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Encyclopedia > Differential (infinitesimal)
The differential dy
The differential dy
Topics in calculus

Fundamental theorem
Limits of functions
Continuity
Vector calculus
Tensor calculus
Mean value theorem Image File history File links Broom_icon. ... Image File history File linksMetadata Dydx. ... Image File history File linksMetadata Dydx. ... Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which use a common and specialized algebraic notation. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse functions of one another. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ...

Differentiation

Product rule
Quotient rule
Chain rule
Implicit differentiation
Taylor's theorem
Related rates
Table of derivatives In mathematics, a derivative is the rate of change of a quantity. ... In mathematics, the product rule of calculus, also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... In mathematics, to give a function implicitly is to give an equation that at least in part has the same graph as . ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... In differential calculus, related rates problems involve ratios of derivatives of two or more related variables that are changing with respect to time. ... The primary operation in differential calculus is finding a derivative. ...

Integration

Lists of integrals
Improper integrals
Integration by: parts, disks,
cylindrical shells, substitution,
trigonometric substitution In calculus, the integral of a function is an extension of the concept of a sum. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ... In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ... Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ...

In calculus, a differential is an infinitesimally small change in a variable. A differential is a change in a variable much like the familiar Δx. The difference is that a differential (dx) is infinitely small and thus does not have an actual value. Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which use a common and specialized algebraic notation. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... The infinity symbol ∞ in several typefaces. ...

Contents

Uses

A derivative (of a single variable equation) is a ratio of two differentials, typically denoted frac{dy}{dx}, which is the Leibniz notation equivalent of y' big( x big) = dot y in Newton's notation for differentiation. This is a consequence of the slope equation m = frac{Delta y}{Delta x} where the only difference is the replacement of Deltas with differentials to reflect the fact that the x and y values are at one point, not across two. For a more in-depth explanation see derivatives. In mathematics, a derivative is the rate of change of a quantity. ... In calculus, the Leibniz notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz (pronounced LIBE nits) was originally the use of dx and dy and so forth to represent infinitely small increments of quantities x and y, just as Δx and Δy represent finite... Newtons notation for differentiation involved placing a dash/dot over the function name, which he termed the fluxion. ... Look up Slope in Wiktionary, the free dictionary. ... In mathematics, a derivative is the rate of change of a quantity. ...


Integrals often use differentials in their notation to indicate the integration variable. In the way the Riemann integral is defined, it refers to some kind of thickness, an idea which is mathematically not correct, for the differential represents a linear transformation. In calculus, the integral of a function is an extension of the concept of a sum. ... If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ...


The differential as a local linear transformation

Several authors have attempted to define the differential without reference to infinitesimals. This definition is based on the definition in Apostol's book:[1] In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ...


Consider a real-valued function f defined on an open subset S of mathbb{R}^n. If mathbf{a} in S is a point in S, then we say that f has a differential at mathbf{a} if there exists g_mathbf{a} satisfying:

  1. g_mathbf{a} is a real-valued function defined in the whole of mathbb{R}^n
  2. g_mathbf{a} is linear. That is given mathbf{t}, mathbf{t'} in mathbb{R}^n and alpha, beta in mathbb{R}:
    g_mathbf{a}left(alpha mathbf{t} + beta mathbf{t'}right) = alpha g_mathbf{a}(mathbf{t}) + beta g_mathbf{a}(mathbf{t'})
  3. For every ε > 0, there exists a neighborhood N(mathbf{a}) of mathbf{a} such that:
    mathbf{t} in N(mathbf{a}) Longrightarrow left|f(mathbf{t})-f(mathbf{a})-g_mathbf{a}(mathbf{t}-mathbf{a})right|<epsilonleft|mathbf{t}-mathbf{a}right|

g_mathbf{a} left(mathbf{t}right) is often thought of as a function of two n-dimensional variables and written g left(mathbf{a}; mathbf{t}right). Note, however, that it may not be defined over the whole of S. It is common to write the variables mathbf{t} = left(t_1,t_2, dots t_nright) as dmathbf{x} = left(dx_1,dx_2, dots dx_nright) and the differential, if it exists as dfleft(mathbf{x}; dmathbf{x}right). It is then possible to prove that it is unique and satisfies:

dfleft(mathbf{x}; dmathbf{x}right) = sum_{k=1}^n D_k fleft(mathbf{x}right)dx_k,

Where D_kfleft(mathbf{x}right) are the n partial derivatives at mathbf{x}. This can be written more briefly using the following notation: 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. ...

df = frac{partial f}{partial x_{1}}dx_1 + frac{partial f}{partial x_{2}}dx_2 + dots +frac{partial f}{partial x_{n}}dx_n.

In the one dimensional case this becomes:

df = frac{df}{dx}dx.

This notation is very suggestive but it should be realised that frac{df}{dx} is a complete symbol whereas dx is a linear transformation of a one dimensional space. Thus there is no question of "cancelling" the dx.


The existence of all the partial derivatives of fleft(mathbf{x}right) at mathbf{x} is a necessary condition for the existence of a differential at mathbf{x}. However it is not a sufficient condition. It is possible to prove that if fleft(mathbf{x}right) has a differential at mathbf{x} then it is continuous at mathbf{x}. However the following function: 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. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

f(x,y)= left{begin{matrix} frac{xy^{2}}{x^{2}+y^{4}}, & mbox{if }xneq0, 0, & mbox{if } x=0 end{matrix}right.

has finite directional derivatives in all directions at left(0,0right), and therefore has all partial derivatives at the origin. However it is not continuous at the origin since it has value frac{1}{2} at every point on the parabola x = y2, except at the origin, where it has value 0. It therefore does not possess a differential at the origin. In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the... 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. ...


Confusion

It is common for math students (especially first year calculus students) to confuse differential and derivative. Although they sound similar, the mathematical meanings are distinct. In mathematics, a derivative is the rate of change of a quantity. ...


Although in some cases certain algebraic functions such as cancellation in fractions are applicable to differentials, it is important not to carry this convenient property too far. Differentials are not numbers and cannot always be treated as numbers. Whenever they are treated as numbers, it is likely that it is done to compute an approximation. Differentials have the same unit as the variable they are associated with. In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ...


History

Differentials were essential to the development of calculus and were discovered in the same time frame. However, the math innovation that made differentials more apparent and visible was Leibniz notation. In calculus, the Leibniz notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz (pronounced LIBE nits) was originally the use of dx and dy and so forth to represent infinitely small increments of quantities x and y, just as Δx and Δy represent finite...


References

  1. ^ Tom M Apostol (1967). Calculus, 2nd Ed. Wiley. ISBN 0-471-00005-1 and ISBN 0-471-00007-8.

See also


 
 

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