1) The document discusses solving partial differential equations using variable separation. It presents the one-dimensional wave equation and solves it using variable separation.
2) It derives three cases for the solution based on whether k is positive, negative, or zero. It then presents the general solution as a summation involving sines and cosines.
3) It applies the general solution to two example problems of a vibrating string, finding the displacement as a function of position and time by satisfying the boundary conditions.
This document contains exercises and solutions for line integrals from a chapter on the topic. It includes 6 exercises evaluating line integrals over various curves defined parametrically or through equations. It also contains exercises using Green's Theorem and evaluating line integrals for conservative vector fields. The solutions provide the parametrizations needed to set up and evaluate the line integrals.
This document provides an overview of the classic game Minesweeper, including game rules, modes, world records, applications, variations, and tips and tricks. It explains the basic rules of the game where the goal is to clear a board of mines without exploding. It also outlines the different game modes for beginner, intermediate, and expert levels that vary the board size and number of mines. Advanced tips are given for patterns, guessing strategies, efficient gameplay, and maximizing openings. World records show completion times on the different skill levels.
Spectral methods for solving differential equationsRajesh Aggarwal
This project report describes Rajesh Aggarwal's summer research fellowship project on spectral methods for solving differential equations under the guidance of Dr. Pravin Kumar Gupta at IIT Roorkee from June 11, 2014 to August 6, 2014. The report provides background on analytical and numerical methods for solving differential equations, specifically conventional finite difference methods and spectral finite difference methods. It then describes the methodology, codes, and results of applying both methods to solve sample differential equations on a half-space and layered earth problems. Tables and graphs comparing the accuracy of the two methods are presented.
1) The document discusses solving partial differential equations using variable separation. It presents the one-dimensional wave equation and solves it using variable separation.
2) It derives three cases for the solution based on whether k is positive, negative, or zero. It then presents the general solution as a summation involving sines and cosines.
3) It applies the general solution to two example problems of a vibrating string, finding the displacement as a function of position and time by satisfying the boundary conditions.
This document contains exercises and solutions for line integrals from a chapter on the topic. It includes 6 exercises evaluating line integrals over various curves defined parametrically or through equations. It also contains exercises using Green's Theorem and evaluating line integrals for conservative vector fields. The solutions provide the parametrizations needed to set up and evaluate the line integrals.
This document provides an overview of the classic game Minesweeper, including game rules, modes, world records, applications, variations, and tips and tricks. It explains the basic rules of the game where the goal is to clear a board of mines without exploding. It also outlines the different game modes for beginner, intermediate, and expert levels that vary the board size and number of mines. Advanced tips are given for patterns, guessing strategies, efficient gameplay, and maximizing openings. World records show completion times on the different skill levels.
Spectral methods for solving differential equationsRajesh Aggarwal
This project report describes Rajesh Aggarwal's summer research fellowship project on spectral methods for solving differential equations under the guidance of Dr. Pravin Kumar Gupta at IIT Roorkee from June 11, 2014 to August 6, 2014. The report provides background on analytical and numerical methods for solving differential equations, specifically conventional finite difference methods and spectral finite difference methods. It then describes the methodology, codes, and results of applying both methods to solve sample differential equations on a half-space and layered earth problems. Tables and graphs comparing the accuracy of the two methods are presented.
Laurent Series ........ By Muhammad Umersialkot123
The document discusses the Laurent series, which represents a complex function as a power series including terms with negative powers, unlike the Taylor series. It explains that the Laurent series can be used when the Taylor series cannot be applied. The document provides an example of calculating the Laurent series for the function f(z)=1, expanding it in the regions inside and outside the circle of radius 1 centered at z=0.
Z-transforms can be used to evaluate discrete functions, similar to how Laplace transforms are used for continuous functions. The z-transform of a discrete function f(n) is defined as the sum of f(n) multiplied by z to the power of -n, from n=0 to infinity. Some standard z-transform results include formulas for exponential, sinusoidal, and polynomial functions. Z-transforms have properties of linearity and shifting, and can be used to solve differential equations with constant coefficients and in applications of signal processing.
Quantum mechanics 1st edition mc intyre solutions manualSelina333
Quantum Mechanics 1st Edition McIntyre Solutions Manual
Download at: https://goo.gl/SdC7Ef
quantum mechanics david mcintyre solutions pdf
quantum mechanics mcintyre pdf
quantum mechanics a paradigms approach solutions pdf
quantum mechanics mcintyre solutions pdf
quantum mechanics a paradigms approach solution manual
quantum mechanics mcintyre solutions manual pdf
hidden life of prayer
Double integral using polar coordinatesHarishRagav10
Maths is aan acknowledge for all humanity's stast of technology and many resources now in all country maths is the father of technology is a wondering..maths comes into play a vital role in all form of chemistry physics and all other subjects...in our day to day life maths playa a role everything and everywhere
This document discusses exponential functions and their properties. It explores exponential growth and decay through graphs of functions like y=2^x and y=0.5^x. It shows that as x increases, exponential growth functions approach infinity, while decay functions approach zero. The document also introduces the irrational number e as the most important base for modeling continuous growth and decay. It shows how the function A=Ce^rt models continuous compound interest as the compounding period approaches infinity.
This document defines Sturm-Liouville boundary value problems (SL-BVPs) and Sturm-Liouville eigenvalue problems (SL-EVPs). It discusses regular, singular, and periodic SL-BVPs. Two examples are presented in detail: one with separated boundary conditions and one with periodic boundary conditions. Properties of regular and periodic SL-BVPs are discussed, including that eigenvalues are real and eigenfunctions corresponding to distinct eigenvalues are orthogonal with respect to the weight function. The document proves several properties and establishes that regular SL-BVPs have an infinite sequence of eigenvalues.
Let's analyze the remainder term R6 using the geometry series method:
|tj+1| = (j+1)π-2 ≤ π-2 = k|tj| for all j ≥ 6 (where 0 < k = π-2 < 1)
Then, |R6| ≤ t7(1 + k + k2 + k3 + ...)
= t7/(1-k)
= 7π-2/(1-π-2)
So the estimated upper bound of the truncation error |R6| is 7π-2/(1-π-2)
This document discusses Gaussian quadrature, a method for numerical integration. It begins by comparing Gaussian quadrature to Newton-Cotes formulae, noting that Gaussian quadrature selects both weights and locations of integration points to exactly integrate higher order polynomials. The document then provides examples of 2-point and 3-point Gaussian quadrature on the interval [-1,1], showing how to determine the points and weights to integrate polynomials up to a certain order exactly. It also discusses extending Gaussian quadrature to other intervals via a coordinate transformation, and provides an example integration problem.
This document discusses Taylor series expansions. It defines Taylor series as the expansion of a complex function f(z) that is analytic inside and on a simple closed curve C in the z-plane. The Taylor series expresses f(z) as a power series centered at a point z0 within C. It provides examples of standard Taylor series expansions and worked illustrations of expanding various functions as Taylor series. The document also notes that the radius of convergence of a Taylor series is defined by the distance to the nearest singularity from the center point z0.
Derivation and solution of the heat equation in 1-DIJESM JOURNAL
Heat flows in the direction of decreasing temperature, that is, from hot to cool. In this paper we derive the heat equation and consider the flow of heat along a metal rod. The rod allows us to consider the temperature, u(x,t), as one dimensional in x but changing in time, t.
The document defines a homogeneous linear differential equation as an equation of the form:
a0(dx/dy)n + a1(dx/dy)n-1 + ... + an-1(dx/dy) + any = X, where a0, a1, ..., an are constants and X is a function of x.
It provides the method of solving such equations by first reducing it to a linear equation with constant coefficients, then taking a trial solution of the form y = emz, and finally solving the resulting auxiliary equation.
It proves identities relating derivatives of y with respect to x and z, then uses these identities to solve two sample homogeneous linear differential equations of orders 2 and
This document discusses orthogonal subspaces and inner products in advanced engineering mathematics. It defines the inner product of two vectors u and v in Rn as the transpose of u dotted with v, which results in a scalar. Two vectors are orthogonal if their inner product is 0. An orthogonal basis for a subspace W is a basis for W that is also an orthogonal set. The document also discusses orthogonal complements, projections, and inner products on function spaces.
Laurent Series ........ By Muhammad Umersialkot123
The document discusses the Laurent series, which represents a complex function as a power series including terms with negative powers, unlike the Taylor series. It explains that the Laurent series can be used when the Taylor series cannot be applied. The document provides an example of calculating the Laurent series for the function f(z)=1, expanding it in the regions inside and outside the circle of radius 1 centered at z=0.
Z-transforms can be used to evaluate discrete functions, similar to how Laplace transforms are used for continuous functions. The z-transform of a discrete function f(n) is defined as the sum of f(n) multiplied by z to the power of -n, from n=0 to infinity. Some standard z-transform results include formulas for exponential, sinusoidal, and polynomial functions. Z-transforms have properties of linearity and shifting, and can be used to solve differential equations with constant coefficients and in applications of signal processing.
Quantum mechanics 1st edition mc intyre solutions manualSelina333
Quantum Mechanics 1st Edition McIntyre Solutions Manual
Download at: https://goo.gl/SdC7Ef
quantum mechanics david mcintyre solutions pdf
quantum mechanics mcintyre pdf
quantum mechanics a paradigms approach solutions pdf
quantum mechanics mcintyre solutions pdf
quantum mechanics a paradigms approach solution manual
quantum mechanics mcintyre solutions manual pdf
hidden life of prayer
Double integral using polar coordinatesHarishRagav10
Maths is aan acknowledge for all humanity's stast of technology and many resources now in all country maths is the father of technology is a wondering..maths comes into play a vital role in all form of chemistry physics and all other subjects...in our day to day life maths playa a role everything and everywhere
This document discusses exponential functions and their properties. It explores exponential growth and decay through graphs of functions like y=2^x and y=0.5^x. It shows that as x increases, exponential growth functions approach infinity, while decay functions approach zero. The document also introduces the irrational number e as the most important base for modeling continuous growth and decay. It shows how the function A=Ce^rt models continuous compound interest as the compounding period approaches infinity.
This document defines Sturm-Liouville boundary value problems (SL-BVPs) and Sturm-Liouville eigenvalue problems (SL-EVPs). It discusses regular, singular, and periodic SL-BVPs. Two examples are presented in detail: one with separated boundary conditions and one with periodic boundary conditions. Properties of regular and periodic SL-BVPs are discussed, including that eigenvalues are real and eigenfunctions corresponding to distinct eigenvalues are orthogonal with respect to the weight function. The document proves several properties and establishes that regular SL-BVPs have an infinite sequence of eigenvalues.
Let's analyze the remainder term R6 using the geometry series method:
|tj+1| = (j+1)π-2 ≤ π-2 = k|tj| for all j ≥ 6 (where 0 < k = π-2 < 1)
Then, |R6| ≤ t7(1 + k + k2 + k3 + ...)
= t7/(1-k)
= 7π-2/(1-π-2)
So the estimated upper bound of the truncation error |R6| is 7π-2/(1-π-2)
This document discusses Gaussian quadrature, a method for numerical integration. It begins by comparing Gaussian quadrature to Newton-Cotes formulae, noting that Gaussian quadrature selects both weights and locations of integration points to exactly integrate higher order polynomials. The document then provides examples of 2-point and 3-point Gaussian quadrature on the interval [-1,1], showing how to determine the points and weights to integrate polynomials up to a certain order exactly. It also discusses extending Gaussian quadrature to other intervals via a coordinate transformation, and provides an example integration problem.
This document discusses Taylor series expansions. It defines Taylor series as the expansion of a complex function f(z) that is analytic inside and on a simple closed curve C in the z-plane. The Taylor series expresses f(z) as a power series centered at a point z0 within C. It provides examples of standard Taylor series expansions and worked illustrations of expanding various functions as Taylor series. The document also notes that the radius of convergence of a Taylor series is defined by the distance to the nearest singularity from the center point z0.
Derivation and solution of the heat equation in 1-DIJESM JOURNAL
Heat flows in the direction of decreasing temperature, that is, from hot to cool. In this paper we derive the heat equation and consider the flow of heat along a metal rod. The rod allows us to consider the temperature, u(x,t), as one dimensional in x but changing in time, t.
The document defines a homogeneous linear differential equation as an equation of the form:
a0(dx/dy)n + a1(dx/dy)n-1 + ... + an-1(dx/dy) + any = X, where a0, a1, ..., an are constants and X is a function of x.
It provides the method of solving such equations by first reducing it to a linear equation with constant coefficients, then taking a trial solution of the form y = emz, and finally solving the resulting auxiliary equation.
It proves identities relating derivatives of y with respect to x and z, then uses these identities to solve two sample homogeneous linear differential equations of orders 2 and
This document discusses orthogonal subspaces and inner products in advanced engineering mathematics. It defines the inner product of two vectors u and v in Rn as the transpose of u dotted with v, which results in a scalar. Two vectors are orthogonal if their inner product is 0. An orthogonal basis for a subspace W is a basis for W that is also an orthogonal set. The document also discusses orthogonal complements, projections, and inner products on function spaces.
This document discusses relative humidity and temperature and their effects on collections. It begins by explaining that relative humidity is the amount of water vapor in the air compared to the total amount the air can hold at a given temperature. Changes in relative humidity and temperature can damage collections, especially fluctuations. Extremes can still stabilize collections if they adjust, but changes are more harmful. The document recommends understanding these concepts and taking steps to minimize harmful fluctuations and protect collections.
This document discusses relative humidity and temperature and their effects on collections. It begins by explaining that relative humidity is the amount of water vapor in the air compared to the total amount the air can hold at a given temperature. Changes in relative humidity and temperature can damage collections, especially fluctuations. Extremes can still stabilize collections if they adjust, but changes are more harmful. The document provides tips for minimizing damage by understanding and controlling humidity and temperature levels.
The document discusses release coats, which are coatings that allow easy removal of adhesives or other materials when desired. It describes different types of release coats including their chemical composition, methods of application, and curing processes. Key details include that silicone and non-silicone release coats exist, with non-silicone types including polyurethane, polyvinyl alcohol, and fluoropolymers. The document also discusses factors that influence release coat performance such as coat weight and drying temperature.
Release coating special additive product selection guideXin Zheng
This document provides a product selection guide for release coatings and special additives from Mayzo Inc. It includes a table listing various product forms including liquids and powders. The table specifies the solubility, substrate compatibility, and chemical composition of each product. A second table lists special additive products and their compatibility with various polymers and rubbers. The guide provides application guidelines and characteristics for selecting the appropriate release coating or additive for different manufacturing applications.
1. 实验七 用 Mathematica 解常微分方程
实验目的:
掌握用 Mathematica 软件求微分方程通解与特解的方法的语句和方法。
实验过程与要求:
教师利用多媒体组织教学,边讲边操作示范。
实验的内容:
一、求微分方程的通解
在 Mathematica 系统中用 DSolve 函数求解微分方程,基本格式为:
DSolve [微分方程,未知函数名称,未知函数的自变量]
实验 1 求微分方程 y ′ = 2 x 的通解.
解 In[1]:= DSolve[ y '[ x ]==2 x , y [ x ], x ]
Out[1]=
求微分方程 y ′′ − 3 y ′ + 2 y = 3 xe 的通解.
2x
实验 2
解 In[2]:= DSolve[y''[x]-3y'[x]+2y[x]==(3x)Exp[2x],y[x],x]
Out[2]=
实验 3 求微分方程 y ′′ + 3 y ′ = 2 sin x 的通解.
解 In[3]:=DSolve[y''[x]+3y'[x]==2Sin[x],y[x],x]
Out[3]=
其中方程中的等号应连输 2 个“=”,二阶导数记号应连输两个单引号.
二、求微分方程的特解
在 Mathematica 系统中求特解的函数仍为 DSolve,而基本格式为:
DSolve [{微分方程,初始条件},未知函数名称,未知函数的自变
量]
实验 4 解微分方程 y ′ = 2 x + y, y x =0 = 0.
解 In[4]:=DSolve[{y'[x]==2x+y[x],y[0]==0},y[x],x]
2. Out[4]=
实验
用笔算和机算两种方法求解下列微分方程:
1. y ′ − 6 y = e 3 x 2. y ′′ − 4 y ′ + 4 y = 2 cos x
3. y ′ = 3xy + x 3 + x 4. y ′′ − 2 y ′ − 3 y = e 4 x
5. y ′ − y tan x = sec x, y (0) = 0 6.(1 + e x ) yy ′ = e x , y x =0 = 0