Further Pure Mathematics 2
A-Level Further Mathematics Topic 2 11:29 English narration · English + 中文 subtitles burned in
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Look at any hanging chain, any power line, any suspension cable.
看任何一条悬挂的链子、任何一根电线、任何一条悬索。
It always settles into one special curve.
它总会落成一条特别的曲线。
You might guess a parabola — but it is not.
你也许会猜是抛物线——但并不是。
It is the catenary, the graph of the hyperbolic cosine.
它是悬链线，也就是双曲余弦的图像。
These hyperbolic functions, built from the exponential, describe the real world in ways ordinary trigonometry cannot.
这些双曲函数由指数函数构造而来，能描述普通三角函数无法刻画的真实世界。
That is where Further Pure Two begins.
这正是进阶纯数学二的起点。
Further Pure Mathematics Two is the calculus and algebra that opens the door to higher mathematics.
进阶纯数学二，是通往更高等数学的微积分与代数。
We will meet the hyperbolic functions, pull eigenvalues out of a matrix, expand functions as infinite series, integrate with powerful new forms, harness de Moivre's theorem for complex powers and roots, and finally, solve differential equations — the language of change itself.
我们会认识双曲函数，从矩阵中取出特征值， 把函数展开成无穷级数，用强大的新形式做积分，借助棣莫弗定理处理复数的幂与根， 最后，求解微分方程——它是描述变化本身的语言。
Let's begin.
让我们开始吧。
Further Maths Two opens with a new family of functions: the hyperbolic functions.
进阶数学二从一族新函数开始：双曲函数。
They are built directly from the exponential.
它们直接由指数函数构造出来。
Cosh of x is the average of e-to-the-x and e-to-the-minus-x.
cosh x 是 e 的 x 次方和 e 的负 x 次方的平均。
Sinh of x is half of their difference.
sinh x 是它们差的一半。
Watch the two exponential curves combine into these smooth new shapes.
看这两条指数曲线如何合成这些平滑的新形状。
They look like the trig functions, and they obey almost the same rules — but they are born from exponentials, not from a circle.
它们看起来像三角函数，遵守几乎相同的规则—— 但它们生于指数，而不是生于圆。
The hyperbolic functions come with their own identities, and one rules them all.
双曲函数有它们自己的恒等式，其中一条统领全部。
Remember, cosh and sinh are both built from the exponential.
记住，cosh 和 sinh 都由指数函数构造。
So let's test a claim: cosh squared minus sinh squared.
我们来检验一个说法：cosh 平方减 sinh 平方。
Write each one as its exponential form, square them, and subtract.
把每一个写成指数形式，平方，再相减。
Almost every term cancels — the e-to-the-two-x terms disappear, the e-to-the-minus-two-x terms disappear — and you are left with four over four.
几乎每一项都抵消了——e 的 2x 次方项消失，e 的负 2x 次方项也消失——最后只剩下四分之四。
So cosh squared minus sinh squared equals one, always.
所以 cosh 平方减 sinh 平方永远等于一。
It is the hyperbolic twin of the famous trig identity.
它是那条著名三角恒等式的双曲孪生兄弟。
Three reciprocals complete the family — sech, cosech and coth — and each inverse hyperbolic function has a logarithmic form: inverse sinh of x is the natural log of x plus the root of x squared plus one.
还有三个倒数函数补齐这一家：sech、cosech 和 coth； 而每一个反双曲函数都有对数形式：反 sinh x 等于 x 加上根号下 x 平方加一的自然对数。
Before the eigenvalues, matrices do something more everyday: they solve simultaneous equations.
在特征值之前，矩阵还有一件更日常的工作：解联立方程。
Three linear equations in three unknowns collapse into one matrix equation, A x equals b.
三个未知数的三个线性方程可以合并成一个矩阵方程：A x 等于 b。
And then the determinant decides everything.
接下来一切都由行列式决定。
If the determinant of A is not zero, A is non-singular, it has an inverse, and there is exactly one solution — x equals A inverse b.
如果 A 的行列式不为零，A 就是非奇异的，存在逆矩阵， 解恰好只有一个——x 等于 A 的逆乘以 b。
Geometrically, the three planes meet at a single point.
从几何上看，三个平面交于一点。
If the determinant is zero, A is singular and there is no inverse, but that splits into two quite different cases.
如果行列式为零，A 就是奇异的，没有逆矩阵，但这里又分成两种很不一样的情况。
The equations may still be consistent, in which case there are infinitely many solutions — geometrically the three planes share a whole line.
方程组可能仍然相容，那就有无穷多个解——几何上三个平面共有一整条直线。
Or they may be inconsistent, in which case there is no solution at all, because no point lies on all three planes.
也可能不相容，那就完全没有解，因为没有任何一点同时落在三个平面上。
So here is the thing to say precisely, because it is where marks go: a zero determinant does not mean no solution. It means no UNIQUE solution — and you must then test the equations to see which of the two singular cases you have.
所以有一句话要说准，因为分数就在这里：行列式为零不等于没有解， 而是没有唯一解——接下来你必须检验方程组，判断属于哪一种奇异情形。
Now, the deepest idea about a matrix: its eigenvalues.
现在，关于矩阵最深刻的想法：它的特征值。
Most vectors, when a matrix acts on them, get turned to a new direction.
大多数向量在矩阵作用下会被转向一个新方向。
But a special few — the eigenvectors — are only stretched, never turned; they keep their own line.
但有一小部分特别的向量——特征向量——只被拉伸，从不转向；它们保持自己的直线。
The factor each one is stretched by is its eigenvalue.
每一个被拉伸的倍数，就是它的特征值。
To find them, solve the characteristic equation: the determinant of A minus lambda times the identity, set equal to zero.
要求它们，解特征方程：A 减去 lambda 乘单位矩阵的行列式， 令它等于零。
For this symmetric matrix, that gives two minus lambda, all squared, minus one, equals zero — so lambda comes out as one, or three. Two numbers that capture how the whole matrix behaves.
对这个对称矩阵，得到二减 lambda 的平方，再减一，等于零—— 于是 lambda 求得为一，或三。
With both in hand you can write A as Q, D, Q inverse, where D is the diagonal matrix of eigenvalues and the columns of Q are the eigenvectors.
这两个数刻画了整个矩阵的行为。
Next, a way to turn almost any function into a polynomial — the Maclaurin series.
接下来，一种把几乎任何函数都变成多项式的方法——麦克劳林级数。
It writes a function as an infinite power series, using repeated differentiation of the function at zero — its derivatives there.
它用函数在零点的各阶导数，把函数写成无穷幂级数。
Each new term adds a higher power of x, divided by a factorial.
每一个新项都加上 x 的更高次幂，再除以一个阶乘。
Take the exponential function: its series is one, plus x, plus x squared over two factorial, and onwards.
取指数函数：它的级数是一，加 x，加 x 平方除以二的阶乘，如此下去。
Now watch the polynomials work: with just a few terms they hug the curve near zero, and with more terms they match the exponential over a wider and wider stretch.
现在看多项式如何起作用：只用几项，它们就在零附近紧贴曲线，而项越多，它们与指数函数吻合的范围就越来越宽。
Further integration rests on a small set of standard results.
进阶积分依赖一小组标准结果。
Two standard integrals are worth memorising by heart: one over a squared plus x squared gives an inverse tangent; and one over the root of a squared minus x squared gives an inverse sine.
有两个标准积分值得牢记于心：一除以 a 平方加 x 平方，得到反正切； 一除以 a 平方减 x 平方的平方根，得到反正弦。
The whole skill is to match your integral to one of these forms.
整个技巧就是把你的积分对上其中一种形式。
For example, one over the root of four minus x squared: here a squared is four, so a is two.
比如，一除以四减 x 平方的平方根：这里 a 平方是四，所以 a 是二。
Match the form, and the answer is simply the inverse sine of x over two.
对上形式，答案就是 x 除以二的反正弦。
Three moves get you to those forms: a trigonometric substitution, a hyperbolic substitution, or completing the square in the denominator.
有三招可以把积分化成这些形式：三角代换、 双曲代换，或者把分母配方。
A reduction formula links I n to I n minus one so you can step down.
递推公式把 I n 和 I n 减一联系起来，让你一步一步降下去。
And the same integral gives the arc length of a curve, and the surface area of revolution when you spin it about an axis.
同一个积分还能给出曲线的弧长，以及绕轴旋转时旋转曲面的面积。
Bounding a sum by an integral turns it into a Riemann sum, which equals the integral in the limit.
用积分去夹住一个和式，就把它变成黎曼和，在极限下它就等于这个积分。
Complex numbers carry a beautiful geometry.
复数带着一种优美的几何。
Write a complex number by its length and its angle.
用长度和角度来表示一个复数。
Now multiply two of them together, and something elegant happens: the lengths multiply, and the angles simply add.
现在把两个复数相乘，优雅的事情发生了： 长度相乘，角度相加。
Watch it on the Argand diagram — one arrow scales the other, and rotates it.
看阿甘图上：一支箭把另一支缩放，并旋转它。
This one simple rule is the engine behind everything we do with the powers and roots of complex numbers.
这条简单的规则，就是我们处理复数的幂与根时背后的引擎。
If multiplying adds the angles, then raising to a power multiplies the angle — and that is de Moivre's theorem.
如果相乘是把角度相加，那么求幂就是把角度相乘——这就是棣莫弗定理。
Its most beautiful use is the roots of unity.
它最美的用途是单位根。
Solve z to the n equals one, and the answers spread out perfectly evenly around the unit circle, forming a regular polygon.
解 z 的 n 次方等于一，答案会在单位圆上完美均匀地散开，形成一个正多边形。
Here are the fifth roots of unity, sitting at the corners of a pentagon, each one seventy-two degrees from the next.
这是五次单位根，坐在一个正五边形的顶点上，每一个与下一个相差七十二度。
De Moivre's theorem is a powerhouse.
棣莫弗定理是一台强力机器。
It says a complex number in polar form, raised to a whole number power n, just multiplies its angle by n.
它说，极坐标形式的复数，取整数 n 次幂，就是把它的角度乘以 n。
Let's use it to build a trig identity.
我们用它来构造一个三角恒等式。
Raise cosine theta plus i sine theta to the third power.
把 cosine theta 加 i sine theta 取三次幂。
By the theorem, that equals cosine three theta plus i sine three theta.
由定理，它等于 cosine 三 theta 加 i sine 三 theta。
Now expand the left side the normal way, and take the real part of both sides.
现在照常展开左边，再取两边的实部。
After replacing sine squared with one minus cosine squared, out drops a clean identity: cosine three theta equals four cosine cubed theta, minus three cosine theta.
把 sine 平方换成一减 cosine 平方之后，掉出一条干净的恒等式： cosine 三 theta 等于四倍 cosine 三次方 theta，减三倍 cosine theta。
Differential equations describe how things change.
微分方程描述事物如何变化。
Start with a first-order linear one.
先从一阶线性方程开始。
The trick is the integrating factor: e to the integral of P.
诀窍是积分因子：e 的 P 的积分次方。
Multiply the whole equation by it, and the left side magically collapses into a single derivative — the derivative of mu times y.
用它乘以整个方程，左边就神奇地收拢成一个单一的导数——mu 乘 y 的导数。
Take our example: dy by dx plus two y equals e to the x. The integrating factor is e to the two x.
看我们的例子：dy 比 dx 加二 y 等于 e 的 x 次方。
Multiplying through, the left side becomes the derivative of y times e to the two x, equal to e to the three x.
积分因子是 e 的 2x 次方。 乘开后，左边变成 y 乘 e 的 2x 次方的导数，等于 e 的 3x 次方。
Integrate both sides, and the solution is y equals one third e to the x, plus a constant times e to the minus two x. Simpler equations with separable variables need none of that — separate and integrate directly.
两边积分， 解就是 y 等于三分之一 e 的 x 次方，加上一个常数乘 e 的负 2x 次方。
For a second-order equation with constant coefficients, replace the derivatives with powers of m — and that gives the auxiliary equation.
对于常系数的二阶方程，把导数换成 m 的幂——就得到辅助方程。
Its roots decide the shape of the whole answer.
它的根决定了整个答案的形状。
Two different real roots give a sum of two exponentials.
两个不同的实根，给出两个指数的和。
A repeated root gives an exponential, times A plus B x.
一个重根，给出一个指数乘以 A 加 B x。
And a complex pair gives an exponential wrapped around a sine and a cosine.
而一对复根，给出一个指数包裹着正弦和余弦。
Take this example: m squared minus four m plus four factors as m minus two, all squared — a repeated root, m equals two. So the general solution is A plus B x, times e to the two x.
看这个例子：m 平方减四 m 加四， 因式分解为 m 减二的平方——一个重根，m 等于二。
Initial conditions then fix A and B and give the particular solution.
所以解是 A 加 B x，乘以 e 的 2x 次方。
That auxiliary equation only gave us half the answer.
刚才那个辅助方程只给了我们一半的答案。
A general solution has two halves: the complementary function, which solves the equation with the right-hand side set to zero, plus a particular integral — any single solution of the full equation.
通解由两部分组成：余函数， 也就是把右边设为零时的解，加上一个特积分——完整方程的任意一个解。
And you find a particular integral by guessing, then fixing the constants.
特积分是先猜、再定常数的。
Match the shape of f of x.
要照着 f of x 的形状猜。
If the right-hand side is a polynomial, try a polynomial of the same degree.
如果右边是多项式， 就试同次数的多项式。
If it is a times e to the b x, try C times e to the b x.
如果是 a 乘以 e 的 b x 次方，就试 C 乘以 e 的 b x 次方。
If it is a cosine plus b sine, try C cosine plus D sine — and include both, even when only one appears on the right.
如果是 a 余弦加 b 正弦，就试 C 余弦加 D 正弦——两项都要写上， 哪怕右边只出现了一项。
Substitute your trial into the equation and compare coefficients to pin down the constants.
把你猜的形式代回方程，比较系数，就能定出常数。
Now the one trap.
现在说那个陷阱。
If your trial already appears in the complementary function, it solves the left-hand side to zero and gives you nothing, so multiply it by x.
如果你猜的形式已经出现在余函数里，它会把左边化为零，什么也得不到， 这时要乘以一个 x。
Watch it on the same equation as before: y double dash minus four y dash plus four y, now with e to the two x on the right.
看看还是刚才那个方程：y 的二阶导减四倍 y 的一阶导加四 y， 现在右边是 e 的二 x 次方。
The complementary function was A plus B x, all times e to the two x — so e to the two x and x e to the two x are both taken already.
余函数是 A 加 B x，整体乘以 e 的二 x 次方—— 所以 e 的二 x 次方和 x 乘 e 的二 x 次方都已经被占用了。
Multiply by x twice, and try C x squared e to the two x.
那就乘两次 x， 试 C 乘 x 平方乘 e 的二 x 次方。
Before you go, four ways to lock in your marks.
结束之前，四个锁住分数的办法。
First, learn the hyperbolic identities and derivatives cold — they unlock the hardest integrals.
第一，把双曲恒等式和导数背得滚瓜烂熟——它们能解开最难的积分。
Second, solve every linear differential equation as complementary function plus particular integral, and then apply the boundary conditions.
第二，把每一个线性微分方程都解成余函数加特积分，然后代入边界条件。
Third, reach for de Moivre's theorem whenever you need powers, roots, or a multiple-angle identity.
第三，凡是需要复数的幂、根，或多倍角恒等式，就用棣莫弗定理。
Fourth, find eigenvalues from the characteristic equation, and remember what an eigenvector truly is — a direction the matrix only stretches.
第四，由特征方程求特征值，并记住特征向量到底是什么——一个矩阵只对它拉伸的方向。