HIGHER MATHEMATICS FOR ENGINEERS AND PHYSICISTS
CONTENTS VECTOR ALGEBRA I: SCALARS AND VECTORS SCALARS AND VECTORS ADDITION OF VECTORS SUM OF TWO VECTORS: GEOMETRICAL ADDITION SUBTRACTION OF VECTORS COMPONENTS AND PROJECTION OF A VECTOR COMPONENT REPRESENTATION IN COORDINATE SYSTEMS POSITION VECTOR UNIT VECTORS COMPONENT REPRESENTATION OF A VECTOR REPRESENTATION OF THE SUM OF TWO VECTORS IN TERMS OF THEIR COMPONENTS SUBTRACTION OF VECTORS IN TERMS OF THEIR COMPONENTS MULTIPLICATION OF A VECTOR BY A SCALAR MAGNITUDE OF A VECTOR EXERCISES VECTOR ALGEBRA II: SCALAR AND VECTOR PRODUCTS SCALAR PRODUCT APPLICATION: EQUATION OF A LINE AND A PLANE SPECIAL CASES COMMUTATIVE AND DISTRIBUTIVE LAWS SCALAR PRODUCT IN TERMS OF THE COMPONENTS OF THE VECTORS VECTOR PRODUCT TORQUE TORQUE AS A VECTOR DE?NITION OF THE VECTOR PRODUCT ANTI-COMMUTATIVE LAW FOR VECTOR PRODUCTS COMPONENTS OF THE VECTOR PRODUCT FUNCTIONS THE MATHEMATICAL CONCEPT OF FUNCTIONS AND ITS MEANING IN PHYSICS AND ENGINEERING INTRODUCTION THE CONCEPT OF A FUNCTION GRAPHICAL REPRESENTATION OF FUNCTIONS COORDINATE SYSTEM, POSITION VECTOR THE LINEAR FUNCTION: THE STRAIGHT LINE GRAPH PLOTTING QUADRATIC EQUATIONS PARAMETRIC CHANGES OF FUNCTIONS AND THEIR GRAPHS INVERSE FUNCTIONS TRIGONOMETRIC OR CIRCULAR FUNCTIONS UNIT CIRCLE SINE FUNCTION COSINE FUNCTION RELATIONSHIPS BETWEEN THE SINE AND COSINE FUNCTIONS TANGENT AND COTANGENT ADDITION FORMULAE INVERSE TRIGONOMETRIC FUNCTIONS FUNCTION OF A FUNCTION (COMPOSITION) APPENDIX: TRIGONOMETRIC FUNCTIONS EXPONENTIAL, LOGARITHMIC AND HYPERBOLIC FUNCTIONS POWERS, EXPONENTIAL FUNCTION POWERS LAWS OF INDICES OR EXPONENTS BINOMIAL THEOREM EXPONENTIAL FUNCTION LOGARITHM, LOGARITHMIC FUNCTION LOGARITHM OPERATIONS WITH LOGARITHMS LOGARITHMIC FUNCTIONS HYPERBOLIC FUNCTIONS AND INVERSE HYPERBOLIC FUNCTIONS HYPERBOLIC FUNCTIONS INVERSE HYPERBOLIC FUNCTIONS CONTENTS EXERCISES DIFFERENTIAL CALCULUS SEQUENCES AND LIMITS THE CONCEPT OF SEQUENCE LIMIT OF A SEQUENCE LIMIT OF A FUNCTION EXAMPLES FOR THE PRACTICAL DETERMINATION OF LIMITS CONTINUITY SERIES GEOMETRIC SERIES DIFFERENTIATION OF A FUNCTION GRADIENT OR SLOPE OF A LINE GRADIENT OF AN ARBITRARY CURVE DERIVATIVE OF A FUNCTION PHYSICAL APPLICATION: VELOCITY THE DIFFERENTIAL CALCULATING DIFFERENTIAL COEF?CIENTS DERIVATIVES OF POWER FUNCTIONS; CONSTANT FACTORS RULES FOR DIFFERENTIATION DIFFERENTIATION OF FUNDAMENTAL FUNCTIONS HIGHER DERIVATIVES EXTREME VALUES AND POINTS OF IN?EXION; CURVE SKETCHING MAXIMUM AND MINIMUM VALUES OF A FUNCTION FURTHER REMARKS ON POINTS OF IN?EXION (CONTRA?EXURE) CURVE SKETCHING APPLICATIONS OF DIFFERENTIAL CALCULUS EXTREME VALUES INCREMENTS CURVATURE DETERMINATION OF LIMITS BY DIFFERENTIATION: L’HÔPITAL’S RULE FURTHER METHODS FOR CALCULATING DIFFERENTIAL COEF?CIENTS IMPLICIT FUNCTIONS AND THEIR DERIVATIVES LOGARITHMIC DIFFERENTIATION PARAMETRIC FUNCTIONS AND THEIR DERIVATIVES PARAMETRIC FORM OF AN EQUATION DERIVATIVES OF PARAMETRIC FUNCTIONS APPENDIX: DIFFERENTIATION RULES INTEGRAL CALCULUS THE PRIMITIVE FUNCTION FUNDAMENTAL PROBLEM OF INTEGRAL CALCULUS THE AREA PROBLEM: THE DE?NITE INTEGRAL FUNDAMENTAL THEOREM OF THE DIFFERENTIAL AND INTEGRAL CALCULUS THE DE?NITE INTEGRAL CALCULATION OF DE?NITE INTEGRALS FROM INDE?NITE INTEGRALS EXAMPLES OF DE?NITE INTEGRALS METHODS OF INTEGRATION PRINCIPLE OF VERI?CATION STANDARD INTEGRALS CONSTANT FACTOR AND THE SUM OF FUNCTIONS INTEGRATION BY PARTS: PRODUCT OF TWO FUNCTIONS INTEGRATION BY SUBSTITUTION SUBSTITUTION IN PARTICULAR CASES INTEGRATION BY PARTIAL FRACTIONS RULES FOR SOLVING DE?NITE INTEGRALS MEAN VALUE THEOREM IMPROPER INTEGRALS LINE INTEGRALS APPENDIX APPLICATIONS OF INTEGRATION AREAS AREAS FOR PARAMETRIC FUNCTIONS AREAS IN POLAR COORDINATES AREAS OF CLOSED CURVES LENGTHS OF CURVES LENGTHS OF CURVES IN POLAR COORDINATES SURFACE AREA AND VOLUME OF A SOLID OF REVOLUTION APPLICATIONS TO MECHANICS BASIC CONCEPTS OF MECHANICS CENTER OF MASS AND CENTROID THE THEOREMS OF PAPPUS MOMENTS OF INERTIA; SECOND MOMENT OF AREA TAYLOR SERIES AND POWER SERIES EXPANSION OF A FUNCTION IN A POWER SERIES INTERVAL OF CONVERGENCE OF POWER SERIES APPROXIMATE VALUES OF FUNCTIONS EXPANSION OF A FUNCTION F (X) AT AN ARBITRARY POSITION APPLICATIONS OF SERIES POLYNOMIALS AS APPROXIMATIONS INTEGRATION OF FUNCTIONS WHEN EXPRESSED AS POWER SERIES EXPANSION IN A SERIES BY INTEGRATING APPENDIX: COMMONLY USED APPROXIMATE POLYNOMIALS COMPLEX NUMBERS DE?NITION AND PROPERTIES OF COMPLEX NUMBERS IMAGINARY NUMBERS FIELDS OF APPLICATION OPERATIONS WITH COMPLEX NUMBERS GRAPHICAL REPRESENTATION OF COMPLEX NUMBERS GAUSS COMPLEX NUMBER PLANE: ARGAND DIAGRAM POLAR FORM OF A COMPLEX NUMBER EXPONENTIAL FORM OF COMPLEX NUMBERS EULER’S FORMULA EXPONENTIAL FORM OF THE SINE AND COSINE FUNCTIONS COMPLEX NUMBERS AS POWERS MULTIPLICATION AND DIVISION IN EXPONENTIAL FORM RAISING TO A POWER, EXPONENTIAL FORM PERIODICITY OF REJ? TRANSFORMATION OF A COMPLEX NUMBER FROM ONE FORM INTO ANOTHER OPERATIONS WITH COMPLEX NUMBERS EXPRESSED IN POLAR FORM MULTIPLICATION AND DIVISION RAISING TO A POWER ROOTS OF A COMPLEX NUMBER DIFFERENTIAL EQUATIONS CONCEPT AND CLASSI?CATION OF DIFFERENTIAL EQUATIONS PRELIMINARY REMARKS GENERAL SOLUTION OF FIRST- AND SECOND-ORDER DES WITH CONSTANT COEF?CIENTS HOMOGENEOUS LINEAR DE NON-HOMOGENEOUS LINEAR DE BOUNDARY VALUE PROBLEMS FIRST-ORDER DES SECOND-ORDER DES SOME APPLICATIONS OF DES RADIOACTIVE DECAY THE HARMONIC OSCILLATOR GENERAL LINEAR FIRST-ORDER DES SOLUTION BY VARIATION OF THE CONSTANT A STRAIGHTFORWARD METHOD INVOLVING THE INTEGRATING FACTOR SOME REMARKS ON GENERAL FIRST-ORDER DES BERNOULLI’S EQUATIONS SEPARATION OF VARIABLES EXACT EQUATIONS THE INTEGRATING FACTOR—GENERAL CASE SIMULTANEOUS DES HIGHER-ORDER DES INTERPRETED AS SYSTEMS OF FIRST-ORDER SIMULTANEOUS DES SOME ADVICE ON INTRACTABLE DES LAPLACE TRANSFORMS THE LAPLACE TRANSFORM DE?NITION LAPLACE TRANSFORM OF STANDARD FUNCTIONS SOLUTION OF LINEAR DES WITH CONSTANT COEF?CIENTS SOLUTION OF SIMULTANEOUS DES WITH CONSTANT COEF?CIENTS FUNCTIONS OF SEVERAL VARIABLES; PARTIAL DIFFERENTIATION; AND TOTAL DIFFERENTIATION FUNCTIONS OF SEVERAL VARIABLES REPRESENTING THE SURFACE BY ESTABLISHING A TABLE OF Z-VALUES REPRESENTING THE SURFACE BY ESTABLISHING INTERSECTING CURVES OBTAINING A FUNCTIONAL EXPRESSION FOR A GIVEN SURFACE PARTIAL DIFFERENTIATION HIGHER PARTIAL DERIVATIVES TOTAL DIFFERENTIAL TOTAL DIFFERENTIAL OF FUNCTIONS APPLICATION: SMALL TOLERANCES GRADIENT TOTAL DERIVATIVE EXPLICIT FUNCTIONS IMPLICIT FUNCTIONS MAXIMA AND MINIMA OF FUNCTIONS OF TWO OR MORE VARIABLES APPLICATIONS: WAVE FUNCTION AND WAVE EQUATION WAVE FUNCTION WAVE EQUATION MULTIPLE INTEGRALS; COORDINATE SYSTEMS MULTIPLE INTEGRALS MULTIPLE INTEGRALS WITH CONSTANT LIMITS DECOMPOSITION OF A MULTIPLE INTEGRAL INTO A PRODUCT OF INTEGRALS MULTIPLE INTEGRALS WITH VARIABLE LIMITS COORDINATE SYSTEMS POLAR COORDINATES CYLINDRICAL COORDINATES SPHERICAL COORDINATES APPLICATION: MOMENTS OF INERTIA OF A SOLID APPENDIX: APPLICATIONS OF DOUBLE INTEGRALS TRANSFORMATION OF COORDINATES; MATRICES PARALLEL SHIFT OF COORDINATES: TRANSLATION ROTATION ROTATION IN A PLANE SUCCESSIVE ROTATIONS ROTATIONS IN THREE-DIMENSIONAL SPACE MATRIX ALGEBRA ADDITION AND SUBTRACTION OF MATRICES MULTIPLICATION OF A MATRIX BY A SCALAR PRODUCT OF A MATRIX AND A VECTOR MULTIPLICATION OF TWO MATRICES ROTATIONS EXPRESSED IN MATRIX FORM ROTATION IN TWO-DIMENSIONAL SPACE SPECIAL ROTATION IN THREE-DIMENSIONAL SPACE SPECIAL MATRICES INVERSE MATRIX SETS OF LINEAR EQUATIONS; DETERMINANTS SETS OF LINEAR EQUATIONS GAUSSIAN ELIMINATION: SUCCESSIVE ELIMINATION OF VARIABLES GAUSS–JORDAN ELIMINATION MATRIX NOTATION OF SETS OF EQUATIONS AND DETERMINATION OF THE INVERSE MATRIX EXISTENCE OF SOLUTIONS DETERMINANTS PRELIMINARY REMARKS ON DETERMINANTS DE?NITION AND PROPERTIES OF AN N-ROW DETERMINANT RANK OF A DETERMINANT AND RANK OF A MATRIX APPLICATIONS OF DETERMINANTS EIGENVALUES AND EIGENVECTORS OF REAL MATRICES TWO CASE STUDIES: EIGENVALUES OF 2 × 2 MATRICES GENERAL METHOD FOR FINDING EIGENVALUES WORKED EXAMPLE: EIGENVALUES OF A 3 × 3 MATRIX IMPORTANT FACTS ON EIGENVALUES AND EIGENVECTORS VECTOR ANALYSIS: SURFACE INTEGRALS, DIVERGENCE, CURL AND POTENTIAL FLOW OF A VECTOR FIELD THROUGH A SURFACE ELEMENT SURFACE INTEGRAL SPECIAL CASES OF SURFACE INTEGRALS FLOW OF A HOMOGENEOUS VECTOR FIELD THROUGH A CUBOID FLOW OF A SPHERICALLY SYMMETRICAL FIELD THROUGH A SPHERE APPLICATION: THE ELECTRICAL FIELD OF A POINT CHARGE GENERAL CASE OF COMPUTING SURFACE INTEGRALS DIVERGENCE OF A VECTOR FIELD GAUSS’S THEOREM CURL OF A VECTOR FIELD STOKES’ THEOREM POTENTIAL OF A VECTOR FIELD SHORT REFERENCE ON VECTOR DERIVATIVES FOURIER SERIES; HARMONIC ANALYSIS EXPANSION OF A PERIODIC FUNCTION INTO A FOURIER SERIES EVALUATION OF THE COEF?CIENTS ODD AND EVEN FUNCTIONS EXAMPLES OF FOURIER SERIES EXPANSION OF FUNCTIONS OF PERIOD 2L FOURIER SPECTRUM FOURIER INTEGRALS AND FOURIER TRANSFORMS TRANSITION FROM FOURIER SERIES TO FOURIER INTEGRAL FOURIER TRANSFORMS FOURIER COSINE TRANSFORM FOURIER SINE TRANSFORM, GENERAL FOURIER TRANSFORM COMPLEX REPRESENTATION OF THE FOURIER TRANSFORM SHIFT THEOREM DISCRETE FOURIER TRANSFORM, SAMPLING THEOREMS FOURIER TRANSFORM OF THE GAUSSIAN FUNCTION PROBABILITY CALCULUS CONCEPT OF PROBABILITY RANDOM EXPERIMENT, OUTCOME SPACE AND EVENTS THE CLASSICAL DE?NITION OF PROBABILITY THE STATISTICAL DE?NITION OF PROBABILITY GENERAL PROPERTIES OF PROBABILITIES PROBABILITY OF STATISTICALLY INDEPENDENT EVENTS. COMPOUND PROBABILITY PERMUTATIONS AND COMBINATIONS PERMUTATIONS COMBINATIONS PROBABILITY DISTRIBUTIONS DISCRETE AND CONTINUOUS PROBABILITY DISTRIBUTIONS DISCRETE PROBABILITY DISTRIBUTIONS CONTINUOUS PROBABILITY DISTRIBUTIONS MEAN VALUES OF DISCRETE AND CONTINUOUS VARIABLES THE NORMAL DISTRIBUTION AS THE LIMITING VALUE OF THE BINOMIAL DISTRIBUTION PROPERTIES OF THE NORMAL DISTRIBUTION DERIVATION OF THE BINOMIAL DISTRIBUTION THEORY OF ERRORS PURPOSE OF THE THEORY OF ERRORS MEAN VALUE AND VARIANCE MEAN VALUE VARIANCE AND STANDARD DEVIATION MEAN VALUE AND VARIANCE IN A RANDOM SAMPLE AND PARENT POPULATION MEAN VALUE AND VARIANCE OF CONTINUOUS DISTRIBUTIONS ERROR IN MEAN VALUE NORMAL DISTRIBUTION: DISTRIBUTION OF RANDOM ERRORS LAW OF ERROR PROPAGATION WEIGHTED AVERAGE CURVE FITTING: METHOD OF LEAST SQUARES, REGRESSION LINE CORRELATION AND CORRELATION COEF?CIENT ANSWERS