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Vector Spaces in Physics San Francisco State University Department of Physics and Astronomy August 6, 2015 Vector Spaces in Physics Notes for Ph 385: Introduction to Theoretical Physics I R. Bland TABLE OF CONTENTS Chapter I. Vectors A. The displacement vector. B. Vector addition. C. Vector products. 1. The scalar product. 2. The vector product. D. Vectors in terms of components. E. Algebraic properties of vectors. 1. Equality. 2. Vector Addition. 3. Multiplication of a vector by a scalar. 4. The zero vector. 5. The negative of a vector. 6. Subtraction of vectors. 7. Algebraic properties of vector addition. F. Properties of a vector space. G. Metric spaces and the scalar product. 1. The scalar product. 2. Definition of a metric space. H. The vector product. I. Dimensionality of a vector space and linear independence. J. Components in a rotated coordinate system. K. Other vector quantities. Chapter 2. The special symbols ij and ijk, the Einstein summation convention, and some group theory. A. The Kronecker delta symbol, ij B. The Einstein summation convention. C. The Levi-Civita totally antisymmetric tensor. Groups. The permutation group. The Levi-Civita symbol. D. The cross Product. E. The triple scalar product. F. The triple vector product. The epsilon killer. Chapter 3. Linear equations and matrices. A. Linear independence of vectors. B. Definition of a matrix. C. The transpose of a matrix. D. The trace of a matrix. E. Addition of matrices and multiplication of a matrix by a scalar. F. Matrix multiplication. G. Properties of matrix multiplication. H. The unit matrix I. Square matrices as members of a group. ii J. The determinant of a square matrix. K. The 3x3 determinant expressed as a triple scalar product. L. Other properties of determinants Product law Transpose law Interchanging columns or rows Equal rows or columns M. Cramer's rule for simultaneous linear equations. N. Condition for linear dependence. O. Eigenvectors and eigenvalues Chapter 4. Practical examples A. Simple harmonic motion - a review B. Coupled oscillations - masses and springs. A system of two masses. Three interconnected masses. Systems of many coupled masses. C. The triple pendulum Chapter 5. The inverse; numerical methods A. The inverse of a square matrix. Definition of the inverse. Use of the inverse to solve matrix equations. The inverse matrix by the method of cofactors. B. Time required for numerical calculations. C. The Gauss-Jordan method for solving simultaneous linear equations. D. The Gauss-Jordan method for inverting a matrix. Chapter 6. Rotations and tensors A. Rotation of axes. B. Some properties of rotation matrices. Orthogonality Determinant C. The rotation group. D. Tensors E. Coordinate transformation of an operator on a vector space. F. The conductivity tensor. G. The inertia tensor. Chapter 6a. Space-time four-vectors. A. The origins of special relativity. B. Four-vectors and invariant proper time. C. The Lorentz transformation. D. Space-time events. E. The time dilation. F. The Lorentz contraction. G. The Maxwell field tensor. iii Chapter 7. The Wave Equation A. Qualitative properties of waves on a string. B. The wave equation. Partial derivatives. Wave velocity. C. Sinusoidal solutions. D. General traveling-wave solutions. E. Energy carried by waves on a string. Kinetic energy. Potential energy. F. The superposition principle. G. Group and phase velocity. Chapter 8. Standing Waves on a String A. Boundary Conditions and Initial Conditions String fixed at a boundary. Boundary between two different strings. B. Standing waves on a string. Chapter 9. Fourier Series A. The Fourier sine series. The general solution. Initial conditions. Orthogonality. Completeness. B. The Fourier sine-cosine Series. Odd and even functions. Periodic functions in time. C. The exponential Fourier series. Chapter 10. Fourier Transforms and the Dirac Delta Function A. The Fourier transform. B. The Dirac delta function (x). The rectangular delta function. The Gaussian delta function. Properties of the delta function. C. Application of the Dirac delta function to Fourier transforms. Basis states. Functions of position x. D. Relation to quantum mechanics. Chapter 11. Maxwell's Equations in Special Relativity Appendix A. Useful mathematical facts and formulae. A. Complex numbers. B. Some integrals and identities C. The small-angle approximation. D. Mathematical logical symbols. Appendix B. Using the P&A Computer System 1. Logging on. 2. Running MatLab, Mathematica and IDL. 3. Mathematica. iv Appendix C. Mathematica 1. Calculation of the vector sum using Mathematica. 2. Matrix operations in Mathematica 3. Speed test for Mathematica. References v Vector Spaces in Physics 8/6/2015 Chapter 1. Vectors We are all familiar with the distinction between things which have a direction and those which don't. The velocity of the wind (see figure 1.1) is a classical example of a vector Figure 1-1. Where is the vector? quantity. There are many more of interest in physics, and in this and subsequent chapters we will try to exhibit the fundamental properties of vectors. Vectors are intimately related to the very nature of space. Euclidian geometry (plane and spherical geometry) was an early way of describing space. All the basic concepts of Euclidian geometry can be expressed in terms of angles and distances. A more recent development in describing space was the introduction by Descartes of coordinates along three orthogonal axes. The modern use of Cartesian vectors provides the mathematical basis for much of physics. A. The Displacement Vector The preceding discussion did not lead to a definition of a vector. But you can convince yourself that all of the things we think of as vectors can be related to a single fundamental quantity, the vector r representing the displacement from one point in space to another. Assuming we know how to measure distances and angles, we can define a displacement vector (in two dimensions) in terms of a distance (its magnitude), and an angle: displacement from r12 point 1 to point 2 (1-1) distance, angle measured counterclockwise from due East (See figure 1.2.) Note that to a given pair of points corresponds a unique displacement, but a given displacement can link many different pairs of points. Thus the fundamental definition of a displacement gives just its magnitude and angle. We will use the definition above to discuss certain properties of vectors from a strictly geometrical point of view. Later we will adopt the coordinate representation of vectors for a more general and somewhat more abstract discussion of vectors. 1 - 1 Vector Spaces in Physics 8/6/2015 distance point 2 r angle east point 1 Figure 1-2. A vector, specified by giving a distance and an angle. B. Vector Addition A quantity related to the displacement vector is the position vector for a point. Positions are not absolute – they must be measured relative to a reference point. If we call this point O (the "origin"), then the position vector for point P can be defined as follows: rP displacement from point O to point P (1-2) It seems reasonable that the displacement from point 1 to point 2 should be expressed in terms of the position vectors for 1 and 2. We are be tempted to write r12 r2 r1 (1-3) A "difference law" like this is certainly valid for temperatures, or even for distances along a road, if 1 and 2 are two points on the road. But what does subtraction mean for vectors? Do you subtract the lengths and angles, or what? When are two vectors equal? In order to answer these questions we need to systematically develop the algebraic properties of vectors. We will let A , B , C , etc. represent vectors. For the moment, the only vector quantities we have defined are displacements in space. Other vector quantities which we will define later will obey the same rules. Definition of Vector Addition. The sum of two vector displacements can be defined so as to agree with our intuitive notions of displacements in space. We will define the sum of two displacements as the single displacement which has the same effect as carrying out the two individual displacements, one after the other. To use this definition, we need to be able to calculate the magnitude and angle of the sum vector. This is straightforward using the laws of plane geometry. (The laws of geometry become more complicated in three dimensions, where the coordinate representation is more convenient.) Let and be two displacement vectors, each defined by giving its length and angle: A (A, ), A (1-4) B (B, B ). 1 - 2 Vector Spaces in Physics 8/6/2015 Here we follow the convention of using the quantity A (without an arrow over it) to represent the magnitude of ; and, as stated above, angles are measured counterclockwise from the easterly direction. Now imagine points 1, 2, and 3 such that represents the displacement from 1 to 2, and represents the displacement from 2 to 3. This is illustrated in figure 1-3. 3 B 2 B A A 1 Figure 1-3. Successive displacements and . Definition: The sum of two given displacements and is the third displacement which has the same effect as making displacements and in succession. It is clear that the sum exists, and we know how to find it. An example is shown in figure 1-4 with two given vectors and and their sum . It is fairly clear that the length and angle of can be determined (using trigonometry), since for the triangle 1-2- 3, two sides and the included angle are known. The example below illustrates this calculation. Example: Let and B be the two vectors shown in figure 1-4: =(10 m, 48), B =(14 m, 20). Determine the magnitude and angle from due east of their A B sum , whereC C A B .
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