Parametric, Vector & Polar
AP Calculus BC Topic 9 6:06 English narration · English + 中文 subtitles burned in
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Transcript
Some curves cannot be written as y equals a function of x.
有些曲线无法写成 y 等于 x 的函数。
Think of a particle looping around, or a path that crosses itself — for one x there would be two y values.
想想一个绕圈运动的质点，或者一条自我相交的路径—— 对同一个 x 会有两个 y 值。
Parametric equations solve this.
参数方程解决了这个问题。
Give x and y each as a function of a third variable, t, usually time.
把 x 和 y 各自写成第三个变量 t 的函数， 通常 t 就是时间。
As t runs forward, the point traces the path — and we learn not just the shape, but the direction and the timing.
当 t 向前推进时，那个点就描出这条路径—— 而我们得到的不只是形状，还有方向和时间。
Today we work with parametric curves, vector-valued functions, and polar coordinates — the BC extensions of everything we already know.
今天我们来处理参数曲线、向量值函数和极坐标——这些是我们已学内容在 BC 中的延伸。
Let's begin.
让我们开始吧。
How do we find the slope of a parametric curve?
怎样求参数曲线的斜率？
Use the chain rule.
用链式法则。
The slope, d y d x, equals d y d t divided by d x d t — as long as d x d t is not zero.
斜率 d y d x，等于 d y d t 除以 d x d t—— 只要 d x d t 不为零。
So differentiate each part with respect to t, then divide.
所以把每一部分分别对 t 求导，然后相除。
For x equals t squared, and y equals t cubed minus t, the bottom is two t, and the top is three t squared minus one.
对 x 等于 t 平方、y 等于 t 立方减 t，分母是 二 t，分子是 三 t 平方减一。
At t equals two, that comes to eleven over four.
当 t 等于二时，结果是四分之十一。
The second derivative has a trap.
二阶导数有一个陷阱。
It is NOT the second derivative of y with respect to t, divided by the second derivative of x.
它不是 y 对 t 的二阶导数，除以 x 对 t 的二阶导数。
Instead: take the first derivative you just found, differentiate that with respect to t, and then divide by d x d t once more.
正确的做法是：取你刚求出的一阶导数，把它再对 t 求导，然后再除以 d x d t 一次。
Same pattern as before — differentiate in t, then divide by d x d t.
和之前是同一个套路——先对 t 求导，再除以 d x d t。
Use it to test the concavity of the curve.
用它来判断曲线的凹凸性。
Now, how long is the curve?
那么这条曲线有多长呢？
Zoom in on a tiny piece of the path.
把路径上极小的一段放大来看。
It is almost a straight line — the hypotenuse of a small triangle with sides d x and d y.
它几乎是一条直线—— 是一个小三角形的斜边，两条直角边是 d x 和 d y。
By Pythagoras, its length is the square root of d x squared plus d y squared.
由勾股定理， 它的长度是 d x 平方加 d y 平方 的平方根。
Add up all those tiny hypotenuses, and you get the arc length: the integral of the square root of d x d t squared, plus d y d t squared.
把所有这些小斜边加起来，就得到弧长： 也就是 d x d t 的平方 加 d y d t 的平方，再开平方根的积分。
The same curve can be written as a vector-valued function: r of t, with components x of t and y of t.
同一条曲线也可以写成一个向量值函数：r t，它的分量是 x t 和 y t。
The rule is simple — differentiate component-wise, component by component.
规则很简单——按分量求导，逐个分量。
The derivative is the velocity vector; its derivative is the acceleration.
导数是速度向量；速度的导数是加速度。
The speed is the magnitude of velocity: the square root of the sum of the squares.
速率是速度的模长：也就是各分量平方和的平方根。
And notice, the total distance traveled — distance travelled — is the integral of the speed, which is exactly the arc-length formula again.
并且注意， 走过的总路程就是速率的积分——这恰好又是弧长公式。
For position t squared, t cubed minus t: at t equals one the velocity is two, two, so the speed is two root two.
对位置 t 平方、t 立方减 t：当 t 等于一时速度是 二，二，所以速率是 二 根号二。
Differentiation went one way; integration runs it back, and it is just as component-wise.
求导是一个方向；积分把它反过来，而且同样是逐分量进行的。
Integrate acceleration and you recover velocity.
对加速度积分，你就还原出速度。
Integrate velocity and you recover position.
对速度积分，你就还原出位置。
But an integral only gives you the CHANGE, so you must add the initial position: the position at a later time equals the position you started at, plus the integral of velocity over that interval.
但积分只给出变化量，所以你必须加上初始位置：后一时刻的位置 等于你出发时的位置，加上速度在这段区间上的积分。
Do each component on its own.
每个分量各做各的。
Work one.
来做一个。
Velocity is two t, and three t squared minus one, starting from the origin.
速度是 二 t，和 三 t 平方 减一，从原点出发。
The x component: the integral of two t from zero to two is t squared, which is four.
x 分量：二 t 从零到二的积分是 t 平方，也就是四。
Now the y component: the integral of three t squared minus one is t cubed minus t, which at two is eight minus two, so six.
再看 y 分量：三 t 平方减一的积分是 t 立方减 t，在二处是八减二，所以是六。
The position is four, six.
位置就是 四，六。
Get fluent with this, because plane motion appears on the BC free-response nearly every year.
要把这一套练熟，因为平面运动几乎每年都出现在 BC 的自由作答题里。
Now polar coordinates.
现在讲极坐标。
Instead of across and up, we locate a point by its distance from the origin, r, and its angle, theta.
我们不用"横向和纵向"，而是用一个点到原点的距离 r， 以及它的角度 theta 来定位。
Convert with x equals r cosine theta, and y equals r sine theta.
转换公式是 x 等于 r 乘以 cos theta， y 等于 r 乘以 sin theta。
And here is the useful part: a polar curve, r as a function of theta, is really just a parametric curve, with theta as the parameter.
而有用的地方在于：一条极坐标曲线，也就是 r 作为 theta 的函数， 其实就是一条以 theta 为参数的参数曲线。
So the slope uses exactly the same rule — d y d theta, over d x d theta.
所以斜率用的是完全相同的法则—— d y d theta，除以 d x d theta。
Polar area is not the usual integral.
极坐标面积不是通常的那种积分。
Sweep a thin wedge out from the origin — a narrow sector, like a slice of pie.
从原点扫出一个细细的楔形——一个窄窄的扇形，像一块派。
Its area is one half r squared, d theta.
它的面积是 二分之一 r 平方 d theta。
Add up the sectors, and the area is one half, times the integral of r squared.
把这些扇形加起来，面积就是二分之一，乘以 r 平方的积分。
The hard part is always the limits: find where the curve starts and stops tracing your region.
困难之处永远在积分限：要找出曲线从哪里开始、到哪里结束地描出你要的区域。
For one petal of the rose r equals two sine two theta, the limits run from zero to pi over two, and the area comes to pi over two.
对玫瑰线 r 等于 二 sin 二 theta 的一片花瓣，积分限从零到二分之派，面积算出来是二分之派。
For the region between two polar curves, subtract the sectors: one half, times the integral of the outer radius squared, minus the inner radius squared.
对于两条极坐标曲线之间的区域，就把扇形相减：二分之一，乘以外半径平方减内半径平方的积分。
Two warnings.
两个提醒。
First, find the intersection angles by setting the two radii equal — those give your limits.
第一，令两个半径相等来求交角——那给出你的积分限。
Second, check which curve is actually the outer one on each interval, because they can swap.
第二，检查在每一段上到底哪条曲线在外侧，因为它们可能会互换。
And subtract the squares — never the difference, squared.
并且要把两个半径分别平方再相减——绝不是把差再平方。
Before you go, three marks to keep.
结束之前，三个要守住的分。
First, for a parametric curve, differentiate each part with respect to t, then divide — and for the second derivative, remember you divide by d x d t again.
第一，对参数曲线，把每部分分别对 t 求导，然后相除—— 而求二阶导数时，记住要再除以一次 d x d t。
Second, speed is the magnitude of the velocity vector, and total distance is the integral of the speed.
第二，速率是速度向量的模长， 总路程是速率的积分。
Third, polar area is one half the integral of r squared — with the correct theta limits, and with the squares subtracted, not the difference squared.
第三，极坐标面积是二分之一乘以 r 平方的积分—— 要用正确的 theta 积分限，而且是把平方相减，不是把差再平方。
This topic is yours.
这个专题就是你的了。