Split the differential and differential forms

[JTC]

In undergraduate dynamics, they do things like this:
--------------------
v = ds/dt
a = dv/dt
Then, from this, they construct: a ds = v dv
And they use that to solve some problems.
--------------------

Now I have read that it is NOT wise to treat the derivative like a fraction: it obliterates the meaning.
And that such tricks like the above one, work only in 1D cases. But it is bad policy to get used to it.

I have a FEELING for that, but no PRECISE explanation of why it is unwise to treat the derivative like a fraction.

Can someone please explain this?

And if you can explain it -- and I hope you can -- then I will come back and ask you to discuss that in the context of differential forms where "dx" is a co-vector.

Because with regard to differential forms, one DOES have these bases from the dual space.

Could someone address this for me?

 

[fresh_42]

As long as you know what you do, you can treat it as a fraction - as always with abbreviations. A closer inspection, however, gives rise to some questions: Where is the limit? In the "d"? But we have only one limit and two "d", and this is the major risk here $\displaystyle \lim \frac{\Delta x(t)}{\Delta t} \neq \frac{\lim \Delta x(t)}{\lim \Delta t}$. Another point is, that it marks a derivative, that is $\displaystyle t_0 \longmapsto \left. \frac{dx(t)}{dt}\right|_{t=t_0}$ which is an entire vector field. I recently counted the ways a derivative can be viewed as and found $10$ different interpretations of only one formula, and the fraction wasn't even among them:

$$
D_{x_0}L_g(v)= \left.\frac{d}{d\,x}\right|_{x=x_0}\,L_g(x).v = J_{x_0}(L_g)(v)=J(L_g)(x_0;v)
$$
can be viewed as

1. first derivative $L'_g : x \longmapsto \alpha(x)$
2. differential $dL_g = \alpha_x \cdot d x$
3. linear approximation of $L_g$ by $L_g(x_0+\varepsilon)=L_g(x_0)+J_{x_0}(L_g)\cdot \varepsilon +O(\varepsilon^2)$
4. linear mapping (Jacobi matrix) $J_{x}(L_g) : v \longmapsto \alpha_{x} \cdot v$
5. vector (tangent) bundle $(p,\alpha_{p}\;d x) \in (D\times \mathbb{R},\mathbb{R},\pi)$
6. $1-$form (Pfaffian form) $\omega_{p} : v \longmapsto \langle \alpha_{p} , v \rangle$
7. cotangent bundle $(p,\omega_p) \in (D,T^*D,\pi^*)$
8. section of $(D\times \mathbb{R},\mathbb{R},\pi)\, : \,\sigma \in \Gamma(D,TD)=\Gamma(D) : p \longmapsto \alpha_{p}$
9. If $f,g : D \mapsto \mathbb{R}$ are smooth functions, then $$D_xL_y (f\cdot g) = \alpha_x (f\cdot g)' = \alpha_x (f'\cdot g + f \cdot g') = D_xL_y(f)\cdot g + f \cdot D_xL_y(g)$$ and $D_xL_y$ is a derivation on $C^\infty(\mathbb{R})$.
10. $L_x^*(\alpha_y)=\alpha_{xy}$ is the pullback section of $\sigma: p \longmapsto \alpha_p$ by $L_x$.

The question about the dimension is a bit tricky. As long as we consider a single derivative, it is a directional differential, in which direction ever. But - as in the examples above - the entire tangent bundle is often considered, and we get
$$
d\, f = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\,dx_i
$$
and then we have $n$ fractions in $n$ directions, which are reduced to a scalar, the component of the corresponding tangent vector. The fraction is o.k. as long as we consider the whole thing as a slope. If we start using it as a real quotient and calculate with it, we have to keep in mind, that it is merely an abbreviation. If it helps to find a solution, fine, but we should check the answer and especially be careful if we'll deal with functions, that are not smooth.

 

[JTC]

As long as you know what you do, you can treat it as a fraction - as always with abbreviations. A closer inspection, however, gives rise to some questions: Where is the limit? In the "d"? But we have only one limit and two "d", and this is the major risk here $\displaystyle \lim \frac{\Delta x(t)}{\Delta t} \neq \frac{\lim \Delta x(t)}{\lim \Delta t}$. Another point is, that it marks a derivative, that is $\displaystyle t_0 \longmapsto \left. \frac{dx(t)}{dt}\right|_{t=t_0}$ which is an entire vector field. I recently counted the ways a derivative can be viewed as and found $10$ different interpretations of only one formula, and the fraction wasn't even among them:

$$
D_{x_0}L_g(v)= \left.\frac{d}{d\,x}\right|_{x=x_0}\,L_g(x).v = J_{x_0}(L_g)(v)=J(L_g)(x_0;v)
$$
can be viewed as

1. first derivative $L'_g : x \longmapsto \alpha(x)$
2. differential $dL_g = \alpha_x \cdot d x$
3. linear approximation of $L_g$ by $L_g(x_0+\varepsilon)=L_g(x_0)+J_{x_0}(L_g)\cdot \varepsilon +O(\varepsilon^2)$
4. linear mapping (Jacobi matrix) $J_{x}(L_g) : v \longmapsto \alpha_{x} \cdot v$
5. vector (tangent) bundle $(p,\alpha_{p}\;d x) \in (D\times \mathbb{R},\mathbb{R},\pi)$
6. $1-$form (Pfaffian form) $\omega_{p} : v \longmapsto \langle \alpha_{p} , v \rangle$
7. cotangent bundle $(p,\omega_p) \in (D,T^*D,\pi^*)$
8. section of $(D\times \mathbb{R},\mathbb{R},\pi)\, : \,\sigma \in \Gamma(D,TD)=\Gamma(D) : p \longmapsto \alpha_{p}$
9. If $f,g : D \mapsto \mathbb{R}$ are smooth functions, then $$D_xL_y (f\cdot g) = \alpha_x (f\cdot g)' = \alpha_x (f'\cdot g + f \cdot g') = D_xL_y(f)\cdot g + f \cdot D_xL_y(g)$$ and $D_xL_y$ is a derivation on $C^\infty(\mathbb{R})$.
10. $L_x^*(\alpha_y)=\alpha_{xy}$ is the pullback section of $\sigma: p \longmapsto \alpha_p$ by $L_x$.

The question about the dimension is a bit tricky. As long as we consider a single derivative, it is a directional differential, in which direction ever. But - as in the examples above - the entire tangent bundle is often considered, and we get
$$
d\, f = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\,dx_i
$$
and then we have $n$ fractions in $n$ directions, which are reduced to a scalar, the component of the corresponding tangent vector. The fraction is o.k. as long as we consider the whole thing as a slope. If we start using it as a real quotient and calculate with it, we have to keep in mind, that it is merely an abbreviation. If it helps to find a solution, fine, but we should check the answer and especially be careful if we'll deal with functions, that are not smooth.

Perfect.
Thank you.