Derivatives and Integrals

Foundational working tools in calculus, the derivative and integral permeate all aspects of modeling nature in the physical sciences.

微积分中的基本工具，导数和积分，贯穿了物理科学中对自然现象建模的所有方面。

The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.

函数的导数可以几何上解释为数学函数 f(x) 随 x 变化而绘制的曲线的斜率。但其对自然建模的意义远不止于此简单的几何应用所能暗示的那样。毕竟，你也可以自己画出有限的三角形来发现斜率，那么为什么导数如此重要呢？其重要性在于许多物理量，如速度、加速度、力等，都是某些其他量的瞬时变化率。导数可以给你这个变化率的精确瞬时值，并导致对所求量的精确建模。

The integral of a function can be geometrically interpreted as the area under the curve of the mathematical function f(x) plotted as a function of x. You can see yourself drawing a large number of blocks to appproximate the area under a complex curve, getting a better answer if you use more blocks. The integral gives you a mathematical way of drawing an infinite number of blocks and getting a precise analytical expression for the area. That's very important for geometry - and profoundly important for the physical sciences where the definitions of many physical entities can be cast in a mathematical form like the area under a curve. The area of a little block under the curve can be thought of as the width of the strip weighted by (i.e., multiplied by) the height of the strip. Many properties of continuous bodies depend upon weighted sums, which to be exact must be infinite weighted sums - a problem tailor-made for the integral. For example, finding the center of mass of a continuous body involves weighting each element of mass by its distance from an axis of rotation, a process for which the integral is necessary if you are going to get a precise value. A vast number of physical problems involve such infinite sums in their solutions, making the integral an essential tool for the physical scientist.

函数的积分可以几何上解释为数学函数f(x)作为x的函数所绘制曲线下的面积。你可以自己想象用大量小方块来近似复杂曲线下的面积，使用更多方块会得到更准确的结果。积分提供了一种数学方法，可以画出无限多个方块并得到面积的精确解析表达式。这对于几何学非常重要——对于物理科学来说，这种重要性更加深远，因为许多物理量的定义可以以类似于曲线下面积的形式表达出来。曲线下一小块方块的面积可以看作是该条带的宽度乘以（即乘以）该条带的高度。许多连续体的性质都依赖于加权和，要精确的话，这些加权和必须是无限的加权和——这正是积分所擅长解决的问题。例如，寻找连续体的质心涉及对每个质量元素乘以其到旋转轴的距离进行加权，如果要得到精确的值，这一过程就离不开积分。大量的物理问题在解题过程中都涉及这样的无限和，因此积分对于物理科学家来说是必不可少的工具。