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MathsNotations

calculus

Calculus notations

Calculus

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

Last Modified: 27 Dec 11 @ 02:18     Page Rendered: 2022-03-14 15:41:54