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MathsNotations

calculus

Calculus notations
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Calculus

\displaystyle \max_{x \in [a,b]} f(x) The maximum of the continuous function f in the range a to b.
\displaystyle \min_{x \in [a,b]} f(x) The minimum of the continuous function f in the range a to b.
\displaystyle\sup_{x \in [a,b]} f(x) The supremum of the function f over the interval [a,b], i.e. the smallest real number that is greater than or equal to every value of f(x) with x \in [a,b].
\displaystyle \inf_{x \in [a,b]} f(x) The infimum of the function f over the interval [a,b], i.e. the biggest real number that is smaller than or equal to every value of f(x) with x \in [a,b].
\displaystyle (x_n)_{n \geq 0} This denotes an infinite sequence of real numbers x_0, x_1, x_2, ... that are given through some particular formula depending on the index of each term. For example the sequence (\sin(n \alpha))_{n \geq 0} is the sequence of terms
\displaystyle \lim_{n \rightarrow \infty} x_n This denotes the limit of the sequence (x_n)_{n \geq 0}, whenever it exists. Intuitively this is the value that the term x_n approaches as the index n gets closer and closer to infinity. It can be shown for example that
\displaystyle \sum_{i=0}^{\infty} x_i This is called an infinite series of the sequence (x_n)_{n \geq 0} and it is defined through a sequence of partial sums (S_n)_{n \geq 0} whose terms are given by the formula Whenever it exists, the limit of the sequence (S_n)_{n \geq 0} is called the value of the infinite series and thus
\displaystyle \prod_{i=0}^{\infty} x_i This is called an infinite product of the sequence (x_n)_{n \geq 0} and it is defined through a sequence of partial products (P_n)_{n \geq 0} whose terms are given by the formula Whenever it exists, the limit of the sequence (P_n)_{n \geq 0} is called the value of the infinite product and thus
\displaystyle \lim_{x \rightarrow a} f(x) This denotes the limit of the function f(x) at point a, whenever it exists. Intuitively this is the value that the function f approaches as the argument x gets closer and closer to a. It can be shown for example that
\displaystyle f`(\alpha), \quad \frac{df}{dx}(\alpha) This denotes the first derivative of the function f:(a,b) \rightarrow \mathbb{R} at point \alpha \in (a,b), whenever it exists. It can be defined through the following formula using limits
\displaystyle f^{(n)}(\alpha), \quad \frac{d^n f}{dx^n}(\alpha) This denotes the n-th derivative of the function f:(a,b) \rightarrow \mathbb{R} at point \alpha \in (a,b), whenever it exists. It can be defined recurrently through the following formulae
\displaystyle \frac{\partial}{\partial x_k} f(a) The partial derivative of a function f:U \rightarrow \mathbb{R}, U \subseteq \mathbb{R}^n with respect to the k-th variable, at a point a = (a_1, a_2, \ldots, a_n) \in \mathbb{R}^n. This is defined through the following formula Basically this evaluates the first derivative of the function g_k:I \rightarrow \mathbb{R}, I \subseteq \mathbb{R} defined by Therefore it will also be valid to write As an example consider the function f:\mathbb{R}^3 \rightarrow \mathbb{R} given by f(x, y, z) = \sin x + \cos y + \tan z. Using the basic differentiation rules it is easy to see that
\displaystyle \frac{\partial^n f}{\partial x_k^n} If f: U \subseteq \mathbb{R}^p \rightarrow \mathbb{R} is a function then the n-th order partial derivative of f with respect to the k-th variable is defined through the following recurrence relation For example if f:\mathbb{R}^2 \rightarrow \mathbb{R} is defined by f(x,y) = x^3 + xy^3, then while
\displaystyle \int_a^b f(x) \,dx The Riemann integral of the non-negative real-valued function f on the interval [a,b]. This basically gives the area below the graph of f calculated from point a to point b.
\displaystyle \int_a^{\infty} f(x) \,dx The improper integral of the non-negative real-valued function f defined through This gives the area below the graph of f calculated from point a to infinity.
\displaystyle \int_{-\infty}^b f(x) \,dx The improper integral of the non-negative real-valued function f defined through This gives the area below the graph of f calculated from minus infinity to point b.