CALCULUS HANDOUT 1 - SEQUENCES
A sequence of real numbers is a function n an whose domain is the set of positive integers N and
whose values belong to the set of real numbers R. Usual notation: (an ).
A sequence (an ) is increasing if an an+1 for all n N.
CALCULUS HANDOUT 9 - HIGHER ORDER DIFFERENTIABILITY. LOCAL EXTREMA.
Let f : A Rn Rm be a partially dierentiable function with respect to every variable xj , j = 1, n on A.
SECOND ORDER PARTIAL DERIVATIVES:
fi
f is two times partially dierentiable at a wi
CALCULUS HANDOUT 10 - DOUBLE AND TRIPLE INTEGRALS
JORDAN MEASURABLE SETS IN R2
Consider the set of one dimensional bounded intervals of the form (a, b), [a, b), (a, b], [a, b], where a, b R.
The cartesian product = I1 I2 of two intervals of this type will
CALCULUS HANDOUT 8 - PARTIAL AND DIRECTIONAL DERIVATIVES. DIFFERENTIA
BILITY. FRECHET DERIVATIVE. LOCAL INVERSION AND IMPLICIT FUNCTION THMS
PARTIAL DERIVATIVES
Let f : A Rn R1 be a real valued function of n variables and a = (a1 , a2 , ., an ) Int(A). Th
CALCULUS HANDOUT 7
FUNCTIONS OF SEVERAL VARIABLES. LIMITS AND CONTINUITY.
THE VECTOR SPACE Rn
Rn = cfw_(x1 , x2 , ., xn )|xi R1 , i = 1, 2, ., n. The elements of Rn are called vectors.
Rn is a n-dimensional vector space with respect to the sum and the sca
CALCULUS HANDOUT 6 - RIEMANN-DARBOUX INTEGRAL. FOURIER SERIES
THE RIEMANN DARBOUX INTEGRAL
A partition P of the interval [a, b] is a nite set of points cfw_x0 , x1 , ., xn satisfying a = x0 < x1 < . <
xn = b. Consider a function f dened and bounded on [a
CALCULUS HANDOUT 3 - LIMITS, CONTINUITY, DIFFERENTIABILITY
LIMITS
Suppose that f (x) is a function dened on an open interval containing the point a (except possibly not
at a itself). We say that L is the limit of the function f as x approaches a if for an
CALCULUS HANDOUT 5 - POWER SERIES. TAYLOR POLYNOMIALS
POWER SERIES
an xn is called power series.
A series of functions of the form
n=0
The Abel-Cauchy-Hadamard theorem: the set of convergence of a power series:
1
,
if = 0
we have:
Considering = lim n |an
CALCULUS HANDOUT 4 - SEQUENCES OF FUNCTIONS, SERIES OF FUNCTIONS
SEQUENCES OF FUNCTIONS
A sequence of real valued functions dened on A R is a function F : N cfw_f | f : A R.
We write F (n) = fn and the sequence of functions is denoted by (fn ).
An element
CALCULUS HANDOUT 2 - SERIES
An innite series is an expression of the form
an = a1 + a2 + . + an + .
n=1
where (an ) is a sequence of real numbers. The number an is called the n-th term of the series.
The n-th partial sum sn of the series
an is the sum of
CALCULUS HANDOUT 11 - LINE AND SURFACE INTEGRALS
ELEMENTARY CURVES
An elementary curve is a set of points C R3 for which there exists a closed interval [a, b] R and a function
: [a, b] C which is bijective on [a, b) and smooth (of class C 1 ).
The points