Differentials

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• Differential forms

Question: Can we interpret the derivative $\frac{dy}{dx}$ as a fraction?

Emphatic answer: NO!!!

Advanced answer: Yes, here's how.

Suppose we have a function $y=f(x)$ and we are to study it behavior around point $x=a$. Then

$\frac{dy}{dx}\bigg| _a$ = the derivative at $a$ = the slope of the tangent line through $(a,f(a))$ = $\frac{rise}{run}$

So it is a fraction if we create new variables, $dx$ and $dy$.

Here's the idea.

Suppose $y = f(x)$, $f^{\prime}(a) = 3$, or $$ \frac{dy}{dx} = 3. $$ Let's re-write! How about this? $$dy = 3 dx. $$ These are the new variables. What is their meaning?

This is where it comes from:

The slope of the secant line is $m = \frac{\Delta y}{\Delta x}$, or $$ \Delta y = m\cdot\Delta x. $$

What if it's the tangent line instead? $$dy = 3 dx. $$ The slope of tangent is 3. Where are $dx$, $dy$?

These are run, $dx$, and rise, $dy$, of the tangent line.

The more advance view is that $dy = 3 dx$ is a differential form of degree $1$.

Question: Why do we need this?

Answer: What if the Universe is curved?

(Einstein: It is).

Then we need to make a careful distinction between the location, $x$, and the direction, $dx$.

Note: Differentials $dx,dy$ formalize Leibniz’s notion of infinitesimals.