calculus
Calculus notations
Calculus
The maximum of the continuous function in the range to . | |||
The minimum of the continuous function in the range to . | |||
The supremum of the function over the interval , i.e. the smallest real number that is greater than or equal to every value of with . | |||
The infimum of the function over the interval , i.e. the biggest real number that is smaller than or equal to every value of with . | |||
This denotes an infinite sequence of real numbers , , , ... that are given through some particular formula depending on the index of each term. For example the sequence is the sequence of terms | |||
This denotes the limit of the sequence , whenever it exists. Intuitively this is the value that the term approaches as the index gets closer and closer to infinity. It can be shown for example that | |||
This is called an infinite series of the sequence and it is defined through a sequence of partial sums whose terms are given by the formula Whenever it exists, the limit of the sequence is called the value of the infinite series and thus | |||
This is called an infinite product of the sequence and it is defined through a sequence of partial products whose terms are given by the formula Whenever it exists, the limit of the sequence is called the value of the infinite product and thus | |||
This denotes the limit of the function at point , whenever it exists. Intuitively this is the value that the function approaches as the argument gets closer and closer to . It can be shown for example that | |||
This denotes the first derivative of the function at point , whenever it exists. It can be defined through the following formula using limits | |||
This denotes the -th derivative of the function at point , whenever it exists. It can be defined recurrently through the following formulae | |||
The partial derivative of a function , with respect to the -th variable, at a point . This is defined through the following formula Basically this evaluates the first derivative of the function , defined by Therefore it will also be valid to write As an example consider the function given by . Using the basic differentiation rules it is easy to see that | |||
If is a function then the -th order partial derivative of with respect to the -th variable is defined through the following recurrence relation For example if is defined by , then while | |||
The Riemann integral of the non-negative real-valued function on the interval . This basically gives the area below the graph of calculated from point to point . | |||
The improper integral of the non-negative real-valued function defined through This gives the area below the graph of calculated from point to infinity. | |||
The improper integral of the non-negative real-valued function defined through This gives the area below the graph of calculated from minus infinity to point . |