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#151 2022-12-05 00:39:38

ganesh
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Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#152 2022-12-06 00:36:39

ganesh
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Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#153 2022-12-09 00:45:33

ganesh
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Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics

Arc Length

Arc length is the distance between two points along a section of a curve.

Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification. A rectifiable curve has a finite number of segments in its rectification (so the curve has a finite length).

If a curve can be parameterized as an injective and continuously differentiable function (i.e., the derivative is a continuous function)

, then the curve is rectifiable (i.e., it has a finite length).

The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases.

Arc Length


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#154 2022-12-12 02:14:44

ganesh
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Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics

Implicit Differentiation.

Implicit differentiation

In calculus, a method called implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.

To differentiate an implicit function y(x), defined by an equation R(x, y) = 0, it is not generally possible to solve it explicitly for y and then differentiate. Instead, one can totally differentiate R(x, y) = 0 with respect to x and y and then solve the resulting linear equation for dy/dx  to explicitly get the derivative in terms of x and y. Even when it is possible to explicitly solve the original equation, the formula resulting from total differentiation is, in general, much simpler and easier to use.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#155 2022-12-16 17:58:23

ganesh
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Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics

Partial Derivatives

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#156 2023-01-01 00:03:19

ganesh
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Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#157 2023-01-01 22:47:38

ganesh
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Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics

Taylor Series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.

The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).

Definition

Definition

The Taylor series of a real or complex-valued function f (x) that is infinitely differentiable at a real or complex number a is the power series

where n! denotes the factorial of n. In the more compact sigma notation, this can be written as

where

denotes the nth derivative of f evaluated at the point a. (The derivative of order zero of f is defined to be f itself and [math]{(x - a)}^0 and 0! are both defined to be 1.)

When a = 0, the series is also called a Maclaurin series.[


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#158 2023-01-06 00:06:25

ganesh
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Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics

Intermediate Value Theorem

Intermediate Value Theorem

In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b], then it takes on any given value between f(a) and f(b) at some point within the interval.

This has two important corollaries:

1. If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem).
2. The image of a continuous function over an interval is itself an interval.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#159 2023-01-07 02:29:25

ganesh
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Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#160 2023-01-09 01:29:47

ganesh
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Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#161 2023-01-09 21:53:25

ganesh
Administrator
Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#162 2023-01-12 20:14:38

ganesh
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Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#163 2023-01-17 00:31:29

ganesh
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Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#164 2023-01-26 21:22:41

ganesh
Administrator
Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#165 2023-02-01 23:00:10

ganesh
Administrator
Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#166 2023-02-04 01:08:02

ganesh
Administrator
Registered: 2005-06-28
Posts: 40,614

Re: Higher Mathematics


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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