Limacon - Calculus
•Download as PPTX, PDF•
1 like•1,106 views
Limacon is also called the Limacon of Pascal. It was first investigated by Dürer, who gave a method for drawing it in book Underweysung der Messung (1525). It was rediscovered by Étienne Pascal, father of Blaise Pascal, and named by Gilles-Personne Roberval in 1650 (MacTutor Archive).
Report
Share
Report
Share
Recommended
functions limits and continuity
This document provides an overview of functions, limits, and continuity. It defines key concepts such as domain and range of functions, and examples of standard real functions. It also covers even and odd functions, and how to calculate limits, including left and right hand limits. Methods for evaluating algebraic limits using substitution, factorization, and rationalization are presented. The objectives are to understand functions, domains, ranges, and how to evaluate limits of functions.
01 alphabets strings and languages
This document discusses the key concepts in the theory of computation including languages, alphabets, strings, and operations on languages. It defines languages as sets of strings over an alphabet, with alphabets as finite sets of symbols. Strings are finite sequences of symbols from an alphabet. Common operations on languages include union, concatenation, and Kleene star. The Kleene star of a language A consists of all possible concatenations of strings in A, including the empty string.
Connected and disconnected graph
This document defines and provides examples of connected and disconnected graphs. It also introduces matrix representations of graphs, including adjacency matrices and incidence matrices. Specifically, it states that a graph is connected if there is a path between every pair of vertices, and disconnected otherwise. Examples of connected and disconnected graphs are given. The document also defines adjacency matrices as representing the number of edges between vertices, and incidence matrices as representing the incidence of edges and vertices.
Boolean algebra
The document discusses Boolean algebra and its basic concepts. Boolean algebra uses 1 to represent True and 0 to represent False. It defines basic Boolean operations like AND, OR, and NOT. The document also shows an example of a Boolean equation to represent an "umbrella" proposition based on weather conditions. It then works through reducing a Boolean function step-by-step using Boolean identities and laws like commutative, associative, and distributive properties.
Flipflop
This document contains lecture slides about flip-flops and registers used in digital electronics. It discusses various types of flip-flops including SR, JK, T and D flip-flops. It also covers how multiple flip-flops can be connected to form registers and how data can be transferred and shifted through registers. Parallel adders with accumulators are also summarized, which allow storing a number in a register and adding a second number to it.
Polynomials and Curve Fitting in MATLAB
This document discusses various polynomial functions in MATLAB. It covers defining and manipulating polynomials, including evaluation, finding roots, addition/subtraction, multiplication/division, derivatives, and curve fitting using polynomial regression. Polynomials in MATLAB are defined as row vectors of coefficients. Key functions include polyval for evaluation, roots for finding roots, conv for multiplication, deconv for division, and polyfit for curve fitting.
Lattices
The document discusses lattices and their applications. It defines a lattice as a set with two binary operations that satisfy certain properties. Lattices are studied in order theory and abstract algebra. The document provides examples of crystal lattices, Bethe lattices, and how lattices are used in computer security and music theory. It also explains lattice multiplication, a method for multiplying multi-digit numbers using an array.
Lesson 16 length of an arc
1. This document discusses methods for calculating the length of an arc of a curve and the surface area of revolution. It provides formulas for finding arc length and surface area when curves are defined by rectangular coordinates, parametric equations, or polar coordinates.
2. Several examples are given of applying the formulas to find the arc length of curves and the surface area when graphs are revolved about axes. This includes revolving curves like y=x^3, y=x^2, and xy=2 about the x-axis and y-axis.
3. The key formulas presented are that arc length can be found using an integral of the form ∫√(dx/dy)^2 + 1 dy or
Recommended
functions limits and continuity
This document provides an overview of functions, limits, and continuity. It defines key concepts such as domain and range of functions, and examples of standard real functions. It also covers even and odd functions, and how to calculate limits, including left and right hand limits. Methods for evaluating algebraic limits using substitution, factorization, and rationalization are presented. The objectives are to understand functions, domains, ranges, and how to evaluate limits of functions.
01 alphabets strings and languages
This document discusses the key concepts in the theory of computation including languages, alphabets, strings, and operations on languages. It defines languages as sets of strings over an alphabet, with alphabets as finite sets of symbols. Strings are finite sequences of symbols from an alphabet. Common operations on languages include union, concatenation, and Kleene star. The Kleene star of a language A consists of all possible concatenations of strings in A, including the empty string.
Connected and disconnected graph
This document defines and provides examples of connected and disconnected graphs. It also introduces matrix representations of graphs, including adjacency matrices and incidence matrices. Specifically, it states that a graph is connected if there is a path between every pair of vertices, and disconnected otherwise. Examples of connected and disconnected graphs are given. The document also defines adjacency matrices as representing the number of edges between vertices, and incidence matrices as representing the incidence of edges and vertices.
Boolean algebra
The document discusses Boolean algebra and its basic concepts. Boolean algebra uses 1 to represent True and 0 to represent False. It defines basic Boolean operations like AND, OR, and NOT. The document also shows an example of a Boolean equation to represent an "umbrella" proposition based on weather conditions. It then works through reducing a Boolean function step-by-step using Boolean identities and laws like commutative, associative, and distributive properties.
Flipflop
This document contains lecture slides about flip-flops and registers used in digital electronics. It discusses various types of flip-flops including SR, JK, T and D flip-flops. It also covers how multiple flip-flops can be connected to form registers and how data can be transferred and shifted through registers. Parallel adders with accumulators are also summarized, which allow storing a number in a register and adding a second number to it.
Polynomials and Curve Fitting in MATLAB
This document discusses various polynomial functions in MATLAB. It covers defining and manipulating polynomials, including evaluation, finding roots, addition/subtraction, multiplication/division, derivatives, and curve fitting using polynomial regression. Polynomials in MATLAB are defined as row vectors of coefficients. Key functions include polyval for evaluation, roots for finding roots, conv for multiplication, deconv for division, and polyfit for curve fitting.
Lattices
The document discusses lattices and their applications. It defines a lattice as a set with two binary operations that satisfy certain properties. Lattices are studied in order theory and abstract algebra. The document provides examples of crystal lattices, Bethe lattices, and how lattices are used in computer security and music theory. It also explains lattice multiplication, a method for multiplying multi-digit numbers using an array.
Lesson 16 length of an arc
1. This document discusses methods for calculating the length of an arc of a curve and the surface area of revolution. It provides formulas for finding arc length and surface area when curves are defined by rectangular coordinates, parametric equations, or polar coordinates.
2. Several examples are given of applying the formulas to find the arc length of curves and the surface area when graphs are revolved about axes. This includes revolving curves like y=x^3, y=x^2, and xy=2 about the x-axis and y-axis.
3. The key formulas presented are that arc length can be found using an integral of the form ∫√(dx/dy)^2 + 1 dy or
Laplace transforms
This document provides an introduction to Laplace transforms. It defines the Laplace transform, lists some of its key properties including how it transforms derivatives and functions, and demonstrates how to use Laplace transforms to solve ordinary differential equations (ODEs). The document contains examples of taking Laplace transforms, applying properties like linearity and shifting, performing inverse Laplace transforms using tables and techniques like partial fractions, and solving a sample ODE using Laplace transforms. It also introduces concepts like the step function, Dirac delta function, and convolution as related topics.
Exponential functions
The document defines exponential functions as functions of the form f(x) = ax, where a is the base and a > 0, a ≠ 1. Exponential functions with bases greater than 1 have graphs that increase rapidly and have a horizontal asymptote of y = 0. Exponential functions with bases between 0 and 1 have graphs that increase more slowly and also have a horizontal asymptote of y = 0. Examples are given of sketching the graphs of various exponential functions, including translations and reflections of f(x) = 2x. The irrational number e, which is approximately 2.718, is important in applications involving growth and decay.
1.9. minimization of dfa
The document describes the process of minimizing a deterministic finite automaton (DFA) using the Myhill-Nerode theorem. It involves 4 steps: 1) creating a table of all state pairs, 2) marking pairs where one state is final and the other is not, 3) transitively marking additional pairs, 4) combining any remaining unmarked pairs into single states. An example is provided where the given 6-state DFA is minimized to a 3-state DFA using this process. Terminology used in DFA minimization is also defined.
Spherical Co-ordinate system (Applications)
This document discusses spherical coordinate systems and their applications. It begins by introducing spherical coordinates and their relationship to Cartesian coordinates. It then discusses how spheroids and spheres are used to model the shape of the Earth. It explains how latitude and longitude are used in geographic coordinate systems to specify locations on the Earth's surface. Finally, it discusses the Military Grid Reference System (MGRS) which provides more precise coordinates than typical GPS and how it divides the Earth into grid squares identified by letter-number combinations.
Digital logic circuit
This document provides an overview of digital logic circuits. It begins with an introduction to logic gates and Boolean algebra. Common logic gates like AND, OR, NOT are explained with their truth tables. Boolean algebra identities and theorems like De Morgan's theorem are listed as useful tools for simplifying logic functions. Karnaugh maps are introduced as a method to simplify Boolean functions into sum of products form. The document discusses various logic circuit design techniques including implementing logic functions from their truth tables or Karnaugh maps using logic gates.
L8 fuzzy relations contd.
The document discusses fuzzy relations and fuzzy composition. It defines the fuzzy Cartesian product of two fuzzy sets as a fuzzy relation contained within the full Cartesian product space, with the membership function equal to the minimum of the membership values from each fuzzy set. Fuzzy composition is defined similarly to crisp composition, with max-min composition taking the maximum of the minimum membership grades and max-product composition taking the maximum of the product membership grades. An example calculates the fuzzy Cartesian product of two sample fuzzy sets and performs max-min and max-product composition on two example fuzzy relations.
My Lecture Notes from Linear Algebra
1. The document provides notes from a linear algebra course, covering topics like matrix factorization, row reduction, column space, nullspace, and solving systems of equations.
2. Key concepts explained include LU, LDU, and row echelon factorizations of matrices. The column space and nullspace of a matrix are defined as important subspaces.
3. Solving systems of equations Ax=b is discussed, noting the solution set is the particular solution plus any vector in the nullspace. The system has a solution if and only if b is in the column space of A.
Eigenvalue eigenvector slides
This document discusses eigenvalue problems for matrices. It begins by defining eigenvalues and eigenvectors for a square matrix A. The eigenvalues are scalar values λ such that Ax = λx, where x is a corresponding eigenvector.
It then provides an example of finding the eigenvalues and eigenvectors for a 2x2 matrix. The characteristic equation is formed by taking the determinant of A - λI. The eigenvalues are the roots of the characteristic equation.
Several types of matrices are discussed, including symmetric, skew-symmetric, and orthogonal matrices. Properties of their eigenvalues are outlined, such as real eigenvalues for symmetric matrices. Applications to problems in physics, chemistry and engineering are mentioned.
13 05-curl-and-divergence
The document defines the operators curl and divergence for vector fields. Curl is defined as the cross product of del (the gradient operator) with the vector field and results in another vector. Divergence is defined as the dot product of del with the vector field and results in a scalar. Several examples of computing curl and divergence are worked out. Green's theorem, which relates line integrals of vector fields to surface integrals of curl and divergence, is also discussed.
25285 mws gen_int_ppt_trapcontinuous
The document discusses the trapezoidal rule of integration. It begins by defining integration as measuring the area under a function plotted on a graph. It then explains that the trapezoidal rule approximates this area as a trapezoid by assuming the integrand can be approximated by a linear polynomial. The rule takes the average of the function values at the limits of integration. The document provides an example problem and solution to demonstrate applying the single-segment and multiple-segment trapezoidal rule. It shows that dividing the interval into more segments reduces the true error of the approximation.
Coordinate system
An orthogonal system is one in which the coordinates arc mutually perpendicular
Examples of orthogonal coordinate systems include the Cartesian (or rectangular), the circular cylindrical, the spherical, the elliptic cylindrical, the parabolic cylindrical, the conical, the prolate spheroidal, the oblate spheroidal, and the ellipsoidal.1
A considerable amount of work and time may be saved by choosing a coordinate system that best fits a given problem
A hard problem in one coordi nate system may turn out to be easy in another system.
In this text, we shall restrict ourselves to the three best-known coordinate systems: the Cartesian, the circular cylindrical, and the spherical.
Fuzzy sets
This document summarizes Lotfi Zadeh's 1965 paper that introduced fuzzy set theory. It defines fuzzy sets as sets with imprecise or unclear boundaries, where elements can partially belong through degrees of membership between 0 and 1. It provides key definitions for fuzzy sets, including complement, subset, union, intersection and algebraic operations. Convex combinations and fuzzy relations are also introduced. The document concludes with a definition for convex fuzzy sets.
Vcla.ppt COMPOSITION OF LINEAR TRANSFORMATION KERNEL AND RANGE OF LINEAR TR...
COMPOSITION OF LINEAR TRANSFORMATION
KERNEL AND RANGE OF LINEAR TRANSFORMATION
INVERSE OF LINEAR TRANSFORMATION
Functional analysis
This document discusses key concepts in functional analysis including function spaces, metric spaces, dense subsets, linear spaces, and linear functionals. It provides examples of different types of function spaces like C[a,b] and L1[a,b]. Metric spaces are defined as pairs consisting of a space X and a distance function satisfying properties like non-negativity and triangle inequality. Examples of metric spaces include R and Rn. Dense subsets are defined as sets whose closure is equal to the entire space. Linear spaces satisfy properties like vector addition and scalar multiplication. Linear functionals are functions that map elements of a linear space to real numbers and satisfy properties like additivity and homogeneity.
Dijkstra c
Dijkstra's algorithm is used to find the shortest path between a source node and all other nodes in a graph. It works by maintaining two sets - S of nodes whose shortest paths have been determined, and V-S of remaining nodes. It initializes distances and predecessors, then iteratively relaxes edges of the closest node in V-S until V-S is empty, determining the shortest path to each node. The algorithm runs in O(|E| + |V|)log|V|) time.
Cubic Equation
The document provides a history of solving cubic equations from ancient Greece to modern times. It describes how Greek mathematicians were trying to construct an angle one-third the size of a given angle, which led to the development of trigonometry and cubic equations. Several mathematicians, including Al-Khayammi, Del Ferro, Tartaglia, and Cardano, made discoveries in solving cubic equations. The document then explains how to derive the formula for solving cubic equations by finding the "depressed cubic" without the quadratic term, and uses the quadratic formula to solve for the solutions.
lattice
A lattice is a partially ordered set where every pair of elements has a supremum (least upper bound) and infimum (greatest lower bound). A lattice must satisfy the properties that (1) any two elements have a supremum and infimum and (2) the supremum of two elements is their join and the infimum is their meet. Common examples of lattices include the natural numbers under the divisibility relation and sets under the subset relation. Lattice theory has applications in many areas of computer science and engineering such as distributed computing, concurrency theory, and programming language semantics.
Linear differential equation with constant coefficient
The document discusses linear differential equations with constant coefficients. It defines the order, auxiliary equation, complementary function, particular integral and general solution. It provides examples of determining the complementary function and particular integral for different types of linear differential equations. It also discusses Legendre's linear equations, Cauchy-Euler equations, and solving simultaneous linear differential equations.
Topology M.Sc. 2 semester Mathematics compactness, unit - 4
Topology M.Sc. 2 semester Mathematics compactness, unit - 4Shri Shankaracharya College, Bhilai,Junwani
This ppt covers following topics of Unit - 4 of M.Sc. Maths 2 sem. Compact spaces & Subspace, Bolzano Weierstrass theorem, Heine borel theorem,Independence, basis and dimension
This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
LIMACON 2014
This document is a registration form and program for the Twenty-Eighth Annual LIMAÇON Long Island Mathematics Conference taking place on March 14, 2014 at SUNY College at Old Westbury. The conference will run from 7:45am to 2:35pm and feature sessions on various mathematics topics as well as a keynote address. The registration form provides information on pricing and deadlines and allows attendees to select which sessions they plan to attend.
10.8
This document provides 5 examples of sketching the graph of polar equations, with exercise problems 2 through 6 and 18 through 46 from page 791 suggested for practice. Solutions are not shown.
More Related Content
What's hot
Laplace transforms
This document provides an introduction to Laplace transforms. It defines the Laplace transform, lists some of its key properties including how it transforms derivatives and functions, and demonstrates how to use Laplace transforms to solve ordinary differential equations (ODEs). The document contains examples of taking Laplace transforms, applying properties like linearity and shifting, performing inverse Laplace transforms using tables and techniques like partial fractions, and solving a sample ODE using Laplace transforms. It also introduces concepts like the step function, Dirac delta function, and convolution as related topics.
Exponential functions
The document defines exponential functions as functions of the form f(x) = ax, where a is the base and a > 0, a ≠ 1. Exponential functions with bases greater than 1 have graphs that increase rapidly and have a horizontal asymptote of y = 0. Exponential functions with bases between 0 and 1 have graphs that increase more slowly and also have a horizontal asymptote of y = 0. Examples are given of sketching the graphs of various exponential functions, including translations and reflections of f(x) = 2x. The irrational number e, which is approximately 2.718, is important in applications involving growth and decay.
1.9. minimization of dfa
The document describes the process of minimizing a deterministic finite automaton (DFA) using the Myhill-Nerode theorem. It involves 4 steps: 1) creating a table of all state pairs, 2) marking pairs where one state is final and the other is not, 3) transitively marking additional pairs, 4) combining any remaining unmarked pairs into single states. An example is provided where the given 6-state DFA is minimized to a 3-state DFA using this process. Terminology used in DFA minimization is also defined.
Spherical Co-ordinate system (Applications)
This document discusses spherical coordinate systems and their applications. It begins by introducing spherical coordinates and their relationship to Cartesian coordinates. It then discusses how spheroids and spheres are used to model the shape of the Earth. It explains how latitude and longitude are used in geographic coordinate systems to specify locations on the Earth's surface. Finally, it discusses the Military Grid Reference System (MGRS) which provides more precise coordinates than typical GPS and how it divides the Earth into grid squares identified by letter-number combinations.
Digital logic circuit
This document provides an overview of digital logic circuits. It begins with an introduction to logic gates and Boolean algebra. Common logic gates like AND, OR, NOT are explained with their truth tables. Boolean algebra identities and theorems like De Morgan's theorem are listed as useful tools for simplifying logic functions. Karnaugh maps are introduced as a method to simplify Boolean functions into sum of products form. The document discusses various logic circuit design techniques including implementing logic functions from their truth tables or Karnaugh maps using logic gates.
L8 fuzzy relations contd.
The document discusses fuzzy relations and fuzzy composition. It defines the fuzzy Cartesian product of two fuzzy sets as a fuzzy relation contained within the full Cartesian product space, with the membership function equal to the minimum of the membership values from each fuzzy set. Fuzzy composition is defined similarly to crisp composition, with max-min composition taking the maximum of the minimum membership grades and max-product composition taking the maximum of the product membership grades. An example calculates the fuzzy Cartesian product of two sample fuzzy sets and performs max-min and max-product composition on two example fuzzy relations.
My Lecture Notes from Linear Algebra
1. The document provides notes from a linear algebra course, covering topics like matrix factorization, row reduction, column space, nullspace, and solving systems of equations.
2. Key concepts explained include LU, LDU, and row echelon factorizations of matrices. The column space and nullspace of a matrix are defined as important subspaces.
3. Solving systems of equations Ax=b is discussed, noting the solution set is the particular solution plus any vector in the nullspace. The system has a solution if and only if b is in the column space of A.
Eigenvalue eigenvector slides
This document discusses eigenvalue problems for matrices. It begins by defining eigenvalues and eigenvectors for a square matrix A. The eigenvalues are scalar values λ such that Ax = λx, where x is a corresponding eigenvector.
It then provides an example of finding the eigenvalues and eigenvectors for a 2x2 matrix. The characteristic equation is formed by taking the determinant of A - λI. The eigenvalues are the roots of the characteristic equation.
Several types of matrices are discussed, including symmetric, skew-symmetric, and orthogonal matrices. Properties of their eigenvalues are outlined, such as real eigenvalues for symmetric matrices. Applications to problems in physics, chemistry and engineering are mentioned.
13 05-curl-and-divergence
The document defines the operators curl and divergence for vector fields. Curl is defined as the cross product of del (the gradient operator) with the vector field and results in another vector. Divergence is defined as the dot product of del with the vector field and results in a scalar. Several examples of computing curl and divergence are worked out. Green's theorem, which relates line integrals of vector fields to surface integrals of curl and divergence, is also discussed.
25285 mws gen_int_ppt_trapcontinuous
The document discusses the trapezoidal rule of integration. It begins by defining integration as measuring the area under a function plotted on a graph. It then explains that the trapezoidal rule approximates this area as a trapezoid by assuming the integrand can be approximated by a linear polynomial. The rule takes the average of the function values at the limits of integration. The document provides an example problem and solution to demonstrate applying the single-segment and multiple-segment trapezoidal rule. It shows that dividing the interval into more segments reduces the true error of the approximation.
Coordinate system
An orthogonal system is one in which the coordinates arc mutually perpendicular
Examples of orthogonal coordinate systems include the Cartesian (or rectangular), the circular cylindrical, the spherical, the elliptic cylindrical, the parabolic cylindrical, the conical, the prolate spheroidal, the oblate spheroidal, and the ellipsoidal.1
A considerable amount of work and time may be saved by choosing a coordinate system that best fits a given problem
A hard problem in one coordi nate system may turn out to be easy in another system.
In this text, we shall restrict ourselves to the three best-known coordinate systems: the Cartesian, the circular cylindrical, and the spherical.
Fuzzy sets
This document summarizes Lotfi Zadeh's 1965 paper that introduced fuzzy set theory. It defines fuzzy sets as sets with imprecise or unclear boundaries, where elements can partially belong through degrees of membership between 0 and 1. It provides key definitions for fuzzy sets, including complement, subset, union, intersection and algebraic operations. Convex combinations and fuzzy relations are also introduced. The document concludes with a definition for convex fuzzy sets.
Vcla.ppt COMPOSITION OF LINEAR TRANSFORMATION KERNEL AND RANGE OF LINEAR TR...
COMPOSITION OF LINEAR TRANSFORMATION
KERNEL AND RANGE OF LINEAR TRANSFORMATION
INVERSE OF LINEAR TRANSFORMATION
Functional analysis
This document discusses key concepts in functional analysis including function spaces, metric spaces, dense subsets, linear spaces, and linear functionals. It provides examples of different types of function spaces like C[a,b] and L1[a,b]. Metric spaces are defined as pairs consisting of a space X and a distance function satisfying properties like non-negativity and triangle inequality. Examples of metric spaces include R and Rn. Dense subsets are defined as sets whose closure is equal to the entire space. Linear spaces satisfy properties like vector addition and scalar multiplication. Linear functionals are functions that map elements of a linear space to real numbers and satisfy properties like additivity and homogeneity.
Dijkstra c
Dijkstra's algorithm is used to find the shortest path between a source node and all other nodes in a graph. It works by maintaining two sets - S of nodes whose shortest paths have been determined, and V-S of remaining nodes. It initializes distances and predecessors, then iteratively relaxes edges of the closest node in V-S until V-S is empty, determining the shortest path to each node. The algorithm runs in O(|E| + |V|)log|V|) time.
Cubic Equation
The document provides a history of solving cubic equations from ancient Greece to modern times. It describes how Greek mathematicians were trying to construct an angle one-third the size of a given angle, which led to the development of trigonometry and cubic equations. Several mathematicians, including Al-Khayammi, Del Ferro, Tartaglia, and Cardano, made discoveries in solving cubic equations. The document then explains how to derive the formula for solving cubic equations by finding the "depressed cubic" without the quadratic term, and uses the quadratic formula to solve for the solutions.
lattice
A lattice is a partially ordered set where every pair of elements has a supremum (least upper bound) and infimum (greatest lower bound). A lattice must satisfy the properties that (1) any two elements have a supremum and infimum and (2) the supremum of two elements is their join and the infimum is their meet. Common examples of lattices include the natural numbers under the divisibility relation and sets under the subset relation. Lattice theory has applications in many areas of computer science and engineering such as distributed computing, concurrency theory, and programming language semantics.
Linear differential equation with constant coefficient
The document discusses linear differential equations with constant coefficients. It defines the order, auxiliary equation, complementary function, particular integral and general solution. It provides examples of determining the complementary function and particular integral for different types of linear differential equations. It also discusses Legendre's linear equations, Cauchy-Euler equations, and solving simultaneous linear differential equations.
Topology M.Sc. 2 semester Mathematics compactness, unit - 4
Topology M.Sc. 2 semester Mathematics compactness, unit - 4Shri Shankaracharya College, Bhilai,Junwani
This ppt covers following topics of Unit - 4 of M.Sc. Maths 2 sem. Compact spaces & Subspace, Bolzano Weierstrass theorem, Heine borel theorem,Independence, basis and dimension
This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
What's hot (20)
Vcla.ppt COMPOSITION OF LINEAR TRANSFORMATION KERNEL AND RANGE OF LINEAR TR...
Vcla.ppt COMPOSITION OF LINEAR TRANSFORMATION KERNEL AND RANGE OF LINEAR TR...
Linear differential equation with constant coefficient
Linear differential equation with constant coefficient
Topology M.Sc. 2 semester Mathematics compactness, unit - 4
Topology M.Sc. 2 semester Mathematics compactness, unit - 4
Viewers also liked
LIMACON 2014
This document is a registration form and program for the Twenty-Eighth Annual LIMAÇON Long Island Mathematics Conference taking place on March 14, 2014 at SUNY College at Old Westbury. The conference will run from 7:45am to 2:35pm and feature sessions on various mathematics topics as well as a keynote address. The registration form provides information on pricing and deadlines and allows attendees to select which sessions they plan to attend.
10.8
This document provides 5 examples of sketching the graph of polar equations, with exercise problems 2 through 6 and 18 through 46 from page 791 suggested for practice. Solutions are not shown.
n-Curving - A Transformation of Curves
A method of transforming plane curves, developed on the basis of Functional Theoretic Algebra.
http://en.wikipedia.org/wiki/Functional-theoretic_algebra
Lec 13 Traffic Light Signals (Transportation Engineering Dr.Lina Shbeeb)
Traffic light signals regulate traffic flow at intersections through different colored lights and timed phases. They aim to improve safety by organizing conflicting traffic streams and increase junction capacity. A signal cycle consists of phases allotted to traffic movements, with intervals for changing lights. Sensors detect vehicle presence and adjust timings. Manual design methods calculate optimal cycle lengths and phase timings based on traffic volumes and pedestrian clearance needs to minimize delays.
Bus 16 (Transportation Engineering Dr.Lina Shbeeb)
This document provides guidelines for designing rural bus stops. It discusses key considerations for bus stop placement and amenities. Recommendations include using driveways or low-traffic streets as informal boarding areas where full pads aren't feasible, requiring paved surfaces, and ensuring adequate sight distances and grades of less than 2%. The guidelines also provide specifications for bus stop signage, shelters, benches and other amenities based on daily boardings. Safety factors for pedestrian crossings and specifications for bus turnouts in rural and urban environments are also outlined.
Lecture 05 Roundabout (Traffic Engineering هندسة المرور & Dr. Usama Shahdah)
This document summarizes key points from a traffic engineering course lecture on roundabouts. It discusses conflict points, geometric characteristics of roundabouts like entry width and exit width. It covers calculating volumes using passenger car equivalents and peak hour factors. An example is provided on calculating entry, exit, and circulating volumes. Finally, it discusses methods for determining roundabout capacity like the HCM 2000 and Kimber methods, and using software like Synchro to analyze roundabouts.
Photodiode
Photodiodes are semiconductor light sensors that generate a current or voltage when illuminated by light. They work by absorbing photons which creates electron-hole pairs, generating a photocurrent. Photodiodes can operate in photovoltaic, reverse bias, or forward bias modes. They are made from materials like silicon, germanium, or indium gallium arsenide and constructed with a PN or PIN junction. Common applications include use in consumer electronics, light sensors, medical devices, and scientific instruments.
Lesson 15 polar curves
This document provides an overview of polar curves and coordinates. It defines polar coordinates, explains how to plot points in a polar coordinate system, and how to convert between polar and rectangular coordinates. Examples are given to plot points with polar coordinates and express rectangular coordinates in terms of polar coordinates. Finally, common polar curves such as the cardiod, limacon, roses, spirals, and lemniscate are listed.
Lecture 04 Capacity for TWSC (Traffic Engineering هندسة المرور & Dr. Usama Sh...
This document discusses gap acceptance theory and its application in determining the capacity of traffic movements at two-way stop controlled (TWSC) intersections. It covers key concepts such as critical gap (tc), follow-up time (tf), and impedance. An example calculation is provided to estimate capacity for different movements based on conflicting traffic volumes and tc/tf values. Adjustments to tc and tf for factors like vehicle type and number of lanes are also outlined. Finally, the document provides the Highway Capacity Manual (HCM) methodology for calculating control delay and level of service at TWSC intersections.
Lec 08 Cross Section (Transportation Engineering Dr.Lina Shbeeb)
This document discusses elements of highway cross sections, including the roadway, median, and roadside. It describes key components of the roadway like travel lanes and shoulders. Travel lanes are meant to guide drivers and support vehicles, and their width depends on factors like traffic volume and environment. Shoulders provide lateral support and space for avoidance or emergency stopping. The document also outlines different types of medians and their functions in separating opposing traffic and providing drainage or space for future lanes. It notes design considerations for elements like cross slope and barriers.
Lecture 01 Introduction (Traffic Engineering هندسة المرور & Dr. Usama Shahdah)
This document provides an overview of a traffic engineering course, including:
- Contact information for the instructor and information about the course website.
- Requirements and grading breakdown, including assignments, exams, and a term project.
- References and software used in the course.
- An introduction to traffic engineering and what traffic engineers do, such as conducting traffic studies, evaluating performance, designing facilities, and controlling traffic.
- Components of a traffic system including diverse road users and vehicles, and the complex roadway network.
- The role of traffic flow theory in modeling and analyzing traffic systems.
Traffic flow model
Adjusting the flow in crucial areas can maximize the overall throughput of traffic along a stretch of road. This is of particular interest in regions of high traffic density, which may be caused by high volume peak time traffic, accidents or closure of one or more lanes of the road.
Lesson 14 a - parametric equations
Parametric equations define a curve where x and y are defined in terms of a third variable called a parameter. The graph of parametric equations consists of all points (x,y) obtained by allowing the parameter to vary over its domain. Eliminating the parameter between the two equations yields a non-parametric equation of the curve. Examples are provided of eliminating parameters between various parametric equations to obtain the curve and sketching the resulting graphs. Exercises are given to further practice eliminating parameters and sketching curves.
Polar Graphs: Limaçons, Roses, Lemniscates, & Cardioids
Learn about different polar graphs, including limaçons (convex, dimpled, looped), lemniscates, rose curves, and cardioids. View compare and contrast between the 4 different types of polar graphs, and view my impressions on this final unit in pre-calculus honors.
Lecture 03 Intersection Control (Traffic Engineering هندسة المرور & Dr. Usama...
This document provides information about different types of intersection control, including unsignalized intersections under basic rules of the road (Level I), intersections with stop or yield control (Level II), and signalized intersections (Level III). It discusses factors that affect intersection operation such as sight distance and gap availability. It also covers warrants and guidelines for implementing different levels of intersection control based on traffic volumes, delays, crashes, and other data. Sample problems are provided to demonstrate how to apply the warrants and determine if signalization is justified.
Lec 10 Traffic Stream Models (Transportation Engineering Dr.Lina Shbeeb)
This document discusses traffic stream flow models and the variables used to describe traffic flow conditions. It introduces concepts like traffic flow, concentration, speed, spacing, headway, and gap. Flow-density relationships are presented, including the Greenshield linear model that shows the relationship between density, flow, and speed in traffic. Car following rules are discussed as the basis for models that determine safe spacing between vehicles based on speed.
Microphone Patterns
There are several microphone patterns that determine the directionality of sound pickup. Omni-directional microphones pick up sound equally from all directions, making them good for interviews where mic angle is variable. Cardioid microphones are most sensitive to sound from the front and pick up less from the sides and back, reducing feedback and ambient noise. Bi-directional microphones pick up sound equally from the two sides but not the front or back. Mid-side stereo techniques use a bi-directional mic together with a cardioid mic to allow control over the left-right stereo image during mixing. Hypercardioid and shotgun microphones have a tight, directional pickup pattern that focuses on sound from directly in front
Application of Integration
The document discusses applications of integration, including calculating the length of a curve and surface area of solids obtained by rotating curves. It provides formulas for finding the arc length of a curve given by y=f(x), and surface area of solids obtained by rotating curves about the x- or y-axes. Examples are worked out applying these formulas to find the arc length of curves and surface area of rotated regions. The document also discusses evaluating triple integrals to find the volume of a three-dimensional region and using triple integrals to find the centroid of a volume.
Curve tracing
Polar coordinates represent points in a plane using an ordered pair (r, θ) where r is the distance from a fixed point called the pole (or origin) and θ is the angle between the line from the pole to the point and a reference axis called the polar axis. Some key properties:
- Points (r, θ) and (-r, θ + π) represent the same location.
- Polar equations like r = a describe circles with radius a centered at the pole.
- Equations can be converted between Cartesian (x, y) and polar (r, θ) coordinates using trigonometric relationships.
- Graphs of polar equations r = f(θ) consist of all points satisfying
S4 E Phy Waves(Tranverse)(T)
Transverse waves transfer energy from one place to another without the physical transfer of the medium between the two points. Transverse waves are waves in which the particles of the medium move at right angles to the direction of wave motion. The key characteristics of transverse waves are that the particles vibrate perpendicular to the direction of wave propagation, and properties like amplitude, wavelength, frequency, and speed can be used to describe wave behavior.
Viewers also liked (20)
Lec 13 Traffic Light Signals (Transportation Engineering Dr.Lina Shbeeb)
Lec 13 Traffic Light Signals (Transportation Engineering Dr.Lina Shbeeb)
Bus 16 (Transportation Engineering Dr.Lina Shbeeb)
Bus 16 (Transportation Engineering Dr.Lina Shbeeb)
Lecture 05 Roundabout (Traffic Engineering هندسة المرور & Dr. Usama Shahdah)
Lecture 05 Roundabout (Traffic Engineering هندسة المرور & Dr. Usama Shahdah)
Lecture 04 Capacity for TWSC (Traffic Engineering هندسة المرور & Dr. Usama Sh...
Lecture 04 Capacity for TWSC (Traffic Engineering هندسة المرور & Dr. Usama Sh...
Lec 08 Cross Section (Transportation Engineering Dr.Lina Shbeeb)
Lec 08 Cross Section (Transportation Engineering Dr.Lina Shbeeb)
Lecture 01 Introduction (Traffic Engineering هندسة المرور & Dr. Usama Shahdah)
Lecture 01 Introduction (Traffic Engineering هندسة المرور & Dr. Usama Shahdah)
Polar Graphs: Limaçons, Roses, Lemniscates, & Cardioids
Polar Graphs: Limaçons, Roses, Lemniscates, & Cardioids
Lecture 03 Intersection Control (Traffic Engineering هندسة المرور & Dr. Usama...
Lecture 03 Intersection Control (Traffic Engineering هندسة المرور & Dr. Usama...
Lec 10 Traffic Stream Models (Transportation Engineering Dr.Lina Shbeeb)
Lec 10 Traffic Stream Models (Transportation Engineering Dr.Lina Shbeeb)
Recently uploaded
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
哪里办理(csu毕业证书)查尔斯特大学毕业证硕士学历原版一模一样
原版一模一样【微信：741003700 】【(csu毕业证书)查尔斯特大学毕业证硕士学历】【微信：741003700 】学位证，留信认证（真实可查，永久存档）offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原海外各大学 Bachelor Diploma degree, Master Degree Diploma
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
Eric Nizeyimana's document 2006 from gicumbi to ttc nyamata handball play
this notes have been created
by Eric36 at nyamata ttc
after readings
pe book for y2sme .
Embedded machine learning-based road conditions and driving behavior monitoring
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Computational Engineering IITH Presentation
This Presentation will give you a brief idea about what Computational Engineering at IIT Hyderabad has to offer.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, SerbiaNew techniques for characterising damage in rock slopes.pdf
rock mass characterization and New techniques for characterising damage in rock slopes
ISPM 15 Heat Treated Wood Stamps and why your shipping must have one
For International shipping and maritime laws all wood must contain the ISPM 15 Stamp. Here is how and why.
5214-1693458878915-Unit 6 2023 to 2024 academic year assignment (AutoRecovere...
Bigdata of technology
学校原版美国波士顿大学毕业证学历学位证书原版一模一样
原版一模一样【微信：741003700 】【美国波士顿大学毕业证学历学位证书】【微信：741003700 】学位证，留信认证（真实可查，永久存档）offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原海外各大学 Bachelor Diploma degree, Master Degree Diploma
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
ACEP Magazine edition 4th launched on 05.06.2024
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
官方认证美国密歇根州立大学毕业证学位证书原版一模一样
原版一模一样【微信：741003700 】【美国密歇根州立大学毕业证学位证书】【微信：741003700 】学位证，留信认证（真实可查，永久存档）offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原海外各大学 Bachelor Diploma degree, Master Degree Diploma
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMS
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
Advanced control scheme of doubly fed induction generator for wind turbine us...
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
Recently uploaded (20)
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024
Eric Nizeyimana's document 2006 from gicumbi to ttc nyamata handball play
Eric Nizeyimana's document 2006 from gicumbi to ttc nyamata handball play
Embedded machine learning-based road conditions and driving behavior monitoring
Embedded machine learning-based road conditions and driving behavior monitoring
Engineering Drawings Lecture Detail Drawings 2014.pdf
Engineering Drawings Lecture Detail Drawings 2014.pdf
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...
New techniques for characterising damage in rock slopes.pdf
New techniques for characterising damage in rock slopes.pdf
ISPM 15 Heat Treated Wood Stamps and why your shipping must have one
ISPM 15 Heat Treated Wood Stamps and why your shipping must have one
5214-1693458878915-Unit 6 2023 to 2024 academic year assignment (AutoRecovere...
5214-1693458878915-Unit 6 2023 to 2024 academic year assignment (AutoRecovere...
Generative AI leverages algorithms to create various forms of content
Generative AI leverages algorithms to create various forms of content
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMS
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMS
Advanced control scheme of doubly fed induction generator for wind turbine us...
Advanced control scheme of doubly fed induction generator for wind turbine us...
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODEL
Limacon - Calculus
- 2. FANTASTIC FOUR PRESENTS 1.MOHIT BALHARA 2.UTKARSH ANAND 3.ABHISHEK CHIIKARA 4.HARIKESH KUMAR
- 3. The Limacon is a polar curve of the form having Equation r=b+acosΘ Cartesian Equation (x2 + y2 - 2ax)2 = b2(x2 + y2) Some basic curves of limacon. INTRODUCTION
- 4. Some other examples of Limacon
- 5. Limacon is also called the Limacon of Pascal. It was first investigated by Dürer, who gave a method for drawing it in book Underweysung der Messung (1525). It was rediscovered by Étienne Pascal, father of Blaise Pascal, and named by Gilles- Personne Roberval in 1650 (MacTutor Archive). The word “Limacon" comes from the Latin limax, meaning "snail." HISTORY
- 6. As we know the equation of Limacon is r=b+acosѲ In it, If,b>=2a the Limacon is convex. If, 2a>b>a the Limacon is dimpled. If, b=a the Limacon degenerates to a cardioid. If, b<a the Limacon has an inner loop. If, b=a/2 it is a trisectrix. SOME BASIC CONDITIONS
- 7. The first one is the concaved. Second one is dimpled. The third one is cardioid. CONSTRUCTION OF LIMACON
- 8. Limacon — Pedal Curve of a Circle
- 9. For , the inner loop has area. = = = where . AREA OCCUPIED BY LIMACON
- 10. For , the outer loop has area = = = Area Contd.
- 11. Thus, the area between the loops is But there is a special case for b = a/2 Area Contd.
- 12. In the special case of b=a/2 , these simplify to Area Contd. = = =
- 13. The parametric form of Limacon is: x= (b + a cos t) cos t y = (b + a cos t) sin t gives the arc length s(t) as a function of t as where E(z,k) is an elliptic integral of the second kind. PARAMETRIC FORM
- 14. Let t=2𝜋 gives the arc length of the entire curve as where E(k) is a complete elliptic integral of the second kind. Contd.
- 15. Limacon Evolute means a limacon which is the locus of the centre of curvature of another limacon. Some examples are, LIMACON EVOLUTE
- 16. The catacaustic of a circle for a radiant point is the limacon evolute. It has parametric equations. = = Contd.
- 18. THANK YOU