DIFFERENTIAL CALCULUS

By F. MAERO

CONTENTS LANGUAGE SETS LANGUAGE SETS REFERENCES NUMBERS AND PROPOSITIONS THE NATURAL NUMBERS COUNTING PROBLEMS PRIME NUMBERS CODES AND DECODING INTEGER NUMBERS RATIONAL NUMBERS REPRESENTATIONS OF RATIONAL NUMBERS THE NUMERATION THE REAL NUMBERS DENSITY CLOSURE OF A SET WITH RESPECT TO AN OPERATIONS ABBREVIATED NOTATIONS THERE IS AT LEAST ONE THE IMPLICATION IMPLICATION AND LOGICAL EQUIVALENCE THE THEOREM TERTIUM NON DATUR PROOFS IN SCIENCE VISUAL PROOFS THE INVERSE THEOREM IRRATIONALITY OF ? 2 THE PYTHAGOREAN SCHOOL SOCRATES AND THE DIAGONAL OF THE SQUARE THE INDUCTIVE METHOD AND THE INDUCTION PRINCIPLE NECESSARY CONDITION SUF?CIENT CONDITION INTUITION MATHEMATICS AND CULTURE ON EDUCATION INDIVIDUAL STUDY AND WORK RELATIONS INTRODUCTION CARTESIAN PRODUCT OF SETS RELATIONS BINARY RELATIONS ORDERINGS THE POWER SET TOTAL ORDER PREFERENCES INDIFFERENCE EQUIVALENCE RELATIONS PARTITIONS OF A SET REMAINDER CLASSES EUCLIDEAN GEOMETRY FIRST AXIOMS THE AXIOMATIC METHOD FURTHER AXIOMS OF EUCLIDEAN GEOMETRY THE REFOUNDATION OF GEOMETRY GEOMETRIC FIGURES CONVEX AND CONCAVE FIGURES ANGLES RELATIONS BETWEEN LINES AND PLANES RELATIONS BETWEEN PLANES PROJECTIONS THE ANGLE OF A LINE AND A PLANE DIHEDRALS PERPENDICULAR PLANES SYMMETRIES SIMILAR POLYGONS THALES’ THEOREM FUNCTIONS EQUIPOTENT SETS IN?NITE SETS, FINITE SETS HOTEL HILBERT COMPOSITE FUNCTIONS RESTRICTION AND EXTENSION OF A FUNCTION BIBLIOGRAPHY THE REAL LINE THE COORDINATE SYSTEM OF THE AXIS THE MEASURE OF A SEGMENT THE COORDINATE SYSTEM OF AN AXIS EQUALITIES AND IDENTITIES EQUIVALENT EQUATIONS EXAMPLES FORMING AN EQUATION FROM GIVEN INFORMATION ORDER IN R EVALUATING AN INEQUALITY TO MAKING A DECISION INTERVALS, NEIGHBORHOODS, ABSOLUTE VALUE EXERCISES THE EXTENDED SET OF REAL NUMBERS UPPER BOUNDS AND LOWER BOUNDS COMMENSURABILITY AND REAL NUMBERS SEPARATE SETS AND CONTIGUOUS SETS REAL-VALUED FUNCTIONS OF A REAL VARIABLE THE LINE THE CARTESIAN PLANE QUADRANTS DISTANCE REAL-VALUED FUNCTIONS OF A REAL VARIABLE EXTREMA OF A REAL-VALUED FUNCTION THE GRAPH OF A REAL-VALUED FUNCTION GRAPH AND CURVE LINES IN THE CARTESIAN PLANE THE CONSTANT FUNCTION THE IDENTICAL FUNCTION THE FUNCTION F : X? KX THE FUNCTION F : X ? KX + N THE LINEAR EQUATION THE PARAMETRIC EQUATIONS OF THE LINE PARALLEL LINES PARALLEL LINES REPRESENTED BY PARAMETRIC EQUATIONS PARALLEL LINES REPRESENTED BY ORDINARY EQUATIONS PARALLEL LINES EXERCISES THE ABSOLUTE VALUE FUNCTION A LINEAR MODEL INVERTIBLE FUNCTIONS AND INVERSE FUNCTIONS CIRCULAR FUNCTIONS THE EQUATION OF THE CIRCUMFERENCE THE GONIOMETRIC CIRCUMFERENCE SINE, COSINE AND TANGENT FURTHER GONIOMETRIC IDENTITIES THE GRAPHS OF SINX, COSX AND TANX GEOMETRIC AND NUMERIC VECTORS N-TUPLES OF REAL NUMBERS LINEAR COMBINATIONS OF N-TUPLES SCALARS AND VECTORS APPLIED VECTORS AND FREE VECTORS ADDITION OF FREE VECTORS MULTIPLICATION OF A SCALAR BY A FREE VECTOR PROPERTIES OF OPERATIONS WITH FREE VECTORS COMPONENT VECTORS OF A PLANE VECTOR SPACE COORDINATE SYSTEM AND VECTORS UNIT VECTORS THE SPHERE SCALAR PRODUCT LINES AND PLANES SCALAR PRODUCT ORTHOGONAL PROJECTIONS OF A VECTOR SCALAR PRODUCT IN TERMS OF THE COMPONENTS SCALAR PRODUCT AND ORTHOGONALITY ANGLES OF LINES AND VECTORS ORTHOGONAL LINES IN THE PLANE THE EQUATION OF THE PLANE PERPENDICULAR LINES AND PLANES SYSTEMS OF LINEAR EQUATIONS REDUCTION LINEAR EQUATIONS SYSTEMS OF LINEAR EQUATIONS EQUIVALENT SYSTEMS ELEMENTARY OPERATIONS REDUCED SYSTEMS VECTOR SPACES COMPLEX NUMBERS OPERATIONS FIELDS LINEAR DEPENDENCE AND LINEAR INDEPENDENCE FINITELY GENERATED VECTOR SPACES BASES VECTOR SUBSPACES SPANNED SUBSPACES DIMENSION ISOMORPHISM IDENTI?CATION OF GEOMETRIC AND NUMERICAL VECTOR SPACES SCALAR PRODUCT IN RN MATRICES FIRST CONCEPTS REDUCED MATRICES RANK MATRIX REDUCTION METHOD AND RANK ROUCHÉ-CAPELLI’S THEOREM COMPATIBILITY OF A REDUCED SYSTEM SQUARE MATRICES DETERMINANTS AND SYSTEMS OF LINEAR EQUATIONS DETERMINANTS PROPERTIES OF THE DETERMINANTS SUBMATRICES AND MINORS COFACTORS MATRIX MULTIPLICATION INVERSE AND TRANSPOSE MATRICES SYSTEMS OF LINEAR EQUATIONS AND MATRICES RANK OF A MATRIX AND MINORS MATRIX A HAS A NON-ZERO MINOR OF MAXIMAL ORDER CALCULATING THE RANK OF A MATRIX VIA KRONECKER’S THEOREM CRAMER’S RULE HOMOGENEOUS LINEAR SYSTEMS ASSOCIATED HOMOGENEOUS LINEAR SYSTEM LINES AND PLANES COPLANAR LINES AND SKEW LINES LINE PARALLEL TO PLANE PERPENDICULAR LINES PERPENDICULAR PLANES INTERSECTION OF PLANES BUNDLE OF PLANES ALGORITHMS GREATEST COMMON DIVISOR: THE EUCLIDEAN ALGORITHM REGULAR SUBDIVISION OF A SEGMENT GAUSS ELIMINATION CONCLUSION