Cauchy's Calcul Infinitésimal

Chapter 1 OF VARIABLES, THEIR LIMITS, AND INFINITELY SMALL QUANTITIES.

Chapter 2 OF CONTINUOUS AND DISCONTINUOUS FUNCTIONS. GEOMETRIC REPRESENTATION OF CONTINUOUS FUNCTIONS.

Chapter 3 DERIVATIVES OF FUNCTIONS OF A SINGLE VARIABLE.

Chapter 4 DIFFERENTIALS OF FUNCTIONS OF A SINGLE VARIABLE.

Chapter 5 THE DIFFERENTIAL OF THE SUM OF SEVERAL FUNCTIONS IS THE SUM OF THEIR DIFFERENTIALS. CONSEQUENCES OF THIS PRINCIPLE. DIFFERENTIALS OF IMAGINARY FUNCTIONS.

Chapter 6 USE OF DIFFERENTIALS AND DERIVED FUNCTIONS IN THE SOLUTION OF SEVERAL PROBLEMS. MAXIMA AND MINIMA OF FUNCTIONS OF A SINGLE VARIABLE. VALUES OF FRACTIONS WHICH ARE PRESENTED UNDER THE FORM $$\frac{0}{0}$$ .

Chapter 7 VALUES OF SOME EXPRESSIONS WHICH ARE PRESENTED UNDER THE INDETERMINATE FORM $$\frac{\infty }{\infty }, \infty ^{0}, \, \dots .$$   A RELATIONSHIP WHICH EXISTS BETWEEN THE RATIO OF FINITE DIFFERENCES AND THE DERIVED FUNCTION.

Chapter 8 DIFFERENTIALS OF FUNCTIONS OF SEVERAL VARIABLES. PARTIAL DERIVATIVES AND PARTIAL DIFFERENTIALS.

Chapter 9 USE OF PARTIAL DERIVATIVES IN THE DIFFERENTIATION OF COMPOSED FUNCTIONS. DIFFERENTIALS OF IMPLICIT FUNCTIONS.

Chapter 10 THEOREM OF HOMOGENEOUS FUNCTIONS. MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES.

Chapter 11 USE OF INDETERMINATE FACTORS IN THE STUDY OF MAXIMA AND MINIMA.

Chapter 12 DIFFERENTIALS AND DERIVATIVES OF VARIOUS ORDERS FOR FUNCTIONS OF A SINGLE VARIABLE. CHANGE OF THE INDEPENDENT VARIABLE.

Chapter 13 DIFFERENTIALS OF VARIOUS ORDERS FOR FUNCTIONS OF SEVERAL VARIABLES.

Chapter 14 METHODS THAT WORK TO SIMPLIFY THE STUDY OF TOTAL DIFFERENTIALS FOR FUNCTIONS OF SEVERAL INDEPENDENT VARIABLES. SYMBOLIC VALUES OF THESE DIFFERENTIALS.

Chapter 15 RELATIONSHIPS WHICH EXIST BETWEEN FUNCTIONS OF A SINGLE VARIABLE AND THEIR DERIVATIVES OR DIFFERENTIALS OF VARIOUS ORDERS. USE OF THESE DIFFERENTIALS IN THE STUDY OF MAXIMA AND MINIMA.

Chapter 16 USE OF DIFFERENTIALS OF VARIOUS ORDERS IN THE STUDY OF MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES.

Chapter 17 CONDITIONS WHICH MUST BE FULFILLED FOR A TOTAL DIFFERENTIAL TO NOT CHANGE SIGN WHILE WE CHANGE THE VALUES ATTRIBUTED TO THE DIFFERENTIALS OF THE INDEPENDENT VARIABLES.

Chapter 18 DIFFERENTIALS OF ANY FUNCTION OF SEVERAL VARIABLES EACH OF WHICH IS IN ITS TURN A LINEAR FUNCTION OF OTHER SUPPOSED INDEPENDENT VARIABLES. DECOMPOSITION OF ENTIRE FUNCTIONS INTO REAL FACTORS OF FIRST OR OF SECOND DEGREE.

Chapter 19 USE OF DERIVATIVES AND DIFFERENTIALS OF VARIOUS ORDERS IN THE EXPANSION OF ENTIRE FUNCTIONS.

Chapter 20 DECOMPOSITION OF RATIONAL FRACTIONS.

Chapter 21 DEFINITE INTEGRALS.

Chapter 22 FORMULAS FOR THE DETERMINATION OF EXACT OR APPROXIMATE VALUES OF DEFINITE INTEGRALS.

Chapter 23 DECOMPOSITION OF A DEFINITE INTEGRAL INTO SEVERAL OTHERS. IMAGINARY DEFINITE INTEGRALS. GEOMETRIC REPRESENTATION OF REAL DEFINITE INTEGRALS. DECOMPOSITION OF THE FUNCTION UNDER THE $$\int $$ SIGN INTO TWO FACTORS IN WHICH ONE ALWAYS MAINTAINS THE SAME SIGN.

Chapter 24 OF DEFINITE INTEGRALS WHOSE VALUES ARE INFINITE OR INDETERMINATE. PRINCIPAL VALUES OF INDETERMINATE INTEGRALS.

Chapter 25 SINGULAR DEFINITE INTEGRALS.

Chapter 26 INDEFINITE INTEGRALS.

Chapter 27 VARIOUS PROPERTIES OF INDEFINITE INTEGRALS. METHODS TO DETERMINE THE VALUES OF THESE SAME INTEGRALS.

Chapter 28 ON INDEFINITE INTEGRALS WHICH CONTAIN ALGEBRAIC FUNCTIONS.

Chapter 29 ON THE INTEGRATION AND THE REDUCTION OF BINOMIAL DIFFERENTIALS AND OF ANY OTHER DIFFERENTIAL FORMULAS OF THE SAME TYPE.

Chapter 30 ON INDEFINITE INTEGRALS WHICH CONTAIN EXPONENTIAL, LOGARITHMIC, OR CIRCULAR FUNCTIONS.

Chapter 31 ON THE DETERMINATION AND THE REDUCTION OF INDEFINITE INTEGRALS IN WHICH THE FUNCTION UNDER THE $$\int $$ SIGN IS THE PRODUCT OF TWO FACTORS EQUAL TO CERTAIN POWERS OF SINES AND OF COSINES OF THE VARIABLE.

Chapter 32 ON THE TRANSITION OF INDEFINITE INTEGRALS TO DEFINITE INTEGRALS.

Chapter 33 DIFFERENTIATION AND INTEGRATION UNDER THE $$\int $$ SIGN. INTEGRATION OF DIFFERENTIAL FORMULAS WHICH CONTAIN SEVERAL INDEPENDENT VARIABLES.

Chapter 34 COMPARISON OF TWO TYPES OF SIMPLE INTEGRALS WHICH RESULT IN CERTAIN CASES FROM A DOUBLE INTEGRATION.

Chapter 35 DIFFERENTIAL OF A DEFINITE INTEGRAL WITH RESPECT TO A VARIABLE INCLUDED IN THE FUNCTION UNDER THE $$\int $$ SIGN AND IN THE LIMITS OF INTEGRATION. INTEGRALS OF VARIOUS ORDERS FOR FUNCTIONS OF A SINGLE VARIABLE.

Chapter 36 TRANSFORMATION OF ANY FUNCTIONS OF x OR OF $$x+h$$ INTO ENTIRE FUNCTIONS OF x OR OF h TO WHICH WE APPEND DEFINITE INTEGRALS. EQUIVALENT EXPRESSIONS TO THESE SAME INTEGRALS.

Chapter 37 THEOREMS OF TAYLOR AND OF MACLAURIN. EXTENSION OF THESE THEOREMS TO FUNCTIONS OF SEVERAL VARIABLES.

Chapter 38 RULES ON THE CONVERGENCE OF SERIES. APPLICATION OF THESE RULES TO THE SERIES OF MACLAURIN.

Chapter 39 IMAGINARY EXPONENTIALS AND LOGARITHMS. USE OF THESE EXPONENTIALS AND OF THESE LOGARITHMS IN THE DETERMINATION OF DEFINITE OR INDEFINITE INTEGRALS.

Chapter 40 INTEGRATION BY SERIES.