This document discusses coordinate system transformations. It introduces three approaches for finding the needed transformations between coordinate systems: direction cosines, Euler angles, and Euler parameters. Direction cosines use the angles between coordinate system axes and the cosine of those angles to derive transformation matrices. Euler angles express the transformation matrix in terms of three successive rotations about coordinate system axes. Euler parameters provide an elegant alternative to Euler angles by avoiding potential problems with Euler angles.
(1) This document discusses ordinary differential equations of first order and first degree. Examples of differential equations are given and defined.
(2) Methods for solving first order differential equations are discussed, including variable separable, homogeneous, and linear methods. Examples of solving differential equations using these methods are provided.
(3) The order and degree of differential equations are defined. The process of forming differential equations from given functions is demonstrated through several examples.
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
The document defines and discusses differential equations and their solutions. It begins by classifying differential equations as ordinary or partial based on whether they involve one or more independent variables. Ordinary differential equations are then classified as linear or nonlinear based on their form. The order and degree of a differential equation are also defined.
Solutions to differential equations can be either explicit functions that directly satisfy the equation or implicit relations that define functions satisfying the equation. Picard's theorem guarantees a unique solution through each point for first-order equations. The general solution to a first-order equation is a one-parameter family of curves, with a particular solution corresponding to a specific value of the parameter. An initial value problem specifies both a differential equation and
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
Integrating factors found by inspectionShin Kaname
1. The document discusses using exact differentials to solve integration problems.
2. It provides examples of using exact differentials and integrating terms to find solutions.
3. The solutions found are particular solutions for the given values of x and y in each problem.
To summarize:
1) To find the derivative of an implicit function y=y(x) defined by an equation F(x,y)=0, take the derivative of both sides with respect to x.
2) This will give a new equation involving x, y, and dy/dx that can be solved for dy/dx.
3) The examples show applying this process to find derivatives and tangent lines for various implicit equations.
This document discusses differential equations. It begins by explaining that differential equations are used to model many physical phenomena in areas like economics, engineering, and more. It then provides examples of ordinary and partial differential equations. The rest of the document defines key terms related to differential equations like order, degree, families of curves, and how to derive the differential equation of a family of curves by eliminating parameters. Several examples are provided to illustrate these concepts.
The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outside function f'(g(x)) multiplied by the derivative of the inside function g'(x). This allows the calculation of derivatives of more complex functions that cannot be solved using basic derivative rules. Several examples are provided to demonstrate how to use the chain rule to calculate derivatives of various composite functions.
(1) This document discusses ordinary differential equations of first order and first degree. Examples of differential equations are given and defined.
(2) Methods for solving first order differential equations are discussed, including variable separable, homogeneous, and linear methods. Examples of solving differential equations using these methods are provided.
(3) The order and degree of differential equations are defined. The process of forming differential equations from given functions is demonstrated through several examples.
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
The document defines and discusses differential equations and their solutions. It begins by classifying differential equations as ordinary or partial based on whether they involve one or more independent variables. Ordinary differential equations are then classified as linear or nonlinear based on their form. The order and degree of a differential equation are also defined.
Solutions to differential equations can be either explicit functions that directly satisfy the equation or implicit relations that define functions satisfying the equation. Picard's theorem guarantees a unique solution through each point for first-order equations. The general solution to a first-order equation is a one-parameter family of curves, with a particular solution corresponding to a specific value of the parameter. An initial value problem specifies both a differential equation and
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
Integrating factors found by inspectionShin Kaname
1. The document discusses using exact differentials to solve integration problems.
2. It provides examples of using exact differentials and integrating terms to find solutions.
3. The solutions found are particular solutions for the given values of x and y in each problem.
To summarize:
1) To find the derivative of an implicit function y=y(x) defined by an equation F(x,y)=0, take the derivative of both sides with respect to x.
2) This will give a new equation involving x, y, and dy/dx that can be solved for dy/dx.
3) The examples show applying this process to find derivatives and tangent lines for various implicit equations.
This document discusses differential equations. It begins by explaining that differential equations are used to model many physical phenomena in areas like economics, engineering, and more. It then provides examples of ordinary and partial differential equations. The rest of the document defines key terms related to differential equations like order, degree, families of curves, and how to derive the differential equation of a family of curves by eliminating parameters. Several examples are provided to illustrate these concepts.
The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outside function f'(g(x)) multiplied by the derivative of the inside function g'(x). This allows the calculation of derivatives of more complex functions that cannot be solved using basic derivative rules. Several examples are provided to demonstrate how to use the chain rule to calculate derivatives of various composite functions.
This document provides examples of solving systems of nonlinear equations in two variables. It begins with definitions, including that a nonlinear system contains at least one equation that is not of the form Ax + By = C. Methods for solving nonlinear systems include substitution and addition. Examples walk through both methods step-by-step for various systems. Key steps are rewriting equations in terms of variables, substituting values, solving resulting equations, back-substituting, and checking solutions satisfy both original equations.
This document discusses operations on multiple random variables, including:
- The expected value of a function of two random variables X and Y is the sum of the expected values of the functions.
- Joint moments describe the relationship between multiple random variables and can be used to find properties like covariance and correlation.
- Two random variables are jointly Gaussian if their joint density function follows a specific form, and properties of Gaussian random variables include being fully defined by their first and second moments.
- Transformations of multiple random variables, such as applying a linear transformation, preserve properties like expected value and covariance if the original variables were Gaussian.
The document discusses using z-transforms to solve difference equations. It provides examples of first and second order linear difference equations and explains how to solve them using z-transforms. The process involves taking the z-transform of each term in the difference equation, resulting in an algebraic equation that can be solved for the z-transform of the solution sequence. The inverse z-transform is then found to obtain the solution sequence. Partial fractions and residues can be used to invert z-transforms.
The document describes putting a system of linear equations into triangular form. It contains a 3x3 system of linear equations. The summary explains that through Gaussian elimination by eliminating variables from equations, the system can be transformed into an upper triangular matrix with equations arranged from top to bottom to easily solve for the variables through back substitution.
Conditional distributions and multiple correlation coefficienthedikuswanto
The document discusses conditional distributions and multiple correlation coefficients. It provides definitions and properties for conditional densities and moments of multivariate distributions with ellipsoidal contours, such as the multivariate t-distribution. Formulas are derived for the conditional mean and covariance matrix given subsets of the data. Moments up to fourth order are obtained. Kurtosis is defined and shown to characterize the fourth cumulant of a standardized component. Several examples of ellipsoidal distributions are also mentioned.
This document discusses applying z-transforms to discrete systems to obtain the system's transfer function and output response. It begins by introducing z-transforms and their use in obtaining transfer functions for discrete systems. Transfer functions allow analyzing combinations of systems. The document provides examples of obtaining transfer functions for first and second order systems. It also covers obtaining the unit impulse and step responses from the transfer function. Finally, it discusses analyzing series combinations of systems by multiplying their individual transfer functions.
- The z-transform is a mathematical tool that converts discrete-time sequences into complex functions, analogous to how the Laplace transform handles continuous-time signals.
- Key properties and sequences that are transformed include the unit impulse δn, unit step un, and geometric sequences an.
- The z-transform is computed by taking the z-transform definition, which is an infinite summation, and obtaining closed-form expressions using properties like linearity and geometric series sums.
- Common transforms include U(z) for the unit step, 1/1-az^-1 for geometric sequences an, and expressions involving z, sinh/cosh, and sin/cos for exponential and trigonometric sequences.
The document provides a review outline for Math 1a Midterm II covering topics including: differentiation using product, quotient, and chain rules; implicit differentiation; logarithmic differentiation; applications such as related rates and optimization; and the shape of curves including the mean value theorem and extreme value theorem. It also lists learning objectives and provides details on key concepts like L'Hopital's rule and the closed interval method for finding extrema.
This document is a master's thesis that studies independence complexes constructed from independent sets of sequences of graphs. It examines generating functions, closed formulas, and homology groups of independence complexes for different graph sequences. The thesis provides formulas for the f-polynomials, Euler characteristics, and dimensions of homology groups for each independence complex studied. It also compares the results to known number sequences and establishes bijections to other problems when possible.
This document contains a problem set in quantitative methods with 17 questions covering topics in linear algebra including: solving systems of linear equations using Gauss-Jordan elimination; determining the inverse of matrices; finding the null space and row/column spaces of matrices; determining if sets of vectors are linearly independent/dependent or span vector spaces; and identifying if sets of vectors form bases. The problem set is assigned by Manimay Sengupta for the Monsoon Semester 2012 at South Asian University.
Integrability and weak diffraction in a two-particle Bose-Hubbard model jiang-min zhang
We report a bound state, which is embedded in the continuum spectrum, of the one-dimensional two-particle (Bose or Fermion) Hubbard model with an impurity potential. The state has the Bethe-ansatz form, although this model is nonintegrable. Moreover, for a wide region in parameter space, its energy is located in the continuum band. A remarkable advantage of this state with respect to similar states in other systems is the simple analytical form of the wave function and eigenvalue. This state can be tuned in and out of the continuum continuously.
This document discusses fundamental solutions of linear homogeneous differential equations. It introduces the concept of a fundamental set of solutions - two solutions whose Wronskian is nonzero at some point. Any linear combination of a fundamental set with arbitrary constants forms the general solution. The principle of superposition and Wronskian determinant are used to show whether a given set of solutions spans all solutions. Examples demonstrate finding fundamental solutions and using them to solve initial value problems. Theorems establish existence and properties of fundamental solutions, including their role in solving initial value problems uniquely.
This document provides examples of constructing the dual problem of a linear programming primal problem and solving it using the two-phase simplex method. It first presents the rules for constructing the dual problem and then works through two examples. The first example derives the dual problem from the primal and solves it using the two-phase method. The second example shows how to find the optimal dual solution given the optimal primal solution using two methods - using the objective coefficients of the primal variables or using the inverse of the primal basic variable matrix.
This document discusses transformations of functions. It defines various types of transformations including vertical and horizontal stretches and shifts, reflections, and periodic transformations. It provides examples of functions and their transformations. It also discusses even and odd functions. The key points are that transformations can stretch, shrink, shift, or reflect the graph of a function and that even functions are symmetric about the y-axis while odd functions are symmetric about the origin.
1. A complex number λ is an eigenvalue of a matrix A if there exists a non-zero vector x such that Ax = λx.
2. If a matrix has complex eigenvalues, it provides important information about the matrix, such as in problems involving vibrations and rotations in space.
3. For a complex eigenvalue λ = a + bi, a is called the real part and b is called the imaginary part. The absolute value |λ| represents the "length" or magnitude of the eigenvalue.
Solution of second order nonlinear singular value problemsAlexander Decker
1) The document presents a method for finding solutions to second order nonlinear singular boundary value problems using Taylor series.
2) The method modifies the differential equation to handle singularities, then obtains recurrence relations for the Taylor series coefficients by taking derivatives at the singular point.
3) Examples are provided to demonstrate the method, yielding exact solutions for problems in astronomy modeling gas spheres and other physical applications.
The document discusses sampled functions and their z-transforms. It begins by explaining how sampling a continuous function at intervals of T produces a sequence of values. The z-transform of this sampled sequence is then related to the Laplace transform of the original continuous function. Specifically, the z-transform of the sampled sequence is equivalent to the Laplace transform with s replaced by z=esT. Examples are provided to illustrate this relationship between the pole locations of the two transforms.
1) The document discusses elastic instability in structures, using the example of buckling of bars and columns under compressive loading. Elastic instability occurs when a structure transitions from stable to unstable deformation modes with increasing load.
2) As an introductory example, the document analyzes a rigid bar with a torsional spring, subject to horizontal and vertical forces. It derives an expression for the critical vertical load that causes instability, equal to the torsional spring stiffness divided by the bar length.
3) Looking ahead, the document notes it will apply these concepts of instability and critical load to analyze the buckling of a compressed column, which has a continuous distribution of stiffness rather than
1. The document discusses various types of transformations in complex analysis, including translation, rotation, stretching, and inversion.
2. Under inversion (1/w=z), a straight line is mapped to a circle if it does not pass through the origin, and to another straight line if it does pass through the origin. A circle is always mapped to another circle.
3. A general bilinear or Möbius transformation can be expressed as a combination of translation, rotation, stretching, and inversion.
This document provides examples of solving systems of nonlinear equations in two variables. It begins with definitions, including that a nonlinear system contains at least one equation that is not of the form Ax + By = C. Methods for solving nonlinear systems include substitution and addition. Examples walk through both methods step-by-step for various systems. Key steps are rewriting equations in terms of variables, substituting values, solving resulting equations, back-substituting, and checking solutions satisfy both original equations.
This document discusses operations on multiple random variables, including:
- The expected value of a function of two random variables X and Y is the sum of the expected values of the functions.
- Joint moments describe the relationship between multiple random variables and can be used to find properties like covariance and correlation.
- Two random variables are jointly Gaussian if their joint density function follows a specific form, and properties of Gaussian random variables include being fully defined by their first and second moments.
- Transformations of multiple random variables, such as applying a linear transformation, preserve properties like expected value and covariance if the original variables were Gaussian.
The document discusses using z-transforms to solve difference equations. It provides examples of first and second order linear difference equations and explains how to solve them using z-transforms. The process involves taking the z-transform of each term in the difference equation, resulting in an algebraic equation that can be solved for the z-transform of the solution sequence. The inverse z-transform is then found to obtain the solution sequence. Partial fractions and residues can be used to invert z-transforms.
The document describes putting a system of linear equations into triangular form. It contains a 3x3 system of linear equations. The summary explains that through Gaussian elimination by eliminating variables from equations, the system can be transformed into an upper triangular matrix with equations arranged from top to bottom to easily solve for the variables through back substitution.
Conditional distributions and multiple correlation coefficienthedikuswanto
The document discusses conditional distributions and multiple correlation coefficients. It provides definitions and properties for conditional densities and moments of multivariate distributions with ellipsoidal contours, such as the multivariate t-distribution. Formulas are derived for the conditional mean and covariance matrix given subsets of the data. Moments up to fourth order are obtained. Kurtosis is defined and shown to characterize the fourth cumulant of a standardized component. Several examples of ellipsoidal distributions are also mentioned.
This document discusses applying z-transforms to discrete systems to obtain the system's transfer function and output response. It begins by introducing z-transforms and their use in obtaining transfer functions for discrete systems. Transfer functions allow analyzing combinations of systems. The document provides examples of obtaining transfer functions for first and second order systems. It also covers obtaining the unit impulse and step responses from the transfer function. Finally, it discusses analyzing series combinations of systems by multiplying their individual transfer functions.
- The z-transform is a mathematical tool that converts discrete-time sequences into complex functions, analogous to how the Laplace transform handles continuous-time signals.
- Key properties and sequences that are transformed include the unit impulse δn, unit step un, and geometric sequences an.
- The z-transform is computed by taking the z-transform definition, which is an infinite summation, and obtaining closed-form expressions using properties like linearity and geometric series sums.
- Common transforms include U(z) for the unit step, 1/1-az^-1 for geometric sequences an, and expressions involving z, sinh/cosh, and sin/cos for exponential and trigonometric sequences.
The document provides a review outline for Math 1a Midterm II covering topics including: differentiation using product, quotient, and chain rules; implicit differentiation; logarithmic differentiation; applications such as related rates and optimization; and the shape of curves including the mean value theorem and extreme value theorem. It also lists learning objectives and provides details on key concepts like L'Hopital's rule and the closed interval method for finding extrema.
This document is a master's thesis that studies independence complexes constructed from independent sets of sequences of graphs. It examines generating functions, closed formulas, and homology groups of independence complexes for different graph sequences. The thesis provides formulas for the f-polynomials, Euler characteristics, and dimensions of homology groups for each independence complex studied. It also compares the results to known number sequences and establishes bijections to other problems when possible.
This document contains a problem set in quantitative methods with 17 questions covering topics in linear algebra including: solving systems of linear equations using Gauss-Jordan elimination; determining the inverse of matrices; finding the null space and row/column spaces of matrices; determining if sets of vectors are linearly independent/dependent or span vector spaces; and identifying if sets of vectors form bases. The problem set is assigned by Manimay Sengupta for the Monsoon Semester 2012 at South Asian University.
Integrability and weak diffraction in a two-particle Bose-Hubbard model jiang-min zhang
We report a bound state, which is embedded in the continuum spectrum, of the one-dimensional two-particle (Bose or Fermion) Hubbard model with an impurity potential. The state has the Bethe-ansatz form, although this model is nonintegrable. Moreover, for a wide region in parameter space, its energy is located in the continuum band. A remarkable advantage of this state with respect to similar states in other systems is the simple analytical form of the wave function and eigenvalue. This state can be tuned in and out of the continuum continuously.
This document discusses fundamental solutions of linear homogeneous differential equations. It introduces the concept of a fundamental set of solutions - two solutions whose Wronskian is nonzero at some point. Any linear combination of a fundamental set with arbitrary constants forms the general solution. The principle of superposition and Wronskian determinant are used to show whether a given set of solutions spans all solutions. Examples demonstrate finding fundamental solutions and using them to solve initial value problems. Theorems establish existence and properties of fundamental solutions, including their role in solving initial value problems uniquely.
This document provides examples of constructing the dual problem of a linear programming primal problem and solving it using the two-phase simplex method. It first presents the rules for constructing the dual problem and then works through two examples. The first example derives the dual problem from the primal and solves it using the two-phase method. The second example shows how to find the optimal dual solution given the optimal primal solution using two methods - using the objective coefficients of the primal variables or using the inverse of the primal basic variable matrix.
This document discusses transformations of functions. It defines various types of transformations including vertical and horizontal stretches and shifts, reflections, and periodic transformations. It provides examples of functions and their transformations. It also discusses even and odd functions. The key points are that transformations can stretch, shrink, shift, or reflect the graph of a function and that even functions are symmetric about the y-axis while odd functions are symmetric about the origin.
1. A complex number λ is an eigenvalue of a matrix A if there exists a non-zero vector x such that Ax = λx.
2. If a matrix has complex eigenvalues, it provides important information about the matrix, such as in problems involving vibrations and rotations in space.
3. For a complex eigenvalue λ = a + bi, a is called the real part and b is called the imaginary part. The absolute value |λ| represents the "length" or magnitude of the eigenvalue.
Solution of second order nonlinear singular value problemsAlexander Decker
1) The document presents a method for finding solutions to second order nonlinear singular boundary value problems using Taylor series.
2) The method modifies the differential equation to handle singularities, then obtains recurrence relations for the Taylor series coefficients by taking derivatives at the singular point.
3) Examples are provided to demonstrate the method, yielding exact solutions for problems in astronomy modeling gas spheres and other physical applications.
The document discusses sampled functions and their z-transforms. It begins by explaining how sampling a continuous function at intervals of T produces a sequence of values. The z-transform of this sampled sequence is then related to the Laplace transform of the original continuous function. Specifically, the z-transform of the sampled sequence is equivalent to the Laplace transform with s replaced by z=esT. Examples are provided to illustrate this relationship between the pole locations of the two transforms.
1) The document discusses elastic instability in structures, using the example of buckling of bars and columns under compressive loading. Elastic instability occurs when a structure transitions from stable to unstable deformation modes with increasing load.
2) As an introductory example, the document analyzes a rigid bar with a torsional spring, subject to horizontal and vertical forces. It derives an expression for the critical vertical load that causes instability, equal to the torsional spring stiffness divided by the bar length.
3) Looking ahead, the document notes it will apply these concepts of instability and critical load to analyze the buckling of a compressed column, which has a continuous distribution of stiffness rather than
1. The document discusses various types of transformations in complex analysis, including translation, rotation, stretching, and inversion.
2. Under inversion (1/w=z), a straight line is mapped to a circle if it does not pass through the origin, and to another straight line if it does pass through the origin. A circle is always mapped to another circle.
3. A general bilinear or Möbius transformation can be expressed as a combination of translation, rotation, stretching, and inversion.
Horizontal Shifts of Quadratic Functions MarkBredin
- The graph of y = (x - h)2 has a vertex at (h,0) that is translated horizontally by h units from the origin.
- If h is positive, the translation is to the right, and if h is negative, the translation is to the left.
- Other quadratic graphs can be obtained by translating the basic graph of y = x2.
This document discusses techniques for calculating electric potential, including:
1. Laplace's equation and its solutions in 1D, 2D, and 3D, including boundary conditions.
2. The method of images, which uses fictitious "image" charges to solve problems involving conductors. The classical image problem and induced surface charge on a conductor are examined.
3. Other techniques like multipole expansion, separation of variables, and numerical methods like relaxation are mentioned but not explained in detail.
12 x1 t07 02 v and a in terms of x (2012)Nigel Simmons
1) The document discusses relating acceleration to velocity and position for particles moving in one dimension.
2) It derives the formula for acceleration as the second derivative of position with respect to time, equal to the first derivative of velocity with respect to position.
3) It provides two examples:
- Example 1 finds the velocity of a particle in terms of its position by solving the differential equation for acceleration.
- Example 2 finds the position of a particle in terms of time by solving the differential equation and using initial conditions.
The document introduces Maxwell's equations in differential geometric formulation. It defines the electromagnetic tensor F, which combines the electric and magnetic fields E and B into a single matrix. F can be represented as a 2-form comprising forms E and B. The Hodge star operator establishes duality between forms. Applying the exterior derivative d to F yields Maxwell's equations dF=0 and applying it to the Hodge dual d*F yields the other set of Maxwell's equations d*F=0, showing the equivalence between the differential geometric and standard formulations.
Advanced Engineering Mathematics Solutions Manual.pdfWhitney Anderson
This document contains 27 multi-part exercises involving differential equations. The exercises cover topics such as determining whether differential equations are linear or nonlinear, solving differential equations, and classifying differential equations by order.
12X1 T07 02 v and a in terms of x (2011)Nigel Simmons
The document discusses the relationship between velocity, acceleration, and position for particles moving in one dimension.
It first shows that if velocity v is a function of position x, the acceleration is equal to the derivative of v squared with respect to x, divided by 2.
It then works through two examples:
1) Finding the velocity of a particle in terms of x, given its acceleration as a function of x.
2) Finding the position x of a particle in terms of time t, given its initial position and velocity and an acceleration function.
12X1 T07 01 v and a In terms of x (2010)Nigel Simmons
The document discusses the relationship between velocity, acceleration, and position for particles moving in one dimension.
It first shows that if velocity v is a function of position x, the acceleration is equal to the derivative of v squared with respect to x, divided by 2.
It then works through two examples:
1) Finding the velocity of a particle given its acceleration of 3 - 2x as a function of x.
2) Finding the position x of a particle in terms of time t, given its acceleration is 3x^2 and its initial position and velocity.
Interpolation functions for linear triangle elements (very elementary)Prashant K. Jha
Shows how to compute interpolation functions for linear triangle elements. Steps involve defining the interpolation function on the reference triangle and map taking point in the given triangle to the reference triangle.
This document discusses differential equations. It defines differential equations and explains that the order refers to the highest derivative. It distinguishes between ordinary and partial differential equations. It also covers topics like the degree of a differential equation, linear vs nonlinear, and methods for solving first-order differential equations like separation of variables and integrating factors. Examples are provided to illustrate various types of first-order differential equations and solution methods.
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.
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.
This document discusses double integrals and their use in calculating volumes. It begins by introducing double integrals as a way to calculate the volume of a solid bounded above by a function f(x,y) over a rectangular region. It then discusses using iterated integrals to evaluate double integrals by first integrating with respect to one variable and then the other. Finally, it provides examples of using double integrals and iterated integrals to calculate volumes.
This document introduces the concept of double integrals and iterated integrals. It defines a double integral as the limit of double Riemann sums that approximate the volume under a function of two variables over a rectangular region. An iterated integral first integrates with respect to one variable, holding the other constant, resulting in a function of the remaining variable which is then integrated. This allows exact calculation of double integrals by integrating in two steps rather than approximating volume with boxes.
The one-dimensional wave equation governs vibrations of an elastic string. It is solved by separating variables, yielding solutions of the form F(x)G(t) where F and G satisfy ordinary differential equations. Boundary conditions require F(x) to be sinusoidal, with wavelengths that are integer multiples of the string length. The general solution is a superposition of these sinusoidal modes, with coefficients determined by the initial conditions. Motions of strings with different initial displacements are expressed as solutions to the one-dimensional wave equation.
This document provides an introduction to linear transformations. It defines a linear transformation as a function that maps one vector space to another while preserving vector addition and scalar multiplication. Key concepts discussed include the domain, co-domain, range, and pre-image of a linear transformation. Examples are given to demonstrate linear transformations and functions that are not linear transformations. The relationship between linear transformations and matrices is also explained.
1) The Wronskian of three functions is evaluated to show they are linearly independent on the interval (1, ∞).
2) The general solution to a nonhomogeneous differential equation is found using the method of undetermined coefficients and applying initial conditions.
3) The method of variation of parameters is used to find a particular solution and the general solution to a nonhomogeneous differential equation.
This document discusses linear transformations. It begins by defining a linear transformation as a function between vector spaces that preserves vector addition and scalar multiplication. It provides examples of linear transformations and functions that are not linear transformations. It then discusses properties of linear transformations including the zero and identity transformations. It introduces the concept that a linear transformation can be represented by a matrix and that matrix multiplication defines a linear transformation. It concludes by stating that a linear transformation defined by a matrix satisfies the properties of being linear.
Rohit Vijay Bapat's resume summarizes his education and professional experience in software development. He has a MS in Mechanical Engineering from Missouri University of Science and Technology and a BE in Mechanical Engineering from University of Pune. His experience includes roles at Tata Consultancy Services, Mindware Engineering, SigmaTEK Systems, and Vaal Triangle developing CAD/CAM and shopfloor software. He has strong skills in C++, C#, Delphi, and CAD tools like AutoCAD, ProE, and NX.
This document is a book titled "Programmer's Heaven: C# School" that provides 14 lessons to learn the C# programming language. It was written by Faraz Rasheed and edited by Tore Nestenius, Jonathan Worthington, and Lee Addy. The book is published by Programmer's Heaven and copyrighted in 2005-2006. It includes chapters that cover C# fundamentals, classes, inheritance, structures, exceptions, delegates, windows forms, data access, multithreading, file systems, new C# 2.0 features, and the future of C#. The book is intended to teach C# to programmers starting with the language.
This chapter introduces OpenGL and provides an overview of its capabilities. It presents a simple OpenGL program to draw a pyramid and explains the basic program structure. The chapter describes how to build up geometric models from primitives like points, lines and polygons. It notes that OpenGL rendering becomes more realistic as objects are lit and textures are added. The color plates provide examples of increasing complexity from wireframe to lit objects with textures.
This document provides an overview of C++ Essentials, a book that introduces the C++ programming language. The book is divided into 12 chapters that cover topics such as variables, expressions, statements, functions, arrays, pointers, classes, inheritance, templates, exceptions, input/output streams, and the preprocessor. Each chapter consists of short sections to simplify learning the material. The book is designed as a concise introductory text to teach C++ programming to beginners.
This document provides notes from a 1996 short course on modern grid generation techniques. It discusses the history and current state of grid generation, as well as mathematical foundations and numerical techniques for generating 2D and 3D grids. Specific techniques covered include generating grids for complex geometries using a multiblock approach, local grid clustering using "clamp" mappings, and grid adaptation. The document also proposes a grid generation meta-language and examines tools for automatic topology definition and parallelization strategies.
Fundamentals of computational_fluid_dynamics_-_h._lomax__t._pulliam__d._zinggRohit Bapat
This document provides an overview of computational fluid dynamics (CFD) and summarizes its key steps and concepts. It discusses the fundamentals of CFD, including conservation laws, governing equations, finite difference approximations, semi-discrete and finite volume methods, and time-marching algorithms. The document is intended to introduce readers to the basic theory and methods in CFD for modeling fluid flow and transport phenomena.
This document provides an introduction to Fourier series. It emphasizes conceptual understanding over mathematical rigor. It recommends that readers work through examples and derivations in the paper to fully understand Fourier series. The paper introduces Fourier series as representing periodic waveforms as sums of harmonically related sinusoids. It discusses Fourier coefficients and their physical interpretation. It also covers properties of Fourier series such as linearity, symmetry, time shifting, differentiation, and integration.
This document provides an overview of Adobe Photoshop CS4 and introduces the basics of using the software. It covers starting Photoshop, setting up documents, opening images, and understanding the main interface components like menus, tools, palettes and panels. The document then explains how to perform basic image editing tasks such as cropping, resizing and correcting images. It also touches on sharpening and softening images as well as saving files. The tutorial is divided into sections that progress from getting started to more advanced functions.
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1. Chapter 3
Coordinate System
Transformations
3.1 Problem Statement
We have met the various coordinate systems that will be of primary interest
in the study of flight dynamics. Now we address the subject of how these
coordinate systems are related to one another. For instance, when we begin
to sum the external forces acting on the aircraft, we will have to relate all
these forces to a common reference frame. If we take some body-axis system
to be the common frame, then we need to be able to take the gravity vector
(weight) from the local horizontal reference frame, the thrust from some other
body-axis frame, and the aerodynamic forces from the wind-axis frame, and
represent all these forces in the body-axis frame.
Some of these relationships are easy because they are fixed: Two body-
axis systems, once defined, are always related by a single rotation around
their common y axis. Others are much more complicated because they vary
with time: The orientation of a given body-axis system with respect to the
wind axes determines certain aerodynamic forces and moments which change
that orientation.
First we must characterize the relationship between two coordinate sys-
tems at some frozen instant in time. The instantaneous relationship between
two coordinate systems will be addressed by determining a transformation
that will take the representation of an arbitrary vector in one system and
convert it to its representation in the other.
23
2. 24 CHAPTER 3. COORDINATE SYSTEM TRANSFORMATIONS
Three approaches to finding the needed transformations will be presented.
The first is called Direction Cosines, a sort of brute-force approach to the
problem. Next we will discuss Euler angles, by far the most common ap-
proach but one with a potentially serious problem. Finally we will examine
Euler parameters, an elegant solution to the problem found with Euler angles.
3.2 Transformations
3.2.1 Definitions
Consider two reference frames, F1 and F2 , and a vector v whose components
are known in F1 , represented as {v}1 :
v x1
{v}1 = v
y1
v z1
We wish to determine the representation of the same vector in F2 , or
{v}2 :
v x2
{v}2 = v
y2
v z2
These are linear spaces, so the transformation of the vector from F1 to
F2 is simply a matrix multiplication which we will denote T2,1 , such that
{v}2 = T2,1 {v}1 . Transformations such as T2,1 are called similarity trans-
formations. The transformations involved in simple rotations of orthogonal
reference frames have many special properties that will be shown.
The order of subscripts of T2,1 is such that the left subscript goes with the
system of the vector on the left side of the equation and the right subscript
with the vector on the right. For the matrix multiplication to be conformal
T2,1 must be a 3x3 matrix:
t11 t12 t13
T2,1 = t21 t22 t23
t31 t32 t33
3. 3.2. TRANSFORMATIONS 25
x1 x2
y1
z2 v
y2
z1
Figure 3.1: Two Coordinate Systems
(The tij probably need more subscripting to distinguish them from those
in other transformation matrices, but this gets cumbersome.)
The whole point of the three approaches to be presented is to figure out
how to evaluate the numbers tij . It is important to note that for a fixed
orientation between two coordinate systems, the numbers tij are the same
quantities no matter how they are evaluated.
3.2.2 Direction Cosines
Derivation
Consider our two reference frames F1 and F2 , and the vector v, shown in
figure 3.1. We claim to know the representation of v in F1 . The vector v
is the vector sum of the three components vx1 i1 , vy1 j1 , and vz1 k1 so we may
replace the vector by those three components, shown in figure 3.2
Now, the projection of v onto x2 is the same as the vector sum of the
projections of each of its components vx1 i1 , vy1 j1 , and vz1 k1 onto x2 , and
similarly for y2 and z2 . Define the angle generated in going from x2 to x1 as
θx2 x1 . The projection of vx1 i1 onto x2 therefore has magnitude vx1 cos θx2 x1
and direction i2 , as shown if figure 3.3.
To the vector vx1 cos θx2 x1 i2 must be added the other two projections,
vy1 cos θx2 y1 i2 and vz1 cos θx2 z1 i2 . Similar projections onto y2 and z2 result in:
4. 26 CHAPTER 3. COORDINATE SYSTEM TRANSFORMATIONS
x1 x2
y1
z2 v x 1i 1
v y 1j1 y2
z1
v z 1k1
Figure 3.2: Vector in Component Form
x1 x2
v x 1i 1
v x 1 cos θ x 2x 1 i 2
θ x 2x 1
Figure 3.3: One Component of the Vector
5. 3.2. TRANSFORMATIONS 27
vx2 = vx1 cos θx2 x1 +vy1 cos θx2 y1 +vz1 cos θx2 z1
vy2 = vx1 cos θy2 x1 +vy1 cos θy2 y1 +vz1 cos θy2 z1
vz2 = vx1 cos θz2 x1 +vy1 cos θz2 y1 +vz1 cos θz2 z1
In vector-matrix notation, this may be written
v x2
cos θx2 x1 cos θx2 y1 cos θx2 z1 vx1
v = cos θy2 x1 cos θy2 y1 cos θy2 z1 vy1
y2
v z2 cos θz2 x1 cos θz2 y1 cos θz2 z1 v z1
Clearly this is {v}2 = T2,1 {v}1 , so we must have tij = cos θ(axis)2 (axis)1 in
which i or j is 1 if the corresponding (axis) is x, 2 if (axis) is y, and 3 if
(axis) is z.
cos θx2 x1 cos θx2 y1 cos θx2 z1
T2,1 = cos θy2 x1 cos θy2 y1 cos θy2 z1 (3.1)
cos θz2 x1 cos θz2 y1 cos θz2 z1
Properties of the Direction Cosine Matrix
The transformation matrix (often called the direction cosine matrix, regard-
less of how it is derived or represented) does not depend on the vector v for
its existence. It is therefore the same no matter what we choose for v. If we
take
1
{v}1 = 0
0
Then {v}2 is just the first column of the direction cosine matrix:
cos θx2 x1
{v}2 = cos θy2 x1
cos θz2 x1
6. 28 CHAPTER 3. COORDINATE SYSTEM TRANSFORMATIONS
Since the length of v is 1, we have shown the well-known property of
direction cosines, that
cos2 θx2 x1 + cos2 θy2 x1 + cos2 θz2 x1 = 1
The same obviously holds true for any column of the direction cosine
matrix.
If we need to go the other way, {v}1 = T1,2 {v}2 , it is clear that T1,2 =
−1
T2,1 , and we might be tempted to invert the direction cosine matrix. On the
other hand, had we begun by assuming {v}2 was known, and figuring out
how to get {v}1 = T1,2 {v}2 using similar arguments to the above, we would
have arrived at
v x1
cos θx1 x2 cos θx1 y2 cos θx1 z2 vx2
v = cos θy1 x2 cos θy1 y2 cos θy1 z2 vy2
y1
v z1 cos θz1 x2 cos θz1 y2 cos θz1 z2 v z2
Here we observe that cos θx1 x2 is the same as cos θx2 x1 , cos θx1 y2 is the
same as cos θy2 x1 , etc., since (even if we had defined positive rotations of
these angles) the cosine is an even function. So the same transformation is
v x1
cos θx2 x1 cos θy2 x1 cos θz2 x1 vx2
v = cos θx2 y1 cos θy2 y1 cos θz2 y1 vy2
y1
v z1 cos θx2 z1 cos θy2 z1 cos θz2 z1 v z2
Obviously the columns of T1,2 are the rows of T2,1 (and have unit length
as well). This leads us to another nice property of these transformation ma-
trices, that the inverse of the direction cosine matrix is equal to its transpose,
−1 T
T1,2 = T2,1 = T2,1
−1
Since T2,1 T2,1 = T2,1 T2,1 = I3 , the 3 × 3 identity matrix, it must be true
T
that the scalar (dot) product of any row of T2,1 with any other row must
−1 T
be zero. It is easy to show that T1,2 T1,2 = T1,2 T1,2 = I3 , so the scalar (dot)
product of any column of T2,1 with any other column must be zero since the
7. 3.2. TRANSFORMATIONS 29
columns of T2,1 are the rows of T1,2 . When the rows (or columns) of the
direction cosine matrix are viewed as vectors, this means the rows (or the
columns) form orthogonal bases for 3-dimensional space.
T
Also, with T2,1 T2,1 = I3 , if we take the determinant of each side and note
that T2,1 = |T2,1 |, we have
T
T2,1 T2,1 = |T2,1 | T2,1
T T
= |T2,1 | |T2,1 |
= |T2,1 |2 = 1
The only conclusion we can reach at this point is that |T2,1 | = ±1. Be-
cause the identity matrix is a transformation matrix with determinant +1,
and since all other transformations may be reached through continuous rota-
tions from the identity transformation, it seems unreasonable to think that
the sign of the determinant would be different for some rotations and not
others. We will show in our discussion of Euler angles that the right answer
is indeed |T2,1 | = +1.
While we have 9 variables in T2,1 = {tij } , i, j = 1 . . . 3, there are 6
nonlinear constraining equations based on the orthogonality of the rows (or
columns) of the matrix. For the rows these are:
t2 + t2 + t2 = 1
11 12 13
t21 + t22 + t2 = 1
2 2
23
t31 + t32 + t2 = 1
2 2
33
t11 t21 +t12 t22 +t13 t23 = 0
t11 t31 +t12 t32 +t13 t33 = 0
t21 t31 +t22 t32 +t23 t33 = 0
The result of these constraints is that there are only 3 independent vari-
ables. In principle all nine may be derived given any three that do not violate
a constraint.
3.2.3 Euler angles
If there are only three independent variables in the direction cosine matrix,
then we should be able to express each of the tij in terms of some set of three
8. 30 CHAPTER 3. COORDINATE SYSTEM TRANSFORMATIONS
x1
θz x′
y1
θz
y′
z1 = z′
Figure 3.4: Rotation Through θz
(not necesarily unique) independent variables. One means of determining
these variables is by use of a famous theorem due to the Swiss mathematician
Leonhard Euler (1707-83). Briefly, this theorem holds that any arbitrarily
oriented reference frame may be placed in alignment with (made to have axes
parallel to) any other reference frame by three successive rotations about the
axes of the reference frame. The order of selection of axes in these rotations
is arbitrary, but the same axis may not be used twice in succesion. The
rotation sequences are usually denoted by three numbers, 1 for x, 2 for y,
and 3 for z. The twelve valid sequences are 123, 121, 131, 132, 213, 212,
231, 232, 312, 313, 321, and 323. The angles through which these rotations
are performed (defined as positive according to the right hand rule for right
handed coordinate systems) are called generically Euler angles.
The rotation sequence most often used in flight dynamics is the 321, or
z − y − x. Considering a rotation from F1 to F2 the first rotation (figure 3.4)
is about z1 through an angle θz which is positive according to the right hand
rule about the z1 axis. With two rotations to go, the resulting alignment in
general is oriented with neither F1 or F2 , but some intermediate reference
frame (the first of two) denoted F . Since the rotation was about z1 , z is
parallel to it but neither of the other two primed axes is.
The next rotation (figure 3.5) is through an angle θy about the axis y
of the first intermediate reference frame to the second intermediate reference
frame, F . Note that y = y , and neither y or z are necessarily axes of
either F1 or F2 .
9. 3.2. TRANSFORMATIONS 31
x″
θy x′
θy y″ = y′
z″
z′
Figure 3.5: Rotation Through θy
The final rotation (figure 3.6) is about x through angle θx and the final
alignment is parallel to the axes of F2 .
Now assuming we know the angles θx , θy , and θz we need to relate them
to the elements of the direction cosine matrix T2,1 . We will do this by seeing
how the arbitrary vector v is represented in each of the intermediate and final
reference frames in terms of its representation in the prior reference frame.
Consider first the rotation about z1 (Figure 3.7). In terms of the direc-
tion cosines previously defined, the angles between the axes are as follows:
between z1 and z it is zero; between either z and any x or y it is 90 deg;
between x1 and x or y1 and y it is θz ; between x1 and y it is 90 deg +θz ;
and between y1 and x it is 90 deg −θz . We therefore may write
10. 32 CHAPTER 3. COORDINATE SYSTEM TRANSFORMATIONS
x″ = x 2
θx y″
z2 θx z″
y2
Figure 3.6: Rotation Through θx
x1 x′
θz
v x ′i′
vx i1 v
v y 1j 1 y1
θz
v y ′j′
y′
Figure 3.7: Rotation from F1 to F
11. 3.2. TRANSFORMATIONS 33
vx
cos θx x1 cos θx y1 cos θx z1 vx1
v = cos θy x1 cos θy y1 cos θy z1 vy1
y
vz cos θz x1 cos θz y1 cos θz z1 v z1
cos θz cos (90 deg −θz ) cos 90 deg vx1
= cos (90 deg +θz ) cos θz cos 90 deg vy1
cos 90 deg cos 90 deg cos 0 v z1
cos θz sin θz 0 vx1
= − sin θz cos θz 0 vy1
0 0 1 v z1
In short, {v} = TF ,1 {v}1 in which
cos θz sin θz 0
TF ,1 = − sin θz cos θz 0 (3.2)
0 0 1
From the rotation about y we get {v} = TF ,F {v} in which
cos θy 0 − sin θy
TF ,F = 0 1 0 (3.3)
sin θy 0 cos θy
From the rotation about x , finally, {v}2 = T2,F {v} with
1 0 0
T2,F = 0 cos θx sin θx (3.4)
0 − sin θx cos θx
We now cascade the relationships: {v}2 = T2,F {v} , {v} = TF ,F {v}
to arrive first at {v}2 = T2,F TF ,F {v} and then at {v}2 = T2,F TF ,F TF ,1 {v}1 ,
or
12. 34 CHAPTER 3. COORDINATE SYSTEM TRANSFORMATIONS
1 0 0 cos θy 0 − sin θy cos θz sin θz 0
{v}2 = 0 cos θx sin θx 0 1 0 − sin θz cos θz 0 {v}1
0 − sin θx cos θx sin θy 0 cos θy 0 0 1
The transformation matrix we seek is the product of the three sequential
transformations (in the correct order!), or T2,1 = T2,F TF ,F TF ,1 . The details
are slightly tedious, but the result is
cos θy cos θz cos θy sin θz − sin θy
sin θx sin θy cos θz sin θx sin θy sin θz
sin θx cos θy
T2,1 =
− cos θx sin θz + cos θx cos θz
cos θx sin θy cos θz cos θx sin θy sin θz
cos θx cos θy
+ sin θx sin θz − sin θx cos θz
Since this is just a different way of representing the direction cosine ma-
trix, it must be true that cos θy cos θz = cos θx2 x1 , cos θy sin θz = cos θx2 y1 ,
etc. At this point we may observe that since T2,1 = T2,F TF ,F TF ,1 , we must
have
|T2,1 | = |T2,F TF ,F TF ,1 | = |T2,F | |TF ,F | |TF ,1 |
The determinant of each of the three intermediate transformations is eas-
ily verified to be +1, so that we have the expected result,
|T2,1 | = +1
It is very important to note that the Euler angles θx , θy , and θz have been
defined for a rotation from F1 to F2 using a 321 rotation sequence. Thus, a
321 rotation from F2 to F1 (T1,2 ) with suitably defined angles (say, φx , φy ,
and φz ) would have the same form as given for T2,1 , but the angles involved
would be physically different from those in T2,1 . So, for the rotation from F1
to F2 using a 321 rotation sequence:
13. 3.2. TRANSFORMATIONS 35
cos φy cos φz cos φy sin φz − sin φy
sin φx sin φy cos φz sin φx sin φy sin φz
sin φx cos φy
T1,2 =
− cos φx sin φz + cos φx cos φz
cos φx sin φy cos φz cos φx sin φy sin φz
cos φx cos φy
+ sin φx sin φz − sin φx cos φz
−1 T
Alternatively we may note that T1,2 = T2,1 = T2,1 and may write the
matrix
sin θx sin θy cos θz cos θx sin θy cos θz
cos θy cos θz
− cos θx sin θz + sin θx sin θz
T1,2 =
cos θ sin θ sin θx sin θy sin θz cos θx sin θy sin θz
y z
+ cos θx cos θz − sin θx cos θz
− sin θy sin θx cos θy cos θx cos θy
The two matrices are the same, only the definitions of the angles are
different. Clearly the relationships among the two sets of angles are non-
trivial.
In short, the definitions of Euler angles are unique to the rotation sequence
used and the decision as to which frame one goes from and which one goes
to in that sequence. We will normally define our Euler angles going in only
one direction using a 321 rotation sequence, and rely on relationships like
−1 T
T1,2 = T2,1 = T2,1 to obtain the other.
3.2.4 Euler parameters
The derivation of Euler parameters is at appendix A. They are based on the
observation that any two coordinate systems are instantaneously related by
a single rotation about some axis that has the same representation in each
system. The axis, called the eigenaxis, has direction cosines ξ, ζ, and χ; the
angle of rotation is η. Then, with the definition of the Euler parameters
14. 36 CHAPTER 3. COORDINATE SYSTEM TRANSFORMATIONS
.
q0 =cos (η/2)
.
q1 =ξ sin (η/2)
. (3.5)
q2 =ζ sin (η/2)
.
q3 =χ sin (η/2)
the transformation matrix becomes:
(q0 + q1 − q2 − q3 )
2 2 2 2
2 (q1 q2 + q0 q3 ) 2 (q1 q3 − q0 q2 )
T2,1 = 2 (q1 q2 − q0 q3 ) (q0 − q1 + q2 − q3 )
2 2 2 2
2 (q2 q3 + q0 q1 )
2 (q1 q3 + q0 q2 ) 2 (q2 q3 − q0 q1 ) (q0 − q1 − q2 + q3 )
2 2 2 2
(3.6)
Euler parameters have one great disadvantage relative to Euler angles:
Euler angles may in most cases be easily visualized. If one is given values of
θx , θy , and θz it is not hard to visualize the relative orientation of two coordi-
nate systems. A given set of Euler parameters, however, conveys almost no
information about how the systems are related. Thus even in applications
in which Euler parameters are preferred (such as in flight simulation), the
results are very often converted to Euler angles for ease of interpretation and
visualization.
The direct approach to convert a set of Euler parameters to the corre-
sponding set of Euler angles is to equate corresponding elements of the two
representations of the transformation matrix. We may combine the (2,3) and
(3,3) entries to obtain
sin θx cos θy 2 (q2 q3 + q0 q1 )
= tan θx = 2
cos θx cos θy q0 − q1 − q2 + q 3
2 2 2
t23 2 (q2 q3 + q0 q1 )
θx = tan−1 = tan−1 , −π ≤ θx < π
t33 2
q0 − q 1 − q 2 + q3
2 2 2
From the (1,3) entry we have
− sin θy = 2 (q1 q3 − q0 q2 )
15. 3.2. TRANSFORMATIONS 37
θy = − sin−1 (t13 ) = − sin−1 (2q1 q3 − 2q0 q2 ) , −π/2 ≤ θy ≤ π/2
Finally we use the (1,1) and (1,2) entries to yield
cos θy sin θz 2 (q1 q2 + q0 q3 )
= tan θz = 2
cos θy cos θz q 0 + q 1 − q2 − q 3
2 2 2
t12 2 (q1 q2 + q0 q3 )
θz = tan−1 = tan−1 , 0 ≤ θz ≤ 2π
t11 2
q0 + q 1 − q2 − q3
2 2 2
With the arctangent functions one has to be careful to not divide by
zero when evaluating the argument. Most software libraries have the two-
argument arctangent function which avoids this problem and helps keep track
of the quadrant. In summary,
θx = tan−1 2(q2 q3 +q0 q1 )
q0 −q1 −q2 +q3
2 2 2 2 , −π ≤ θx < π
θy = − sin−1 (2q1 q3 − 2q0 q2 ) , −π/2 ≤ θy ≤ π/2 (3.7)
θz = tan−1 2(q1 q2 +q0 q3 )
q0 +q1 −q2 −q3
2 2 2 2 , 0 ≤ θz ≤ 2π
Going the other way, from Euler angles to Euler parameters, has been
studied extensively. It may be done in a somewhat similar manner to equat-
ing elements of the transformation matrix. Another more elegant method
due to Junkins and Turner 1 yields a very nice result, here presented without
proof:
q0 = cos (θz /2) cos (θy /2) cos (θx /2) + sin (θz /2) sin (θy /2) sin (θx /2)
q1 = cos (θz /2) cos (θy /2) sin (θx /2) − sin (θz /2) sin (θy /2) cos (θx /2)
(3.8)
q2 = cos (θz /2) sin (θy /2) cos (θx /2) + sin (θz /2) cos (θy /2) sin (θx /2)
q3 = sin (θz /2) cos (θy /2) cos (θx /2) − cos (θz /2) sin (θy /2) sin (θx /2)
1
Junkins, J.L. and Turner, J.D., “Optional Continuous Torque Attitude Maneuvers,”
AIAA-AAS Astrodynamics Conference, Palo Alto, California, August 1978.
16. 38 CHAPTER 3. COORDINATE SYSTEM TRANSFORMATIONS
Just one final note on Euler parameters. Frequently in the literature
Euler parameters are referred to as quaternions. However, quaternions are
actually defined as half-scalar, half-vector entities in some coordinate system
˜
as Q = q0 + q1 i + q2 j + q3 k. The algebra of quaternions is useful for proving
theorems regarding Euler parameters, but will not be used in this course.
3.3 Transformations of Systems of Equations
Suppose we have a linear system of equations in some reference frame F1 :
{y}1 = A1 {x}1
We know the transformation to F2 (T2,1 ) and wish to represent the same
equations in that reference frame. Each side of {y}1 = A1 {x}1 is a vector in
F1 , so we transform both sides as T2,1 {y}1 = {y}2 = T2,1 A1 {x}1 . Then we
T
may insert the identity matrix in the form of T2,1 T2,1 in the right hand side
to yield {y}2 = T2,1 A1 T2,1 T2,1 {x}1 . The reason for doing this is to transform
T
the vector {x}1 to {x}2 = T2,1 {x}1 . Grouping terms we have the result,
{y}2 = T2,1 A1 T2,1 {x}2 = A2 {x}2
T
With the conclusion that the transformation of a matrix A1 from F1 to
F2 (or the equivalent operation performed by A1 in F2 ) is given by
T
A2 = T2,1 A1 T2,1 (3.9)
This is sometimes spoken of as the transformation of a matrix. Such
transformations may be very useful. For example, if we could find F2 and
T2,1 such that A2 is diagonal, then the system of equations {y}2 = A2 {x}2
is very easy to solve. The original variables can then be recovered by {y}1 =
T2,1 {y}2 , {x}1 = T2,1 {x}2 .
T T
17. 3.4. CUSTOMS AND CONVENTIONS 39
3.4 Customs and Conventions
3.4.1 Names of Euler angles
Some transformation matrices occur frequently, and the 321 Euler angles
associated with them are given special symbols. These are summarized as
follows:
TF2 ,F1 θx θ y θz
TB,H φ θ ψ
(3.10)
TW,H µ γ χ
TB,W 0 α −β
3.4.2 Principal Values of Euler angles
The principal range of values of the Euler angles is fixed largely by convention
and may vary according to the application. In this course the convention is
−π≤θx < π
−π/2≤θy ≤ π/2 (3.11)
0≤θz < 2π
These ranges are most frequently used in flight dynamics, although occa-
sionaly one sees −π < θz ≤ π and 0 < θx ≤ 2π.