Titlebook: Applied Mathematics: Body and Soul; Volume 2: Integrals Kenneth Eriksson,Donald Estep,Claes Johnson Textbook 2004 Springer-Verlag Berlin H

habile

,Analytic Geometry in ℝ,,We now generalize the discussion of analytic geometry to ℝ., where n is an arbitrary natural number. Following the pattern set above for ℝ. and ℝ., we define ℝ. to be the set of all possible ordered .-tuples of the form (.., .., ... , ..) with .. ∈ ℝ for . = 1, ... , .. We refer to ℝ. as ..

寻找

Linear Algebra Tool Bag,. of two vectors . = (.., ..) and . = (.., ..) in ℝ.:

LIEN

枪支

DAFT

The Logarithm log(,),ntinuous on any given interval [., .] with 0 < . < ., we know by the Fundamental Theorem that there is a unique function .(.) which satisfies u′(.) = 1/. for a ≤ x ≤ b and takes on a specific value at some point in [., .], for example .(1) = 0. Since . > 0 may be chosen as small as we please and . a

Essential

Numerical Quadrature,a for the primitive function in terms of known functions. For example we can give a formula for a primitive function of a polynomial as another polynomial. We will return in Chapter . to the question of finding analytical formulas for primitive functions of certain classes of functions. The Fundamen

RALES

Trigonometric Functions,initial conditions because the problem involves a second order derivative. We may compare with the first order initial value problem: .′(.) = −.(.) for . > 0, .(0) = .., with the solution .(.) = exp(−.), which we studied in the previous chapter.

光明正大

Techniques of Integration,the polynomials, rational functions, root functions, exponentials and trigonometric functions along with their inverses and combinations. It is not even true that the primitive function of an elementary function is another elementary function.

描绘

Improper Integrals,r unbounded intervals. We call such integrals ., or sometimes (more properly) . integrals. We compute these integrals using the basic results on convergence of sequences that we have already developed.

GLIB