This document provides two examples of using double and triple integrals to calculate the moment of inertia and volume of solids. The first example calculates the moment of inertia of a solid inside a cylinder using cylindrical coordinates. The second example finds the volume of a solid inside a sphere and outside a cone using spherical coordinates. It converts the equations to spherical coordinates and sets up the integral to evaluate the volume.
The document summarizes key concepts and applications of integration. It discusses:
1) Important historical figures like Archimedes, Gauss, Leibniz and Newton who contributed to the development of integration and calculus.
2) Engineering applications of integration like in the design of the Petronas Towers and Sydney Opera House.
3) The integration by parts formula and examples of using it to evaluate integrals of composite functions.
The document discusses different methods for calculating the volume of a solid of revolution: disk method, washer method, and shell method. It provides examples of applying each method to find the volume generated when an area bounded by curves is revolved around an axis. The disk method calculates volume by summing the volumes of thin circular disks. The washer method accounts for holes by subtracting the inner circular area from the outer. The shell method imagines the solid as nested cylindrical shells and sums their individual volumes.
The document discusses convex functions and related concepts. It defines convex functions and provides examples of convex and concave functions on R and Rn, including norms, logarithms, and powers. It describes properties that preserve convexity, such as positive weighted sums and composition with affine functions. The conjugate function and quasiconvex functions are also introduced. Key concepts are illustrated with examples throughout.
The document discusses the history and uses of mapping and conformal mapping. It provides examples of some of the oldest known maps dating back to 6200 BC. It then discusses how conformal mapping originated from rejecting the idea that the Earth is flat, and how conformal mapping preserves angles to give a realistic view of the physical world on a map. Some key uses of conformal mapping mentioned are finding inverses of complex numbers, solving Laplace equations, and determining heat conduction.
This document discusses antiderivatives and indefinite integrals. It begins by introducing the concept of an antiderivative, which is a function whose derivative is a known function. It then defines the indefinite integral as representing the set of all antiderivatives. Several properties of antiderivatives and indefinite integrals are presented, including: the constant of integration; basic integration rules like power, exponential, and logarithmic rules; and notation used to represent indefinite integrals. Examples are provided to illustrate key concepts and properties.
In this lecture, we will discuss:
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
The area under a curve between two x-values is the definite integral of the function. This area can be positive if above the x-axis and negative if below. To find the total area under a curve, the curve is broken into sections where the function is either above or below zero and the integral is evaluated over each section adding or subtracting areas as appropriate. The example problem demonstrates finding the total area under the curve defined by y=x^2-x-2 between -2 and 3 by breaking it into three sections and evaluating the integral over each.
This document provides two examples of using double and triple integrals to calculate the moment of inertia and volume of solids. The first example calculates the moment of inertia of a solid inside a cylinder using cylindrical coordinates. The second example finds the volume of a solid inside a sphere and outside a cone using spherical coordinates. It converts the equations to spherical coordinates and sets up the integral to evaluate the volume.
The document summarizes key concepts and applications of integration. It discusses:
1) Important historical figures like Archimedes, Gauss, Leibniz and Newton who contributed to the development of integration and calculus.
2) Engineering applications of integration like in the design of the Petronas Towers and Sydney Opera House.
3) The integration by parts formula and examples of using it to evaluate integrals of composite functions.
The document discusses different methods for calculating the volume of a solid of revolution: disk method, washer method, and shell method. It provides examples of applying each method to find the volume generated when an area bounded by curves is revolved around an axis. The disk method calculates volume by summing the volumes of thin circular disks. The washer method accounts for holes by subtracting the inner circular area from the outer. The shell method imagines the solid as nested cylindrical shells and sums their individual volumes.
The document discusses convex functions and related concepts. It defines convex functions and provides examples of convex and concave functions on R and Rn, including norms, logarithms, and powers. It describes properties that preserve convexity, such as positive weighted sums and composition with affine functions. The conjugate function and quasiconvex functions are also introduced. Key concepts are illustrated with examples throughout.
The document discusses the history and uses of mapping and conformal mapping. It provides examples of some of the oldest known maps dating back to 6200 BC. It then discusses how conformal mapping originated from rejecting the idea that the Earth is flat, and how conformal mapping preserves angles to give a realistic view of the physical world on a map. Some key uses of conformal mapping mentioned are finding inverses of complex numbers, solving Laplace equations, and determining heat conduction.
This document discusses antiderivatives and indefinite integrals. It begins by introducing the concept of an antiderivative, which is a function whose derivative is a known function. It then defines the indefinite integral as representing the set of all antiderivatives. Several properties of antiderivatives and indefinite integrals are presented, including: the constant of integration; basic integration rules like power, exponential, and logarithmic rules; and notation used to represent indefinite integrals. Examples are provided to illustrate key concepts and properties.
In this lecture, we will discuss:
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
The area under a curve between two x-values is the definite integral of the function. This area can be positive if above the x-axis and negative if below. To find the total area under a curve, the curve is broken into sections where the function is either above or below zero and the integral is evaluated over each section adding or subtracting areas as appropriate. The example problem demonstrates finding the total area under the curve defined by y=x^2-x-2 between -2 and 3 by breaking it into three sections and evaluating the integral over each.
1. The document provides information on multiple integrals including double integrals, triple integrals, and integrals in spherical and cylindrical coordinates. It defines each type of integral and gives their general formulas.
2. Examples are provided for calculating double and triple integrals over different regions in rectangular, cylindrical, and spherical coordinate systems. The order of integration can be changed by considering strips or slices of the region.
3. Properties of the integrals include applying Fubini's theorem to change the order of integration, and relating the triple integral over a region to the double integral over the bounds and integrating over the third variable.
This document provides an overview of triple integrals and their applications. It defines triple integrals as the limit of triple Riemann sums for functions of three variables, analogous to double integrals. Triple integrals can be evaluated over rectangular boxes by expressing them as iterated integrals in any of six orders, as stated by Fubini's theorem. The document also describes how to evaluate triple integrals over more general bounded solid regions, including type 1 regions bounded by two graphs, type 2 regions bounded between two planes, and type 3 regions. It provides examples of evaluating triple integrals over a tetrahedron and other specific regions.
Exact Matrix Completion via Convex Optimization Slide (PPT)Joonyoung Yi
Slide of the paper "Exact Matrix Completion via Convex Optimization" of Emmanuel J. Candès and Benjamin Recht. We presented this slide in KAIST CS592 Class, April 2018.
- Code: https://github.com/JoonyoungYi/MCCO-numpy
- Abstract of the paper: We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M. Can we complete the matrix and recover the entries that we have not seen? We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys
𝑚≥𝐶𝑛1.2𝑟log𝑛
for some positive numerical constant C, then with very high probability, most n×n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.
This document provides an overview of integral calculus, including its history, definition, techniques, and applications. It traces integration back to ancient Egypt and developments by Archimedes, Liu Hui, Ibn al-Haytham, Newton, Leibniz, Cauchy, and Riemann. Key techniques discussed are integration by general rule, integration by parts, and integration by substitution. Applications mentioned include designing tall buildings and the Sydney Opera House to withstand forces, and historically calculating wine cask volumes.
The document defines definite integrals and discusses their properties. It states that a definite integral evaluates to a single number by integrating a function over a closed interval from a lower limit to an upper limit. It gives examples of using definite integrals to find areas bounded by curves. The mean value theorem for integrals is also introduced, which states that there is a rectangle between the inscribed and circumscribed rectangles with an area equal to the region under the curve. Exercises are provided on evaluating definite integrals and applying the mean value theorem.
The document discusses two methods for evaluating integrals: integration by substitution and integration by parts. Integration by substitution involves setting up an integral in a way that allows substituting a new variable u for an expression involving x, making the integral easier to evaluate. Integration by parts is a method for evaluating integrals of products of functions by breaking it into multiple integrals using the formula ∫u v dx = u∫v dx −∫u' (∫v dx) dx. The document provides examples of applying both methods to evaluate integrals of trigonometric, logarithmic, and exponential functions. It also briefly mentions partial fractions as a method to decompose rational functions into simpler fractions.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It provides objectives for understanding rules like the derivative of a constant function, the constant multiple rule, sum rule, and derivatives of sine and cosine functions. It then gives examples of finding the derivative of a squaring function using the definition, and introduces the concept of the second derivative.
This document provides an overview of basic discrete mathematical structures including sets, functions, sequences, sums, and matrices. It begins by defining a set as an unordered collection of elements and describes various ways to represent sets, such as listing elements or using set-builder notation. It then discusses operations on sets like unions, intersections, complements, and Cartesian products. Finally, it introduces functions as assignments of elements from one set to another. The document serves as an introduction to fundamental discrete structures used throughout mathematics.
This document provides examples and explanations of double integrals. It defines a double integral as integrating a function f(x,y) over a region R in the xy-plane. It then gives three key points:
1) To evaluate a double integral, integrate the inner integral first treating the other variable as a constant, then integrate the outer integral.
2) The easiest regions to integrate over are rectangles, as the limits of integration will all be constants.
3) For non-rectangular regions, the limits of integration may be variable, requiring more careful analysis to determine the limits for each integral.
The document discusses vector fields and line integrals. Some key points:
- A vector field associates a vector with each point in a region, such as a velocity vector field showing wind patterns.
- Line integrals generalize the idea of integration over an interval to integration over a curve. The line integral of a function f over a curve C is defined as the limit of Riemann sums that multiply f by the length of curve segments.
- Line integrals can be used to calculate properties like work done by a force field or circulation in fluid flow. Their value does not depend on the parametrization of the curve C.
Using matrices to transform geometric figures, including translations, dilations, reflections, and rotations. Translations use a matrix with the distances of movement in each row. Dilations multiply coordinates by a scalar factor. Reflections across an axis involve changing the sign of coordinates on one side of the axis. Rotation matrices involve trigonometric functions to rotate the figure a specified number of degrees clockwise or counterclockwise. Examples show setting up and performing each type of transformation on sample polygons.
The document discusses how to find the domain and range of ordered pairs, graphs, and functions. It provides examples of finding the domain, which is the set of first coordinates for ordered pairs or the values of x included in the solution set for graphs, and the range, which is the set of second coordinates for ordered pairs or the values of y included in the solution set for graphs. The document gives the domain and range for several examples of ordered pairs, graphs and functions.
The document presents information on partial differentiation including:
- Partial differentiation involves a function with more than one independent variable and partial derivatives.
- Notation for partial derivatives is presented.
- Methods for computing first and higher order partial derivatives are explained with examples.
- The concepts of homogeneous functions and the chain rule for partial differentiation are defined.
This document provides information about multiple integrals and examples of evaluating them. It begins by defining multiple integrals and iterated integrals. It then gives 4 examples of evaluating double and triple integrals over different regions. These regions include rectangles, triangles, and solids. The document also discusses Fubini's theorem, which allows reversing the order of integration in certain cases. It concludes by providing an example of converting an integral from Cartesian to polar coordinates.
The document discusses partial differential equations (PDEs). It defines PDEs and gives their general form involving independent variables, dependent variables, and partial derivatives. It describes methods for obtaining the complete integral, particular solution, singular solution, and general solution of a PDE. It provides examples of types of PDEs and how to solve them by assuming certain forms for the dependent and independent variables and their partial derivatives.
This document discusses complex variables and functions. It covers topics such as:
- Cauchy-Riemann conditions, which must be satisfied for a complex function to be analytic/differentiable
- Cauchy's integral theorem, which states that the integral of an analytic function around a closed contour is zero
- Harmonic functions, whose real and imaginary parts satisfy the 2D Laplace equation
- Examples of calculating contour integrals of functions like zn along circular and square paths
The document provides proofs of theorems like Cauchy's integral theorem using techniques like Stokes' theorem. It also discusses simply and multiply connected regions in relation to contour integrals.
This document discusses subspace clustering with missing data. It summarizes two algorithms for solving this problem: 1) an EM-type algorithm that formulates the problem probabilistically and iteratively estimates the subspace parameters using an EM approach. 2) A k-means form algorithm called k-GROUSE that alternates between assigning vectors to subspaces based on projection residuals and updating each subspace using incremental gradient descent on the Grassmannian manifold. It also discusses the sampling complexity results from a recent paper, showing subspace clustering is possible without an impractically large sample size.
1) The document discusses partial derivatives, which involve differentiating functions of two or more variables with respect to one variable while holding the others fixed. It provides definitions, examples of computing partial derivatives, and interpretations as rates of change.
2) Techniques covered include implicit differentiation, using the chain rule to find derivatives of implicitly defined functions, and computing second order partial derivatives.
3) Diagrams and tables are referenced to illustrate level curves and contour maps for functions of two variables.
1. The document provides information on multiple integrals including double integrals, triple integrals, and integrals in spherical and cylindrical coordinates. It defines each type of integral and gives their general formulas.
2. Examples are provided for calculating double and triple integrals over different regions in rectangular, cylindrical, and spherical coordinate systems. The order of integration can be changed by considering strips or slices of the region.
3. Properties of the integrals include applying Fubini's theorem to change the order of integration, and relating the triple integral over a region to the double integral over the bounds and integrating over the third variable.
This document provides an overview of triple integrals and their applications. It defines triple integrals as the limit of triple Riemann sums for functions of three variables, analogous to double integrals. Triple integrals can be evaluated over rectangular boxes by expressing them as iterated integrals in any of six orders, as stated by Fubini's theorem. The document also describes how to evaluate triple integrals over more general bounded solid regions, including type 1 regions bounded by two graphs, type 2 regions bounded between two planes, and type 3 regions. It provides examples of evaluating triple integrals over a tetrahedron and other specific regions.
Exact Matrix Completion via Convex Optimization Slide (PPT)Joonyoung Yi
Slide of the paper "Exact Matrix Completion via Convex Optimization" of Emmanuel J. Candès and Benjamin Recht. We presented this slide in KAIST CS592 Class, April 2018.
- Code: https://github.com/JoonyoungYi/MCCO-numpy
- Abstract of the paper: We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M. Can we complete the matrix and recover the entries that we have not seen? We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys
𝑚≥𝐶𝑛1.2𝑟log𝑛
for some positive numerical constant C, then with very high probability, most n×n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.
This document provides an overview of integral calculus, including its history, definition, techniques, and applications. It traces integration back to ancient Egypt and developments by Archimedes, Liu Hui, Ibn al-Haytham, Newton, Leibniz, Cauchy, and Riemann. Key techniques discussed are integration by general rule, integration by parts, and integration by substitution. Applications mentioned include designing tall buildings and the Sydney Opera House to withstand forces, and historically calculating wine cask volumes.
The document defines definite integrals and discusses their properties. It states that a definite integral evaluates to a single number by integrating a function over a closed interval from a lower limit to an upper limit. It gives examples of using definite integrals to find areas bounded by curves. The mean value theorem for integrals is also introduced, which states that there is a rectangle between the inscribed and circumscribed rectangles with an area equal to the region under the curve. Exercises are provided on evaluating definite integrals and applying the mean value theorem.
The document discusses two methods for evaluating integrals: integration by substitution and integration by parts. Integration by substitution involves setting up an integral in a way that allows substituting a new variable u for an expression involving x, making the integral easier to evaluate. Integration by parts is a method for evaluating integrals of products of functions by breaking it into multiple integrals using the formula ∫u v dx = u∫v dx −∫u' (∫v dx) dx. The document provides examples of applying both methods to evaluate integrals of trigonometric, logarithmic, and exponential functions. It also briefly mentions partial fractions as a method to decompose rational functions into simpler fractions.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It provides objectives for understanding rules like the derivative of a constant function, the constant multiple rule, sum rule, and derivatives of sine and cosine functions. It then gives examples of finding the derivative of a squaring function using the definition, and introduces the concept of the second derivative.
This document provides an overview of basic discrete mathematical structures including sets, functions, sequences, sums, and matrices. It begins by defining a set as an unordered collection of elements and describes various ways to represent sets, such as listing elements or using set-builder notation. It then discusses operations on sets like unions, intersections, complements, and Cartesian products. Finally, it introduces functions as assignments of elements from one set to another. The document serves as an introduction to fundamental discrete structures used throughout mathematics.
This document provides examples and explanations of double integrals. It defines a double integral as integrating a function f(x,y) over a region R in the xy-plane. It then gives three key points:
1) To evaluate a double integral, integrate the inner integral first treating the other variable as a constant, then integrate the outer integral.
2) The easiest regions to integrate over are rectangles, as the limits of integration will all be constants.
3) For non-rectangular regions, the limits of integration may be variable, requiring more careful analysis to determine the limits for each integral.
The document discusses vector fields and line integrals. Some key points:
- A vector field associates a vector with each point in a region, such as a velocity vector field showing wind patterns.
- Line integrals generalize the idea of integration over an interval to integration over a curve. The line integral of a function f over a curve C is defined as the limit of Riemann sums that multiply f by the length of curve segments.
- Line integrals can be used to calculate properties like work done by a force field or circulation in fluid flow. Their value does not depend on the parametrization of the curve C.
Using matrices to transform geometric figures, including translations, dilations, reflections, and rotations. Translations use a matrix with the distances of movement in each row. Dilations multiply coordinates by a scalar factor. Reflections across an axis involve changing the sign of coordinates on one side of the axis. Rotation matrices involve trigonometric functions to rotate the figure a specified number of degrees clockwise or counterclockwise. Examples show setting up and performing each type of transformation on sample polygons.
The document discusses how to find the domain and range of ordered pairs, graphs, and functions. It provides examples of finding the domain, which is the set of first coordinates for ordered pairs or the values of x included in the solution set for graphs, and the range, which is the set of second coordinates for ordered pairs or the values of y included in the solution set for graphs. The document gives the domain and range for several examples of ordered pairs, graphs and functions.
The document presents information on partial differentiation including:
- Partial differentiation involves a function with more than one independent variable and partial derivatives.
- Notation for partial derivatives is presented.
- Methods for computing first and higher order partial derivatives are explained with examples.
- The concepts of homogeneous functions and the chain rule for partial differentiation are defined.
This document provides information about multiple integrals and examples of evaluating them. It begins by defining multiple integrals and iterated integrals. It then gives 4 examples of evaluating double and triple integrals over different regions. These regions include rectangles, triangles, and solids. The document also discusses Fubini's theorem, which allows reversing the order of integration in certain cases. It concludes by providing an example of converting an integral from Cartesian to polar coordinates.
The document discusses partial differential equations (PDEs). It defines PDEs and gives their general form involving independent variables, dependent variables, and partial derivatives. It describes methods for obtaining the complete integral, particular solution, singular solution, and general solution of a PDE. It provides examples of types of PDEs and how to solve them by assuming certain forms for the dependent and independent variables and their partial derivatives.
This document discusses complex variables and functions. It covers topics such as:
- Cauchy-Riemann conditions, which must be satisfied for a complex function to be analytic/differentiable
- Cauchy's integral theorem, which states that the integral of an analytic function around a closed contour is zero
- Harmonic functions, whose real and imaginary parts satisfy the 2D Laplace equation
- Examples of calculating contour integrals of functions like zn along circular and square paths
The document provides proofs of theorems like Cauchy's integral theorem using techniques like Stokes' theorem. It also discusses simply and multiply connected regions in relation to contour integrals.
This document discusses subspace clustering with missing data. It summarizes two algorithms for solving this problem: 1) an EM-type algorithm that formulates the problem probabilistically and iteratively estimates the subspace parameters using an EM approach. 2) A k-means form algorithm called k-GROUSE that alternates between assigning vectors to subspaces based on projection residuals and updating each subspace using incremental gradient descent on the Grassmannian manifold. It also discusses the sampling complexity results from a recent paper, showing subspace clustering is possible without an impractically large sample size.
1) The document discusses partial derivatives, which involve differentiating functions of two or more variables with respect to one variable while holding the others fixed. It provides definitions, examples of computing partial derivatives, and interpretations as rates of change.
2) Techniques covered include implicit differentiation, using the chain rule to find derivatives of implicitly defined functions, and computing second order partial derivatives.
3) Diagrams and tables are referenced to illustrate level curves and contour maps for functions of two variables.
1) The document discusses partial derivatives, which involve differentiating functions of two or more variables with respect to one variable while holding the others constant. It provides examples of computing first and second partial derivatives.
2) Implicit differentiation is introduced as a way to find partial derivatives of functions defined implicitly rather than explicitly. The chain rule is also discussed.
3) Methods are presented for finding partial derivatives of functions of two or three variables, including using implicit differentiation and the chain rule. Examples are provided to illustrate these concepts.
This document discusses various properties and concepts related to the z-transform, which is used to analyze discrete-time signals. It defines the z-transform and region of convergence. It then provides examples of calculating the z-transform for different signals. Key properties discussed include time shifting, scaling, time reversal, differentiation, and convolution. Theorems regarding the initial value and final value are also covered. Worked examples are provided to demonstrate applying the various properties and concepts.
This document discusses various properties and concepts related to the z-transform, which is used to analyze discrete-time signals. It defines the z-transform and region of convergence. It then provides examples of calculating the z-transform for different signals. Key properties discussed include time shifting, scaling, time reversal, differentiation, and convolution. Theorems regarding the initial value and final value are also covered. Worked examples are provided to demonstrate applying the various properties and concepts.
INFLUENCE OF OVERLAYERS ON DEPTH OF IMPLANTED-HETEROJUNCTION RECTIFIERSZac Darcy
In this paper we compare distributions of concentrations of dopants in an implanted-junction rectifiers in a
heterostructures with an overlayer and without the overlayer. Conditions for decreasing of depth of the
considered p-n-junction have been formulated.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Relative superior mandelbrot and julia sets for integer and non integer valueseSAT Journals
Abstract
The fractals generated from the self-squared function,
2 zz c where z and c are complex quantities have been studied
extensively in the literature. This paper studies the transformation of the function , 2 n zz c n and analyzed the z plane and
c plane fractal images generated from the iteration of these functions using Ishikawa iteration for integer and non-integer values.
Also, we explored the drastic changes that occurred in the visual characteristics of the images from n = integer value to n = non
integer value.
Keywords: Complex dynamics,
Relative Superior Julia set, Relative Superior Mandelbrot set.
- The document discusses quantum wires and quantum dots.
- For quantum wires, electrons are confined in two directions and free to move in the third, resulting in a 1D electron gas. The wave function and energy levels depend on the confinement potential.
- For quantum dots, the potential confines electrons in all three dimensions, resulting in discrete energy levels. The wave function is a product of sine waves and the energy depends on quantum numbers in each dimension.
This document summarizes key concepts about the particle in a rigid one-dimensional box:
1. It finds the energy eigenstates and discusses the wave functions and their properties like orthogonality.
2. It calculates the probability and expected values for the particle's position and discusses the physical interpretation of the wave function and coefficients when expanding an arbitrary function in the eigenstates.
3. It addresses several questions about normalized wave functions, time-dependent wave functions, energy measurements, and the wave function after a measurement.
This document provides lecture notes on high voltage engineering. It introduces the concepts of electric potential, electric field intensity, electric flux density, and volume charge density. It describes how Poisson's equation and Laplace's equation relate these concepts and can be used to determine potential distributions. It then discusses several numerical methods for solving Laplace's equation, including the finite difference method (FDM), finite element method (FEM), and others. FDM uses a grid to approximate derivatives and solve for potential values iteratively. FEM seeks to minimize the total electric field energy by dividing the region into discrete elements and solving a system of equations relating node potentials.
The document discusses the Z-transform, which is a tool for analyzing and solving linear time-invariant difference equations. It defines the Z-transform, provides examples of common sequences and their corresponding Z-transforms, and discusses properties such as the region of convergence. Key topics covered include the difference between difference and differential equations, properties of linear time-invariant systems, and mapping between the s-plane and z-plane.
1) The document discusses double integrals and methods for calculating them, including using iterated integrals and Riemann sums. Double integrals can represent volumes under surfaces.
2) Examples are provided to demonstrate calculating double integrals over rectangles and general regions using iterated integrals and partitioning the region.
3) There are two types of general regions: type I defined by ≤≤≤≤ and type II defined by ≤≤≤≤. The document provides methods for calculating double integrals over these region types.
This document describes the Kumaraswamy generalized (Kw-G) distribution, a new family of continuous probability distributions defined on the interval (0,1). The Kw-G distribution is constructed by applying the Kumaraswamy distribution to an existing parent distribution with cumulative distribution function G(x). Properties of the Kw-G distribution such as its probability density function, moments, order statistics, and L-moments are expressed in terms of the parent distribution G(x). Several special cases of the Kw-G distribution are also discussed, including the Kw-normal, Kw-Weibull, and Kw-gamma distributions.
This document provides information on graphing and determining equations for straight lines on different types of graph paper, including:
1) Arithmetic graph paper, where linear functions plot as straight lines and the slope and y-intercept can be determined from any two points.
2) Logarithmic graph paper, where power functions plot as straight lines by taking the logarithm of both sides, making the relationship between log(x) and log(y) linear. The exponent is the slope and the y-intercept gives the constant k.
3) Semi-logarithmic paper, where exponential functions plot as straight lines by taking the logarithm of y. The slope then gives the exponential constant and the y-intercept
Principal component analysis (PCA) is a technique used to reduce the dimensionality of data by transforming correlated variables into a smaller number of uncorrelated variables called principal components. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible. PCA involves computing the covariance matrix of the data and then determining the eigenvectors with the highest eigenvalues, which become the principal components.
1. The document describes the finite element formulation for 2D problems using constant strain triangles.
2. It involves dividing the body into finite elements connected at nodes, then approximating displacements within each element using shape functions of the nodes.
3. Strains and stresses are then approximated based on the displacements. This allows setting up the element stiffness matrix and load vector to solve for the unknown node displacements.
This document describes an approach to optimize the manufacturing of a sense-amplifier based flip-flop by increasing the density of field-effect heterotransistors. It analyzes the diffusion of dopants in heterostructures using Fick's laws and introduces an approach to decrease stress between heterostructure layers. It considers the distribution of point defects like vacancies and interstitials over space and time using a system of equations. The goal is to optimize annealing conditions to decrease the dimensions of transistors for use in a broadband power amplifier within a specific heterostructure configuration.
Text Book: An Introduction to Mechanics by Kleppner and Kolenkow
Chapter 1: Vectors and Kinematics
-Explain the concept of vectors.
-Explain the concepts of position, velocity and acceleration for different kinds of motion.
References:
Halliday, Resnick and Walker
Berkley Physics Volume-1
Similar to Double & triple integral unit 5 paper 1 , B.Sc. 2 Mathematics (20)
The document discusses the relationship between economics, environment, and ethics. It summarizes that we are facing issues today because of ignoring the fundamental relationship between the three. The economy relies on ecosystem services provided by the environment, but the environment is being degraded by waste and emissions. Ethical practices also constitute an unseen force guiding economic behavior.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms for those who already suffer from conditions like anxiety and depression.
Scientific temper and attitude refer to traits like critical thinking, objectivity, open-mindedness, and respect for evidence. Developing a scientific attitude in students is the aim of science teaching. Some key aspects of scientific attitude are questioning beliefs, reasoning logically, honestly reporting observations, and accepting ideas that are supported by evidence. Fostering skills like curiosity, perseverance, and skepticism in students can help cultivate their scientific temper.
This document discusses the aims and objectives of teaching biological science. It begins by defining biological science as the study of life and living organisms. It then lists several objectives of teaching biological science, including developing students' scientific outlook, curiosity about their surroundings, and respect for nature. The document also discusses the values of teaching biological science, which include encouraging curiosity and knowledge, and keeping an open mind. It emphasizes that teaching biological science should help students become responsible democratic citizens and appreciate diverse perspectives. Overall, the document provides an overview of the goals and importance of teaching biological science.
This presentation discusses using information and communication technologies (ICT) applications in biology learning. It introduces the topic, noting the presenter and institution. The document provides references on the advantages and limitations of ICT in education, using ICT to integrate science teaching and learning, and the impact of ICT in education.
The term isolation refers to the separation of a strain from a natural, mixed population of living microbes, as present in the environment. It becomes necessary to maintain the viability and purity of the microorganism by keeping the pure culture free from contamination.
1) The document discusses oxidation-reduction (redox) reactions and concepts related to solution concentrations. It defines oxidizing and reducing agents and gives examples of each.
2) A redox reaction involves the simultaneous oxidation and reduction of reactants. In redox reactions, the total increase in oxidation number equals the total decrease.
3) Disproportionation reactions involve the same element in a compound being both oxidized and reduced. The reverse is called a comproportionation reaction.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise boosts blood flow, releases endorphins, and promotes changes in the brain which help enhance one's emotional well-being and mental clarity.
The document discusses the concept of equilibrium in economics. It defines equilibrium as a state of balance where opposing forces neutralize each other. In microeconomics, market equilibrium occurs when supply equals demand. In macroeconomics, equilibrium is reached when aggregate demand equals aggregate supply. The document provides examples of economic disequilibrium and equilibrium, and examines how prices adjust via demand and supply mechanisms to reach equilibrium. Key terms in Hindi are also defined.
This document summarizes Crystal Field Theory, which considers the electrostatic interactions between metal ions and ligands. It describes ligands and metal ions as point charges that can have attractive or repulsive forces. This causes the d orbitals of the metal ion to split into two sets depending on if the field created by the ligands is weak or strong. The theory explains color in coordination compounds as being caused by d-d electron transitions under the influence of ligands. However, it has limitations like not accounting for other metal orbitals or the partial covalent nature of metal-ligand bonds.
Dr. Laxmi Verma teaches Microeconomics at the BA-1 level and her topic is on utility in Unit 1 of the course. She teaches at Shri Shankracharya Mahavidyalya in Junwani.
Dr. Laxmi Verma is teaching a class of B.A-1 students. The subject is Indian Economy and the topic being covered is New Economic Reform. The document provides basic context about an economics lecture being given to undergraduate students on recent reforms in the Indian economy.
An iso-product curve shows the different combinations of two factors of production, such as labor and capital, that result in the same level of output. It is represented graphically, with the two factors on the x and y axes and points of equal output connected to form an iso-product curve. Key properties are that iso-product curves slope downward to the right, are convex to the origin, and do not intersect, as each curve represents a different output level. Higher iso-product curves correspond to higher output levels. Iso-product curves allow producers to identify input combinations that achieve maximum output efficiently.
This document discusses demand theory and the relationship between supply and demand. It covers the following key points:
1) Demand theory explains how consumer demand for goods and services relates to their prices in the market. It forms the basis for the demand curve, which shows that as price increases, demand decreases.
2) Demand depends on the utility of goods in satisfying wants and needs as well as a consumer's ability to pay. Supply and demand determine market prices and reach equilibrium when supply equals demand.
3) The demand curve has a negative slope, showing an inverse relationship between price and quantity demanded. A change in non-price factors like income can shift the demand curve. The law of supply and
Land reform in India has involved abolishing intermediaries like rent collectors and establishing ceilings on land ownership to redistribute surplus land to the landless. The goals were to remove impediments to agricultural production from the previous feudal system and eliminate exploitation. Key reforms included abolishing rent collectors, regulating tenancy, imposing landholding ceilings, consolidating fragmented holdings, and promoting cooperative farming. Impacts included reducing disparities, giving ex-landlords other work, increasing revenue, and empowering small farmers and laborers. Land reform aimed to promote social justice and economic growth through a more equitable distribution of agricultural land.
This document discusses different types of structural isomerism that can occur in coordination compounds. It defines structural isomerism as compounds having the same molecular formula but different physical and chemical properties due to different structures or orientations. The types of structural isomerism discussed include ionization isomerism, solvate/hydrate isomerism, linkage isomerism, coordination isomerism, ligand isomerism, polymerization isomerism, geometrical isomerism (cis/trans), and optical isomerism. Examples are provided to illustrate each type of isomerism.
More from Shri Shankaracharya College, Bhilai,Junwani (20)
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...TechSoup
Whether you're new to SEO or looking to refine your existing strategies, this webinar will provide you with actionable insights and practical tips to elevate your nonprofit's online presence.
A Free 200-Page eBook ~ Brain and Mind Exercise.pptxOH TEIK BIN
(A Free eBook comprising 3 Sets of Presentation of a selection of Puzzles, Brain Teasers and Thinking Problems to exercise both the mind and the Right and Left Brain. To help keep the mind and brain fit and healthy. Good for both the young and old alike.
Answers are given for all the puzzles and problems.)
With Metta,
Bro. Oh Teik Bin 🙏🤓🤔🥰
CapTechTalks Webinar Slides June 2024 Donovan Wright.pptxCapitolTechU
Slides from a Capitol Technology University webinar held June 20, 2024. The webinar featured Dr. Donovan Wright, presenting on the Department of Defense Digital Transformation.
🔥🔥🔥🔥🔥🔥🔥🔥🔥
إضغ بين إيديكم من أقوى الملازم التي صممتها
ملزمة تشريح الجهاز الهيكلي (نظري 3)
💀💀💀💀💀💀💀💀💀💀
تتميز هذهِ الملزمة بعِدة مُميزات :
1- مُترجمة ترجمة تُناسب جميع المستويات
2- تحتوي على 78 رسم توضيحي لكل كلمة موجودة بالملزمة (لكل كلمة !!!!)
#فهم_ماكو_درخ
3- دقة الكتابة والصور عالية جداً جداً جداً
4- هُنالك بعض المعلومات تم توضيحها بشكل تفصيلي جداً (تُعتبر لدى الطالب أو الطالبة بإنها معلومات مُبهمة ومع ذلك تم توضيح هذهِ المعلومات المُبهمة بشكل تفصيلي جداً
5- الملزمة تشرح نفسها ب نفسها بس تكلك تعال اقراني
6- تحتوي الملزمة في اول سلايد على خارطة تتضمن جميع تفرُعات معلومات الجهاز الهيكلي المذكورة في هذهِ الملزمة
واخيراً هذهِ الملزمة حلالٌ عليكم وإتمنى منكم إن تدعولي بالخير والصحة والعافية فقط
كل التوفيق زملائي وزميلاتي ، زميلكم محمد الذهبي 💊💊
🔥🔥🔥🔥🔥🔥🔥🔥🔥
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
3. Introduction:-
In elementary calculus, students have studied the
definite integral of a function of one variable . In this
chapeter we show how the notion of the definite integral
can be extended to functions of several variables. In
particular, we shall discuss the double integral of a
functions of two variable and the triple integral of a
function of three variables.
4. Define double integrals:-
Let f(x,y) be a function of two independent
variables x,y in - plane defined at all points in a finite
region A. let the region A be divided into n subregions
(k=1,2,…,n) of areas
let be any point inside the kth element
Let us form the sum
let us the number of subdivision becomes
infinite in such a way that the dimensions of each subdivision
approaches to zero.
kR nAAA ,....,, 21
kk , kR
n
k
kkk Af
1
),(
xy
5. If under these conditions, the limit
which is independent of the way in which the point
are chosen exists, then this limit is called the double
integral of over the region A, written
is defined by
n
k
kkk
n
Af
1
),(lim
kk ,
),( yxf
A
dxdyyxf ),(
kk
n
k
k
A n
Afdxdyyxf
),(lim),(
1
6. Properties of double integral:-
if f and g are continuous over the bounded
region R, then:
where R is composed of two subregions R1 and
R2 or
R R
dAyxfkdAyxkfP ),(),(.
1
R RR
dAyxgdAyxfdAyxgyxfP ),(),(),(),(.2
R R R
dAyxfdAyxfdAyxfP 1 23
),(),(),(.
21 1 2
),(),(),(
RR R R
dAyxfdAyxfdAyxf
dxdydA
8. Example 2- when the region of integration R is the triangle
bounded by y= 0, y= x and x = 1 , show that
sol. The region of interation is shown shaded in the
adjoining figure. Let us divide the triangle OAB into
vertical strips. Then it is evident that in an elementary
strips y varies from y = 0 to y = x while x varies from
x = 0 to x = 1
).
2
3
3
(
3
1
²)²4(
R
dxdyyx
10. Thus the given doudle integral can be expressed as the
repeated integral
dxdyyx
R ²)²4(
dxdyyx
x
x
y
1
0 0
²)²4(
x
y
x x
y
xyxy
0
1
0
1
2
sin².4.
2
1
²)²4(
2
1
dx
x
x
xxxx
x
2
sin²4²)²4(
2
1 1
1
0
a
x
axax 1
sin²..
2
1
²)²(
2
1 dxxa ²)²(
1
0 0
²)²4(
x
x
y
dxdyyx yxxa ,2
12. DEFINE TRIPLE INTEGRALS0:-
Let be a function
of three independent variables x,y,z defined for all points
in a finite closed three dimensional region V of space.
Divide V into n sub- regions of volumes , k= 1,2,….,n.
let us select an arbitrary point in each
and form the sum
let the number of sub-division become infinite in such a
way that the maximum dimensions of each
approaches to zero.
zyxf ,,
kV)( ,, kkkkP
kV
k
n
k
kkk V1
,, )(
kV
13. If under these conditions, the limit
Exists, which is independent of the way in which the points
are chosen, then this limit is called the triple
integral of over the region V, written
is defined by
kkk
n
k
k
n
Vf
)( ,
1
,lim
)( ,, kkk
),,( zyxf
,),,( dVzyxf
v
dVzyxf
v ),,(
n
k
kkkk
n
Vf
1
, )(lim
14. Example of triple integrals:-
(1).Evaluate:
Sol:- let the given triple integral be denoted by . Then
3
0
2
0
1
0
)( dxdydzzyx
I
3
0
2
0
1
0
)( dxdydzzyxI
dxdydzzyx
3
0
2
0
1
0
)(
dxdy
z
yzxz
3
0
2
0
1
02
²
dxdyyx
3
0
2
0
0
2
²1
16. (2).evaluate:-
where the region of integration V is a cylinder,which is
bounded by the following surfaces:
z = 0, z = 1,x²+y² = 4
Sol:-
form the adjoining figure it is evident that in the region
of integration V, z varies from z = 0 to z = 1, y varies from
y = to y = and x varies from x = -2 to
x = 2
dxdydzz
v
²)4( x ²)4( x