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Calculus

Absolute operation:
- |x|: absolute value of a number x
- |f(x)|: absolution function or absolute value of f(x)
- max |f(x)|: maximum of absolute function values |f(x)|
Let S and T be nonempty sets.

f:S -> T is an injective function if f(s1) = f(s2) implies s1 = s2. (This is often called a "one-to-one" function.)

f is a surjective function if for all t in T, there is an s in S such that f(s) = t. i.e., f(S)=T. (This is often called an "onto" function.)

f is a bijective function if it is both injective and surjective. (This is often called a "one-to-one correspondence".)
Point of Inflection:

Point of inflection of a graph of a function is the point on the graph such that the concavity of the function changes at that point.

Point of inflection occur when the second derivative of a function is zero or undefined.
Multiple integral (n-fold integral)
A set of integrals taken over n > 1 variables
　

(1)

　

is called a multiple integral. An nth order integral corresponds, in general, to an n-dimensional volume, with n = 2 corresponding to an area. In an indefinite multiple integral, the order in which the integrals are carried out can be varied at will; for definite multiple integrals, care must be taken to correctly transform the limits if the order is changed. Double integral: $\displaystyle \int$ $\displaystyle \int_{R}^{}$ f (x, y) dA.
Repeated Integral
A repeated integral is an integral taken multiple times over a single variable (as distinguished from a multiple integral, which consists of a number of integrals taken with respect to different variables). The following theorem tells how to convert multiple integral into repeated integral.
Theorem (Fubini's theorem for Rectangles) Let f (x, y) be continuous on the rectangular region R : a $\le$ x $\le$ b;c $\le$ y $\le$ d. Then
　

$\displaystyle \int$ $\displaystyle \int_{R}^{}$ f (x, y) dA = $\displaystyle \int_{c}^{d}$ $\displaystyle \left(\vphantom{\int_a^bf(x,y) dx}\right.$ $\displaystyle \int_{a}^{b}$ f (x, y) dx $\displaystyle \left.\vphantom{\int_a^bf(x,y) dx}\right)$ dy = $\displaystyle \int_{a}^{b}$ $\displaystyle \left(\vphantom{\int_c^df(x,y) dy}\right.$ $\displaystyle \int_{c}^{d}$ f (x, y) dy $\displaystyle \left.\vphantom{\int_c^df(x,y) dy}\right)$ dx