ABSTRACT
The description of groundwater flow and transport of solutes and heat require mathematics. In this Appendix, we present the main results needed to understand several mathematical aspects of this book. To go deeper into every one of these subjects, interested readers may consult specialized literature. In appliedmathematics, it is necessary to approximate functions for different reasons (Abramowitz and Stegun 1965). Newton and his pupil Taylor discovered around 1670 that many important functions could be approximated by simple polynomials, if these functions are sufficiently differentiable in a given domain [a, x]. This is Taylor’s theorem (Haaser et al. 2001):
f (x) = n∑
(x − a)k k! f
(k)(a) + b∫
f (n+1)(t) (x − t) n
n ! dt (A.0)
The integral formula in equation A.0 is an expression of the error or residual in Taylor’s expansion. The following are some examples of approximations of classical functions obtained from Taylor’s theorem. They are useful in different parts of the book:
ex ≈ n∑
i! = 1 + x + x2
2! + x
3! + · · · · · · + x
n! , ∀ x ∈ R
1 + x ≈ n∑
k=0 (−1)k xk = 1 − x + x2 − x3 + · · · + (−1)nxn, ∀ |x| < 1
(1 + x)r ≈ 1 + rx + r(r − 1) 2!
