Mathematical Physics II By Dr. Issakha YOUM African Virtual university Université Virtuelle Africaine Universidade Virtual Africana African Virtual University NOTICE This document is published under the conditions of the Creative Commons http://en.wikipedia.org/wiki/Creative_Commons Attribution http://creativecommons.org/licenses/by/2.5/ License (abbreviated “cc-by”), Version 2.5. African Virtual University Table OF CONTeNTS I. Mathematical Physics II _____________________________________ 3 II. Prerequisite _______________________________________________ 3 III. Time ____________________________________________________ 3 IV. Materials _________________________________________________ 3 V. Module Rationale __________________________________________ 3 VI. Content __________________________________________________ 4 6.1 Overview ___________________________________________ 4 6.2 Outline _____________________________________________ 4 6.3 Graphic Organizer _____________________________________ 5 VII. General Objectives _________________________________________ 6 VIII. Learning Objective(s) _______________________________________ 6 IX. Pre-assessment ___________________________________________ 8 X. Learning Activities _________________________________________ 20 XI. Glossary of Key Concepts ___________________________________ 69 XII. List of Useful links ________________________________________ 73 XIII. Synthesis of the Module ____________________________________ 87 XIV. Summative Evaluation ______________________________________ 88 XV. References _____________________________________________ 117 XVI. Main Author of the Module ________________________________ 119 African Virtual University I. Mathematical Physics II By Dr. Issakha Youm, Full Professor II. Prerequisites Before taking this module, the learner should have mastered the following notions: elements of vector calculus (vector sum, scalar multiplication of vectors, scalar product of two vectors), elements of analysis and series, partial derivatives, deri- vative of an implicit function, and total and exact differential equations. III. Time The duration of this module is 120 hours, allocated as follows: • Activity 1 (20 hours): Vector algebra in R3 • Activity 2 (30hours): Vector functions of one and two variables • Activity 3 (30 hours): Physical fields • Activity 4 (40 hours): Spatial integrals – applications in physics. IV. Materials In addition to the usual materials required for taking notes and making geometric drawings (e.g., notebook, pens, pencils, compass, square, protractor, ruler), the learner will need to have access to a computer with an Internet connection and a graphing calculator. V. Module rationale In physics, mathematical tools are used to formulate the laws of nature. Neverthe- less, the fact is that teachers frequently lack the mathematical knowledge required to properly apply these tools in physics. In order to teach physics as a subject, it is important to take this Mathematical Physics course. It will provide you with the calculus rules that you need without having to take additional mathematics courses. African Virtual University VI. Content 6.1 Overview This Mathematical Physics II module builds on the Mathematical Physics I module. It addresses differential and integral calculus tools for functions (scalar and vector) of multiple variables. It reviews the areas of vectors, spatial geo- metry, vector functions, curves, surfaces, partial derivatives, multiple integrals and diverse applications such as surface and volume calculations. It also covers the notions of curvilinear integrals and surface integrals as well as the theorems of Gauss, Green and Stokes. It concludes with applications in wave theory and magneto-electric wave propagation. This last section, which explains some ap- plications in the field of physics, gives the learner an idea of how mathematics is applied in practice. 6.2 Outline The module is organized as follows: • Geometric vectors and vectors in R3: Cartesian coordinate system for a three-dimensional space, vector position of a point; scalar, vector and triple (or mixed) products: properties and geometric and physical interpre- tations. • Vector functions of one and two variables: derivatives and derivation rules, George Stokes defined integral and vector integral; parametric curves, tangent vector, arc (1819–1903) length, curvature, torsion and the Serret-Frenet system; parametric surfaces, tangent plane to a surface; cinematic application: velocity and acceleration vectors. • Physical fields: scalar and vector fields, level curve and level surface, vector field, gradient field, and differential operators: divergence, rotational and Laplacian. • Spatial integrals: curvilinear integrals, surface and volume integrals; theorems of Gauss, Green and Stokes. • Applications in physics: wave equations, electromagnetic wave propaga- tion in a vacuum and material media. African Virtual University Johann Peter Gustav Lejeune Dirichlet (1805–1859) 6.3 Graphic Organizer AlgèbreAlgebra ofdes vectors vecteurs in R3 dans R3 Les Fonctions VectorVectorielles functions Les champsPhysical fields physiques Les intégrales spatiales Spatial integrals Application en Applications in physics Physique African Virtual University VII. General Objectives • Understand basic notions of differential calculus and vector calculus. • Understand tools of differential and integral calculation applied to functions (scalar and vector) of multiple variables • Use vector derivatives • Use the gradient and other vector analysis operators, with a focus on geo- metric and physical interpretations. VIII. Specific learning Objectives Unit Learning objectives 1. Vector algebra in R3 Learners should be able to: Define the modulus, support and standard of a vector. Calculate the sum of two vectors. Understand the relationships between a vector space and a physical space and its vectors. Define a vector position and a radius vector. Determine the components of a vector and the coordinates of a point. Determine the unit vector in a given direction. Calculate a scalar product, a vector product and a mixed product. Provide the formula for a double vector product. Geometrically interpret a scalar product, a vector product and a triple vector (mixed) product. African Virtual University 2. Vector functions • Define vector value functions (of one or two variables) and under- stand the notions of vector limits and vector continuity. • Calculate the derivatives and integrals of vector functions. • Apply the derivation rules for vector functions, particularly the derivation of single vector products. • Geometrically interpret the notions of the derivation and integra- tion of vector functions. • Geometrically interpret vector functions for parametric curves and surfaces. • Define a normal vector to a surface. • Define curvature and torsion of a spatial curve. • Calculate the velocity and acceleration vectors for a trajectory in a Frenet-frame (moving frame). 3. Physical fields and • Define a scalar field and a vector field. their derivatives • Define a level curve, a level surface and field lines. • Determine the equations that define these notions and solve simple cases. • Define the gradient of a scalar field. • Calculate the gradient of a scalar function. • Define the divergence and rotational of a vector field. • Calculate the divergence and rotational of a vector field. • Define the nabla operator (or del). • Determine the conditions for a vector field to be a gradient field. • Calculate a curvilinear integral by curve parameterization. • Calculate the circulation of a vector field. • Define the integral condition for a vector field to be a gradient 4. Spatial integration field and its operation. • Determine the derivation function for a gradient field using the circulation method. • Calculate a double integral for any parameterized surface. • Calculate a vector field flux • Calculate volume integrals. • Geometrically interpret the nabla operator. • Explain the notion of an electromagnetic field. 5. Applications in • Apply mathematical theories of integrals. physics • Apply the local laws of electromagnetism. • Describe wave propagation in media. • Describe the phenomena of reflection, refraction, dispersion and absorption. African Virtual University IX. Pre assessment 9.1 Predictive assessment Title of the predictive assessment: Test on the prerequisites for the Mathe- matical Physics II module Purpose: This test assesses whether the learner has the required knowledge to understand the Mathematical Physics II module. QUESTIONS 1. In a two-dimensional space, a straight line is represented by the equation . A direction vector for this straight line is the vector with 3x − 2y + 6 = 0 the components: A - (3,2) B - (2,3) C - (3,-2) D - (-2,-3) 2. In a three-dimensional space, we have the Cartesian base vectors a (3,−1,2) ; ( 1,3,3) ; (5,4, 1) . The coordinates of 3 2 are: b − c − A = a − b− c A - (3,-2-1) B - (3,-1,2) C - (6,-13,1) D - (5,3,-2) 3. The vector sum of BC − 2BA + CD − AD is equal to: A - AB B - BD C - 0 D - CD 4. In a Cartesian system, the vectors ( 1, 2,1) and (1,2,5) are: a − − b A - collinear B - opposed C - orthogonal D - none of these 1 1 2 5. In a Cartesian system, the vectors a (−3,−1,4) and b( , ,− ) are: 2 6 3 A - collinear B - opposed C - orthogonal D - none of these African Virtual University 6. Consider a parallelogram ABCD. Which of the following obtains exact equality? A - DA+ AB = DB B - DA+ DC = DB C - AD+