Limits

极限

Most of the applications of calculus deal with continuous functions, but the visualization of processes often starts with finite changes in the variables of interest. One power of calculus is that it can deal with changes which are infinitesmally small. For example, an average velocity is obtained by taking a distance traveled and dividing by the finite time interval it took to travel that distance. But suppose you wanted the instantaneous velocity at a given instant of time. You can approach that with a "limit", i.e., you can take the smaller distances traveled in shorter intervals and divide by those shorter times. If this process is allowed to continue until the time interval approaches zero, then this is called "taking the limit as the time interval approaches zero". This is the way a derivative of distance with respect to time is defined and the way that an instantaneous velocity is calculated using calculus methods. Limits are also used in the formation of integrals, or anti-derivatives.

微积分的大多数应用都涉及连续函数，但过程的可视化通常从感兴趣的变量的有限变化开始。微积分的一个优点是它可以处理 infinitesimally 小的变化。例如，平均速度是通过将旅行的距离除以旅行该距离所用的有限时间间隔得到的。但假设你想要某个时间点的瞬时速度，你可以用“极限”来接近，即取更短时间内旅行的更短距离并除以那些更短的时间。如果这个过程持续到时间间隔接近零，那么这就被称为“当时间间隔接近零时取极限”。这是通过微积分方法定义距离对时间的导数的方式，也是用微积分方法计算瞬时速度的方式。极限也用于积分或反导数的形成。

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HyperPhysics*****HyperMath*****Calculus

HyperPhysics ***** HyperMath ***** 微积分

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Differential Form

微分形式

As a precursor to carrying out calculus operations such as derivatives and integrals, a mathematical expression will have to be placed in the differential form for the application of the methods of continuous variables. For example, a finite difference expression occurs in the development of the relationship for radioactive decay:

作为进行微积分运算（如导数和积分）的前奏，数学表达式必须以微分形式呈现，以便应用连续变量的方法。例如，有限差分表达式出现在放射性衰变关系式的推导中：

It is standard practice to use the Greek letter Δ to represent a finite difference. Often the finite difference relationship is only approximately true and is exactly true only in the limit where the differences become infinitesmally small. In this limit, the time interval Δt becomes vanishingly small and as a result, the number of decays N also becomes vanishingly small. The Δ is replaced by the differential symbol d and the resulting form

使用希腊字母Δ来表示有限差是标准做法。通常，有限差的关系仅近似成立，只有在差值变得 infinitesimally 小的极限情况下才精确成立。在这种极限下，时间间隔Δt变得极其微小，因此衰变的数目N也变得极其微小。Δ被替换为微分符号d，从而得到

is said to be the differential form of the expression.

Application to radioactive decay

被称作该表达式的微分形式。应用于放射性衰变

Index

索引

HyperPhysics*****HyperMath*****Calculus

HyperPhysics ***** HyperMath ***** 微积分

R Nave

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