call delete or delete [] it will change how those bytes are interpreted). If you call the delete on memory you used new [], the allocator will think the 4 bytes after the header are a pointer into the segment table (which they're not; those 4 bytes represent the number of elements in the array), and boom, you've successfully corrupted your memory (or hopefully crashed). If you call delete [] on memory you used new, the allocator will think the 4 bytes after the header are the number of elements in the array (which they're not; those 4 bytes represent a pointer into the segment table), and boom, you've successfully corrupted your memory (or hopefully crashed).

If different block formats and allocation strategies are used, it's possible to completely destroy the allocator's linked list of free/allocated blocks by calling the wrong delete, and now you've got memory leaks, you're overwriting data and double allocating the same blocks of memory for different uses, etc. Bad, bad, bad mojo.

Mixing delete with new [] or delete [] with new is not safe! Yes, it's possible the allocator implements new as new [1] (which would be a valid way to do it), and delete as delete [], but banking on that idea is like playing Russian roulette (with a gun that has more than one round in it).

Writing a correct, proper, fast, and efficient allocator is very hard work and involves lots of tricks. The last thing you need as an allocator is someone screwing up your tricks and hacks by freeing memory wrong.

### #1Cornstalks

Posted 15 January 2013 - 11:27 AM

SiCrane mentions some very good points. I'm going to dive a little deeper into the internals of memory management and try to explain why the above is such a big problem.

There are various strategies for managing allocated memory. The allocator is what actually allocates the memory and frees it when you call new/delete. One very common allocation strategy is to request memory from the operating system when more is needed, but when its freed, don't immediately hand it back to the operating system. Keep it around. Why? Well, chances are the programmer is going to allocate more memory some time soon. If you give it all back to the OS right now, you'll just have to request more form the OS soon anyway (and these can all be pretty slow operations), so if the allocator hangs onto it, when you request more memory the allocator has some already available to give you. Yay, increase in speed! (there are other practical reasons why, but this is a big one)

Now, as you allocate memory and free it, your memory becomes fragmented. If you looked at it as one continuous block of memory, you'd have free chunks and allocated chunks mixed all over. How does the allocator keep track of what's allocated and what's free? Remember, this is the allocator! It can't just ask some other allocator for memory and keep a nice std::vector<ptr> of free and allocated chunks of memory. It's a pain in the butt to do, but here's a common way it's done:

Linked lists. In memory. When you allocate memory, the allocator does something like this:

void* malloc(size_t bytes) // same idea for new/new[] (horribly simplified, though)
{
void* ptr = RequestBytesFromOperatingSystem(bytes + HEADER_SIZE);

return (char*)ptr + HEADER_SIZE; // skip the header and give return the usable block to the user
}

The allocator allocates an extra few bytes to keep a header for each block of memory. This header usually says how large the block is, if the block is free or allocated and in use, if the next block (at ptr + bytes + HEADER_SIZE) is free or allocated and in use, and if the previous block is free or allocated and in use. If the block is free, the allocator might use some of the space in the block to store pointers to the previous and next free blocks (maintaining a linked list).

So now let's take this idea for memory blocks, and what SiCrane said above, and document them to figure out how a block of memory might look to the allocator

Potential structure of a memory block allocated by new (that is, this block is allocated and in use):
Upper 29 bits: size of this block in memory
Bit 2: 1 if previous block is free, 0 if previous block is allocated

    Bit 1: 1 if next block is free, 0 if next block is allocated

    Bit 0: 1 if this block is free, 0 if this block is allocated

Pointer into segment table (4 bytes) (also acts as padding to maintain 8-byte alignment, like many CPUs need)

    Instead of wasting the padding, we'll store which segment into our segment table this block should be

    inserted into when its freed (so we don't have compute it when we free it, yay speed!)

User data (N bytes)

    This is the memory the programmer is actually able to use

Potential structure of a memory block allocated by new [] (that is, this block is allocated and in use):

    Upper 29 bits: size of this block in memory

    Bit 2: 1 if previous block is free, 0 if previous block is allocated

    Bit 1: 1 if next block is free, 0 if next block is allocated

    Bit 0: 1 if this block is free, 0 if this block is allocated

Number of elements (4 bytes) (also acts as padding to maintain 8-byte alignment, like many CPUs need)

    Instead of wasting the padding, we'll use it to store the number of elements in this array

User data (N bytes)

    This is the memory the programmer is actually able to use

Potential structure of a free block of memory (after it's been freed by delete or delete []):

    Upper 29 bits: size of this block in memory

    Bit 2: 1 if previous block is free, 0 if previous block is allocated

    Bit 1: 1 if next block is free, 0 if next block is allocated

    Bit 0: 1 if this block is free, 0 if this block is allocated

Pointer to the next free block (4 bytes)

    This way we can maintain a linked list of free blocks

Block data (N bytes)

    Junk

There are various allocation strategies and memory block formats, but the above encapsulates some common practices and ideas shared by many allocators.

Now, when the allocator tries to free a block of memory, it's going to try to use the block's header (the first 4 bytes of the block), and the following 4 bytes (what those bytes represent depends on whether or not you called new or new [], and depending on whether you call delete or delete [] it will change how those bytes are interpreted). If you call the delete on memory you used new [], the allocator will think the 4 bytes after the header are a pointer into the segment table (which they're not; those 4 bytes represent the number of elements in the array), and boom, you've successfully corrupted your memory (or hopefully crashed). If you call delete [] on memory you used new, the allocator will think the 4 bytes after the header are the number of elements in the array (which they're not; those 4 bytes represent a pointer into the segment table), and boom, you've successfully corrupted your memory (or hopefully crashed).

If different block formats and allocation strategies are used, it's possible to completely destroy the allocator's linked list of free/allocated blocks by calling the wrong delete, and now you've got memory leaks, you're overwriting data and double allocating the same blocks of memory for different uses, etc. Bad, bad, bad mojo.

Mixing delete with new [] or delete [] with new is not safe! Yes, it's possible the allocator implements new as new [1] (which would be a valid way to do it), and delete as delete [], but banking on that idea is like playing Russian roulette (with a gun that has more than one round in it).

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