6.1Better Multiplication And Division

Multiplication makes addition easier.Instead of grinding through questions like 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2, we can rewrite it as: 2\( \times \)13.

Boomshakalaka.If you wanted 13 copies of a number, just write it like that!

乘法使重复的加法更容易。But there’s a big limitation: we must use identical, average-sized pieces.

• What’s 2\( \times \)13?It’s 13 copies ofthe same element.
• What’s 100 / 5?It’s 100 split into 5equal parts.

相同的部件在教科书的场景中是没问题的，即你以30英里每小时的速度毫不动摇地行驶3小时。现实世界并不是那么顺利。Calculus lets us accumulate or separate shapes according to theiractual, not average, amount:

• The derivative is a better type of divisionthat splits a shape along a path (into possibly different-sized slices)
• The integral is a better type of multiplicationthat accumulates a sequence of steps (which could be different sizes)

Operation	Example	Notes
Division	\( \dfrac{y}{x} \)	Split whole into identical parts
Differentiation	\( \dfrac{d}{dx} y \)	Split whole into (possibly different) parts
Multiplication	\( y \cdot x \)	Accumulate identical steps
Integration	\( \int y \ dx \)	Accumulate (possibly different) steps

我们再来分析一下环到环的例子。How does arithmetic/algebra compare to calculus?

Division spits back the averaged-sized ring in our pattern.The derivative gives a formula (\( 2 \pi r \)) that describeseveryring (just plug in r).Similarly, multiplication lets us scale up the average element (once we’ve found it) into the full amount.积分让我们直接加起来。

有时我们想使用平均项，而不是花哨的演算步骤，因为它是整体的更简单的表示(“平均交易大小是多少?”我不需要完整的列表。”)。That’s fine, as long as it’s a conscious choice.

6.2Better Formulas

If calculus provides better, more-specific version of multiplication and division, shouldn’t we rewrite formulas with it?You bet.

Algebra	Calculus
\( \textit{distance} = \textit{speed} \cdot \textit{time} \)	\( \textit{distance} = \int \textit{speed} \ dt \)
\( \textit{speed} = \dfrac{\textit{distance}}{\textit{time}} \)	\( \textit{speed} = \dfrac{d}{dt} \textit{distance} \)
\( \textit{area} = \textit{height} \cdot \textit{width} \)	\( \textit{area} = \int \textit{height} \ dw \)
\( \textit{weight} = \textit{density} \cdot \textit{length} \cdot \textit{width} \cdot \textit{height} \)	\( \textit{weight} = \iiint \textit{density} \ dx \ dy \ dz \)

An equation like\( \textit{distance} = \textit{speed} \cdot \textit{time} \)解释如何在假定平均速度的情况下计算总距离。An equation like\( \textit{distance} = \int \textit{speed} \ dt \)tells us how to find total distance by breaking time into instants (split along the “t” axis), and accumulating the (potentially unique) distance traveled each instant (\( \textit{speed} \cdot dt \)).

Similarly,\( \textit{speed} = \frac{d}{dt} \textit{distance} \)解释说，我们可以将轨迹分割成时间片段，我们在该时间片段中移动的(可能是唯一的)量就是速度。

The overused “integrals are area under the curve” explanation becomes more clear.Multiplication, because it deals with static quantities, can only measure the area of rectangles.积分让测量曲线和波动，我们将添加它们的贡献，无论如何。

A series of multiplications becomes a series of integrals (called a triple integral).It’s beyond this primer, but your suspicion was correct: we can mimic the multiplications and integrate several times in a row.

Math, and specifically calculus, is the language of science because it describes relationships extremely well.When I see a formula with an integral or derivative, I mentally convert it to multiplication or division (remembering we can handle differently-sized elements).

6.3Better Algebra

Algebra lets us start with one fact and systematically work out others.假设我想知道一个未知正方形的面积。I can’t measure the area, but I overheard someone saying it was 13.3 inches on a side.

Algebra	Thinking Process
\( \textit{Area of square} = ? \)	The area of this square is unknown…
\( \sqrt{\textit{Area}} = 13.3 \)	但我知道平方根
\( \left(\sqrt{\textit{Area}}\right)^2 = \left(13.3\right)^2 \)	Square both sides…
\( \textit{Area} = 176.89 \)	…and I can recreate the original area

Remember learning that along with add/subtract/multiply/divide, we could take powers and roots?我们增加了两种变换方程的新方法。

微积分扩展了代数，增加了两个运算:积分和导数。Now we can work out the area of a circle, algebra-style:

Algebra + Calculus	Thinking Process
\( \textit{Area of circle} = ? \)	The area of a circle is unknown…
\( \dfrac{d}{dr} \textit{Area} = 2 \pi r \)	…but I know it splits into rings (along the radius)
\( \int \dfrac{d}{dr} \textit{Area} = \int 2 \pi r \)	Integrate both sides…
\( \textit{Area} = \pi r^2 \)	…and I can recreate the original area

The abbreviated notation helps see the big picture.If the integrand only uses a single variable (as in\( 2 \pi r \)), we can assume we’re using\( dr \)from\( r=0 \)to\( r=r \).This helps us think of integrals and derivatives like squares and square roots: operations that cancel!

It’s pretty neat: “gluing together” and “splitting apart” should behave like opposites, right?

With our simpler notation, we can write\( \int \frac{d}{dr} \textit{Area} = \textit{Area} \)instead of the bulky\( \int_0^r \left( \frac{d}{dr} \textit{Area} \right) \ dr = \textit{Area} \).

6.4Learning The Rules

在算术方面，我们学习了组合整数、小数、分数和根号/幂的特殊技巧。Even though\( 3 + 9 = 12 \), we can’t assume\( \sqrt{3} + \sqrt{9} = \sqrt{12} \).

类似地，我们需要学习积分/导数在相加、相乘等情况下如何工作的规则。Yes, there are fancy rules for special categories (what to do with ex, natural log, sine, cosine, etc.), but I’m not concerned with that.Let’s get extremely comfortable with the basics.The fancy stuff can wait.