Operating with dx and dy?

[pivoxa15]

1. Why can dx and dy be moved around in DE (a bit but not exactly like fractions)?


2. What is the justification that allows one to operate with dx and dy after the integral signs as if they were differentials?

 

[EnumaElish]

1. they represent "small changes."
2. they are differentials. For example, G(x) = $\int$g(t)F'(t)dt means G(x) = $\int$g(t)dF(t) where dF(t)/dt = F'(t).

 

[Gib Z]

1. Why can dx and dy be moved around in DE (a bit but not exactly like fractions)?

Because their quotient is the limit of a fraction.

 

[WWGD]

Overall, in a rigurous way, the dx's are differential forms. There is a whole theory
of differential form that explains them in detail.
Still, In some cases this movement of the dy's and dx's is more heuristic and
loose than rigorous. In the case I think you are describing, it is the case of
separation of variables.
Wikipedia has a nice page on them:


http://en.wikipedia.org/wiki/Separation_of_variables

 

[ice109]

Because their quotient is the limit of a fraction.

that doesn't explain to me why you can break them up. in fact it implies you cannot because the limit does not break up across division/multiplication.

 

[HallsofIvy]

That's true. However, any calculus book I have seen defines the differentials, dx and dy, separately from dy/dx. That is, given a derivative dy/dx, dx is defined simply as a "symbol" and then dy is defined by dy= (dy/dx) dx.

I'm a bit puzzled at your assertion "the limit does not break up across division/multiplication." It is certainly true that
$$\lim_{x\rightarrow a}f(x)g(x)= (\lim_{x\rightarrow a}f(x))(\lim_{x\rightarrow a}g(x))$$
and that
$$\lim_{x\rightarrow a} \frac {f(x)} {g(x)} = /frac{\lim_{x\rightarrow a}f(x)}{\lim_{x\rightarrow a}g(x)}$$
as long as the denominator is not 0. Of course that is true for all derivative calculations but the point is still that we can go back "before" the limit, use the fraction property and then take the limit again. That is, we can always treat a derivative as if it were a fraction and the "dy", "dx" notation is defined to make use of that.

 

[ice109]

That's true. However, any calculus book I have seen defines the differentials, dx and dy, separately from dy/dx. That is, given a derivative dy/dx, dx is defined simply as a "symbol" and then dy is defined by dy= (dy/dx) dx.

I'm a bit puzzled at your assertion "the limit does not break up across division/multiplication." It is certainly true that
$$\lim_{x\rightarrow a}f(x)g(x)= (\lim_{x\rightarrow a}f(x))(\lim_{x\rightarrow a}g(x))$$
and that
$$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}= \frac{\lim_{x\rightarrow a}f(x)}{\lim_{x\rightarrow a}g(x)}$$
as long as the denominator is not 0. Of course that is true for all derivative calculations but the point is still that we can go back "before" the limit, use the fraction property and then take the limit again. That is, we can always treat a derivative as if it were a fraction and the "dy", "dx" notation is defined to make use of that.

hmm i guess i forgot that limit property. but what do you mean there in bold?

 

[HallsofIvy]

For example to prove the chain rule, dy/dx= dy/du du/dx, when u is a differentiable function of x, you would set up the "difference quotient" defining dy/dx:
$$\frac{dy}{dx}= \lim_{h\rightarrow 0}\frac{y(x+h)- y(x)}$$
write that as
$$\lim_{h\rightarrow 0}\frac{y(x+h)- y(x)}{u(x+h)- u(x)}\frac{u(x+h)- u(x)}{h}$$
where I have used exactly the "property of fractions", that we can cancel the two "u(x+h)- u(x)" terms, that we "want" to use in saying dy/dx= (dy/du)(du/dx), that we can cancel the two "du" terms.
But since u is differentiable it is also continuous: as h goes to 0, u(x+h)-u(x)= u(x)- u(x)= 0. Renaming the "u(x+h)- u(x)" term in the denominator "k", we have y(x)= y(u(x)) so y(x+h)= y(u(x+h))= y(u(x)+ u(x+h)- u(x))= y(u+ k). We can separate that limit into two limits:
$$\left[\lim_{k\rightarrow 0}\frac{y(u+k)- y(u)}{k}\right]\left[\lim_{h\rightarrow 0}\frac{u(x+h)- u(x)}{h}\right]$$
Taking each of those limits give (dy/du)(du/dx).

We use the "fraction property" on the difference quotient and then take the limit. A derivative is not a fraction but can always be treated like one. Using the "differential" notation, defining dy and dx as separate (symbolic) things with the property that dy divided by dx is the derivative, makes that "formal".

 

[arildno]

However, a potent reminder that derivatives in general cannot be regarded naively as fractions is given by the classical relationship:
$$\frac{\partial{x}}{\partial{y}}\frac{\partial{y}}{\partial{z}}\frac{\partial{z}}{\partial{x}}=-1$$

 

[OrderOfThings]

1. Why can dx and dy be moved around in DE (a bit but not exactly like fractions)?

Because they are numbers. $dx$ and $dy$ are tangent space coordinates.

 

[Valayar]

Arildno : why is this true ?

 

[arildno]

Arildno : why is this true ?

It is a consequence of the implicit function theorem:

Let G(x,y,z) be a function, and consider the equation:

$$G(x,y,z)=0 (*)$$

The implicit function theorem now states that in a vicinity (i.e some open neighbourhood U) of a particular solution of (*) $(x_{0},y_{0},z_{0})$, we may solve for ONE of the variables in terms of the other two (under fairly mild restrictions), for example:
$$x=X(y,z), x_{0}=X(y_{0},z_{0})$$, or for that matter y=Y(x,z), z=Z(x,y), where X, Y, Z are functions.

Now, within U, the following expressions are identities:
$$G(X(y,z),y,z)=0,G(x,Y(x,z),z)=0,G(x,y,Z(x,y)=0$$

Because they are identities, we may differentiate them, and gain, for example:
$$\frac{\partial{G}}{\partial{x}}\frac{\partial{X}}{\partial{y}}+\frac{\partial{G}}{\partial{y}}=0\to\frac{\partial{X}}{\partial{y}}=-\frac{\frac{\partial{G}}{\partial{y}}}{\frac{\partial{G}}{\partial{x}}}$$

$$\frac{\partial{G}}{\partial{y}}\frac{\partial{Y}}{\partial{z}}+\frac{\partial{G}}{\partial{z}}=0\to\frac{\partial{Y}}{\partial{z}}=-\frac{\frac{\partial{G}}{\partial{z}}}{\frac{\partial{G}}{\partial{y}}}$$
$$\frac{\partial{G}}{\partial{z}}\frac{\partial{Z}}{\partial{x}}+\frac{\partial{G}}{\partial{x}}=0\to\frac{\partial{Z}}{\partial{x}}=-\frac{\frac{\partial{G}}{\partial{x}}}{\frac{\partial{G}}{\partial{z}}}$$

Thus, we gain:
$$\frac{\partial{X}}{\partial{y}}\frac{\partial{Y}}{\partial{z}}\frac{\partial{Z}}{\partial{x}}=(-\frac{\frac{\partial{G}}{\partial{y}}}{\frac{\partial{G}}{\partial{x}}})(-\frac{\frac{\partial{G}}{\partial{z}}}{\frac{\partial{G}}{\partial{y}}})(-\frac{\frac{\partial{G}}{\partial{x}}}{\frac{\partial{G}}{\partial{z}}}=-1$$
as stated.


As an example, let G(x,y,z)=ax+by+cz, so that the equation ax+by+cz=0 describes a plane.

We thereby have:
$$X(y,z)=-\frac{b}{a}y-\frac{c}{a}z,Y(x,z)=-\frac{a}{b}x-\frac{c}{b}z,Z(x,y)=-\frac{a}{c}x-\frac{b}{c}y$$
And we have:
$$\frac{\partial{X}}{\partial{y}}=-\frac{b}{a},\frac{\partial{Y}}{\partial{z}}=-\frac{c}{b},\frac{\partial{Z}}{\partial{x}}=-\frac{a}{c}$$
and you see that the product of these three quantities equals -1.

 

[WWGD]

Nice explanations. But there are restrictions to the manipulation:

We cannot, e.g. conclude from y=x^2 and dy/dx=2x, that dx/dy=1/2x.

 

[WWGD]

Because they are numbers. $dx$ and $dy$ are tangent space coordinates.

Aren't they cotangent space coordinates, dual to del/delx and del/dely ?.

 

[arildno]

Nice explanations. But there are restrictions to the manipulation:

We cannot, e.g. conclude from y=x^2 and dy/dx=2x, that dx/dy=1/2x.

Except at x=0, we can:
Let
$$G(x,y)=y-x^{2}$$
Then, we have at G=0:
$$\frac{\partial{G}}{\partial{y}}+\frac{\partial{G}}{\partial{x}}\frac{dx}{dy}=0\to{1}-2x\frac{dx}{dy}=0\to\frac{dx}{dy}=\frac{1}{2x}$$

This is meaningful at all points except for x=0

 

[colby2152]

Strangely enough, the small change terms like dx and dy can be treated as fractions in some cases, but not all.

 

[HallsofIvy]

Can you give an example?

 

[OrderOfThings]

Aren't they cotangent space coordinates, dual to del/delx and del/dely ?.

No, $dx$ and $dy$ are linear coordinates on the tangent space. More precisely, $dx,dy$ is the linearised version of the original $x,y$-coordinate system.

Example:
Take polar coordinates $r,\theta$ in the plane. The coordinate system picks out basis vectors $\mathbf{e}_r$ and $\mathbf{e}_\theta$ at every point. So any tangent vector $\mathbf{v}$ decomposes into

[tex]\mathbf{v} = dr\,\mathbf{e}_r + d\theta\,\mathbf{e}_\theta[/itex]

 

[WWGD]

Nice explanations. But there are restrictions to the manipulation:

We cannot, e.g. conclude from y=x^2 and dy/dx=2x, that dx/dy=1/2x.

Yes, sorry for my ignorance, I realized this --dy/dx=f =>dx/dy=1/f --is _always_true in a neighborhood of every x for which f'(x)=0, by the inverse function theorem, and as someone above said, it is true everywhere for f(x)=x^2, except at a point (x=0),
and it may at times be globally true , e.g, for f(x)=x .
I hope I will be more carefully after I sleep, now that the finals are over.

I guess the truth of the statement also agrees with another perspective:
dy/dx near a point is the slope of the tangent line at x to f(x), so that at
each point we approximate f(x) by dy/dx=lim(deltax->0) delta(y)/delta(x).


Sorry, I am still trying to learn latex. Hopefully for my next post.

 

[WWGD]

No, $dx$ and $dy$ are linear coordinates on the tangent space. More precisely, $dx,dy$ is the linearised version of the original $x,y$-coordinate system.

Example:
Take polar coordinates $r,\theta$ in the plane. The coordinate system picks out basis vectors $\mathbf{e}_r$ and $\mathbf{e}_\theta$ at every point. So any tangent vector $\mathbf{v}$ decomposes into

[tex]\mathbf{v} = dr\,\mathbf{e}_r + d\theta\,\mathbf{e}_\theta[/itex]

So, Order of Things, please let me see if I understand this well:

The linearization gives you coordinates along the tangent plane (since you

have dx, dy ; two coordinates) of the curve (embedded in IR^2), at each tangent

space. If this is the case, then I assume your curve (of which you are giving the

coordinates) is differentiable?. e.g, for y=x^2 , you work with tangent spaces

at each point and dy=2xdx, so your linearized coordinates in this tangent space

(xo,yo) would be (yo,2xo), i.e, the linearized version gives the line y=2x

as local coordinates?

 

[OrderOfThings]

So, Order of Things, please let me see if I understand this well:

The linearization gives you coordinates along the tangent plane (since you

have dx, dy ; two coordinates) of the curve (embedded in IR^2), at each tangent

space. If this is the case, then I assume your curve (of which you are giving the

coordinates) is differentiable?. e.g, for y=x^2 , you work with tangent spaces

at each point and dy=2xdx, so your linearized coordinates in this tangent space

(xo,yo) would be (yo,2xo), i.e, the linearized version gives the line y=2x

as local coordinates?

Yes, start with the curve $y=x^2$, and take a point $(x,y)$ on this curve. Now define a new two-dimensional coordinate system with its origin at $(x,y)$. The coordinates of this system are named $dx, dy$. The tangent to the curve at this point $(x,y)$ is then given by the equation $dy=2x\,dx$.

 

[WWGD]

Because they are numbers. $dx$ and $dy$ are tangent space coordinates.

So, let me see, sorry for insisting so much of this. Then, continuing with the
idea of linearization, the reason why we can only move dx, dy a little is that
the linearization gives us a good _local_ approximation, but not necessarily
a global one (except for cases where the linearized function is itself linear
to start with), so, the more we go on the linearized version of f, the more
we distort , since approx. is only local?.

I think it would be useful too, if we could come up with more examples of
when it is _not_ possible to move the dx's, dy's around. What would be
the answer for the case of separation of variables that we use to solve
separable PDE's?. I have read that this is just a heuristic, but not rigorous
use, tho I cannot explain well why this is not legit; I cannot even understand
the motivation for it.

We have, e.g:

dy/dx=yx^2 . How do we conclude that dy/y=x^2dx ?.

This seems to be a different situation than that of the linearization,

which would seem to apply for a case dy/dx=2x we mentioned before.

Maybe we could bring up implicit differentiation , but I don't see how

to make sense of it. Anyone? . Thanks.


What Would Gauss Do ?

 

[OrderOfThings]

Take again the plane equipped with $x,y$-coordinates. At each point there is a tangent space with $dx,dy$-coordinates. In this simple case the tangent space coordinate systems are just translated copies of the $x,y$-system. Given an equation

$$\frac{dy}{dx}=yx^2,$$

this has the meaning of a plain simple equation of a line through the origin in each tangent space. It is a direction field: at each point there is a line and the solution curves have these lines as their tangents.

You can rearrange $x,y,dx$ and $dy$ just as in any other equation. The equations

$$\frac{dy}{dx}=yx^2\qquad\frac{dy}{y}=x^2dx$$

both denote a line with slope $yx^2$ in $dx,dy$-space, just as

$$\frac{y}{x}=ba^2\qquad\frac{y}{b}=a^2x$$

both denote a line with slope $ba^2$ in $x,y$-space. (Assuming $b\neq0$ etc.)

 

[WWGD]

Thanks, nice explanation. Makes perfect sense now.