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Algebra and Calculus: Class notesJosé María Usategui
September 2023
Index
1- Preliminaries......................................................................................3
Introduction. Real numbers. Absolute value and distance between
real numbers. Intervals. Bounded and unbounded sets. Neighbourhoods.
The Plane 2 . Implications. Mathematical proofs.
2- Algebra.............................................................................................10
The vector space . Sum of vectors and product of a vector by
a scalar. Product of vectors. Orthogonality. Norm and distance between
two points. Linearly dependent/independent vectors. Basis. Matrices.
Determinants. Minors and cofactors. Rank of a matrix. Systems of
linear equations. Solutions of systems of linear equations. Inverse matrix.
Definiteness of matrices.
3- Real-valued functions of a single variable...........................................24
Real-valued functions of one real variable. Basic functions.
Continuity. Concave and convex functions. Appendix: Limit and continuity.
4- Differential calculus with one variable.................................................29
The derivative: Definition and meaning. The derivative: calculation.
Linear approximation of a function. Second derivative and second order
1
,approximation of a function. Higher order derivatives and higher order
approximations of a function. Differentiability and continuity. Mean-value
Theorem. Global maximum and global minimum. Weierstrass’ theorem.
Local maximum and local minimum. Conditions for a local extreme.
5- Integral calculus with one variable......................................................38
Indefinite integrals. Integration by parts. Integration by substitution
or change of variable. Definite integrals. Introduction to differential
equations.
6- Real-valued functions of several variables...........................................42
Introduction. Continuity. Partial derivatives. Gradient. Hessian
matrix. Differentiability. Total differential. The chain rule. Concavity and
convexity of real-valued functions of several variables. Approximation of
a function of several variables. Homogeneous functions. Euler’s theorem.
Multiple integrals. Appendix: Theorem of the implicit function and implicit
differentiation.
7- Functions of into ..................................................................50
The Jacobian matrix and the Jacobian determinant. Composition of
functions.
2
,1 Preliminaries
1.1 Introduction
Natural numbers: N = {1 2 3 }
Integer numbers: Z = N∪{0 −1 −2 −3 } = {0 1 −1 2 −2 3 −3 }
© ª
Rational numbers: Q = all fractions of the form
, where ∈ Z and ∈ N
+ +
Sum of rational numbers:
+
=
(warning: 6= +
)
Product of rational numbers: ( )( ) =
Division of rational numbers: ( ) : ( ) =
=
Order in Q:
⇔
Powers of rational numbers: a rational number multiplied by itself
times ( ∈ N) is written as . If ∈ Q, ∈ Z and ∈ N we
have:
1 √
i) If 0 then = ⇔ = = ,
ii) (−) 6= − (example: (−3)2 6= −32 ),
1
iii) := ( ) ,
+ 2 −3 2 3 −1
iv) = ( )+( ) = (example: 3 3 3 4 = 3 3 − 4 = 3 12 =
1
1 ),
(3) 12
v) ( ) = ( )( ) = ,
vi) = () and
vii) ( + ) 6= + .
3
, But powers of rational numbers do not always are rational numbers! For
1 √
instance, 2 2 = 2 is not a rational number. The rational numbers are not
enough to express the exact measure of all magnitudes (another example:
the quotient between the length of a circumference and its radius is not a
rational number).
1.2 Real numbers
The real numbers R are required to express the exact measure of all
magnitudes. The real numbers are the result of adding the irrational numbers
to the rational numbers. The rational numbers may be defined as the set of
decimal numbers with a finite number of decimals or with infinite decimals
such that beyond a certain digit a finite sequence of digits repeats itself
forever. The numbers with infinite decimals and such that there is not a
finite sequence of digits that repeats itself forever beyond a certain digit are
the irrational numbers.
Remark: there is no such thing like the number “next to” or “preceding
to” any rational number! There are irrational numbers between any two
consecutive rational numbers.
We have: N ⊂ Z ⊂ Q ⊂ R.
In the “real line” every point represents a real number.
1.3 Absolute value and distance between real numbers
½
, if 0
The absolute value of ∈ R is || :=
−, if 0
Properties of the absolute value:
4
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