Second Order Derivative Notation (mingled with)

[BHL 20]

I've been thinking about something recently:

The notation d²x/d²y actually represents something as long as x and y are both functions of some third variable, say u. Then you can take the second derivatives of both with respect to u and evaluate d²x/du² × 1/(d²y/du²).

Now I think it's also reasonable to express d²x/d²y as the product of dx/d and d/dy. Although dx/d is a notation I've never seen before I assume it's an operator. So how can the combination of two operators give an actual function? I think what I'm assuming is wrong so please explain why this does not work. If the notation dx/d isn't defined, why has no one defined it?

If dx/d is not an operator, what is it? Because d/dy is certainly one, I've found it possible to calculate d/dx for various functions x and y by integrating the expression for d²x/d²y. There doesn't seem to be any sort of pattern, but I probably haven't looked hard enough.

 

[Mark44]

I've been thinking about something recently:

The notation d²x/d²y actually represents something as long as x and y are both functions of some third variable, say u.

This isn't the standard notation for the second derivative of x with respect to y. That notation looks like this:
$$ \frac{d^2 x}{d y^2}$$
In this case, x is considered to be a (twice-differentiable) function of y. No third variable is required.

Then you can take the second derivatives of both with respect to u and evaluate d²x/du² × 1/(d²y/du²).

Now I think it's also reasonable to express d²x/d²y as the product of dx/d and d/dy.

I don't. Both of these are meaningless, since they don't convey the information about what variable we're differentiating with respect to. IOW, they don't indicate which is the independent variable. The reasoning behind the notation that I showed above is this:
$$\frac{d}{dy} \frac{dx}{dy} = \frac{d^2 x}{dy^2}$$
It's important to remember that this is a notational device.

Although dx/d is a notation I've never seen before I assume it's an operator. So how can the combination of two operators give an actual function?

They don't. You can combine two operators to get another operator. In both cases, the operators have to be applied to a function.

I think what I'm assuming is wrong so please explain why this does not work. If the notation dx/d isn't defined, why has no one defined it?

If dx/d is not an operator, what is it? Because d/dy is certainly one, I've found it possible to calculate d/dx for various functions x and y by integrating the expression for d²x/d²y. There doesn't seem to be any sort of pattern, but I probably haven't looked hard enough.

 

[BHL 20]

This isn't the standard notation for the second derivative of x with respect to y. That notation looks like this:
$$ \frac{d^2 x}{d y^2}$$
In this case, x is considered to be a (twice-differentiable) function of y. No third variable is required.

Oh please, I've studied calculus in both high school and college, and I've been applying it to problems in physics for the past 3 years. I think I know that much.

I used a different notation on purpose because I was wondering if it had any meaning.

 

[Mark44]

Oh please, I've studied calculus in both high school and college, and I've been applying it to problems in physics for the past 3 years. I think I know that much.

Your experience wasn't evident to me when you asked whether dx/d had any meaning.

I used a different notation on purpose because I was wondering if it had any meaning.

 

[BHL 20]

Your experience wasn't evident to me when you asked whether dx/d had any meaning.

? Experience doesn't rule out the possibility of an unusual notation being defined. I don't know if you have heard of fractional calculus but the notation d^1/2x/dt^1/2 has actually been defined, and it has some esoteric applications in science, despite looking absurd to most maths students.

 

[Mark44]

? Experience doesn't rule out the possibility of an unusual notation being defined.

That's true, but we get a lot of posts here from people who have little or no experience with calculus. I had no way of knowing what your background was, and what you wrote (for example, about the need for both x and y to be functions of a third variable, as well as your dx/d and dy/d operators) suggested to me that you didn't have much experience.

I don't know if you have heard of fractional calculus but the notation d^1/2x/dt^1/2 has actually been defined, and it has some esoteric applications in science, despite looking absurd to most maths students.

Yes, I have heard of fractional derivatives.

 

[Stephen Tashi]

Oh please, I've studied calculus in both high school and college, and I've been applying it to problems in physics for the past 3 years. I think I know that much.

The way you phrased your question gives the impression that you think notation has hidden properties and that certain strings of symbols have specific meanings even though they have not been given definitions. This is an imaginative way to think and occasionally it may lead to inventing useful definitions, but it is only a way of daydreaming about things. Thinking that strings of symbols must have meaning is the typical thought pattern for people who have no understanding of the logical structure that mathematics requires - for example, people who start threads with questions like "Is 0/0 equal to 1" or "0.9999... is not equal to 1" Such questions begin with the fallacy that strings of symbols have a specific meaning even when no definition for them has been stated. If you want to be recognized as a sophisticated thinker, don't say things like "Although dx/d is a notation I've never seen before I assume it's an operator." What you should ask is "Is there a useful mathematical definition for the notation dx/d?".

 

[BHL 20]

The way you phrased your question gives the impression that you think notation has hidden properties and that certain strings of symbols have specific meanings even though they have not been given definitions. This is an imaginative way to think and occasionally it may lead to inventing useful definitions, but it is only a way of daydreaming about things. Thinking that strings of symbols must have meaning is the typical thought pattern for people who have no understanding of the logical structure that mathematics requires - for example, people who start threads with questions like "Is 0/0 equal to 1" or "0.9999... is not equal to 1" Such questions begin with the fallacy that strings of symbols have a specific meaning even when no definition for them has been stated. If you want to be recognized as a sophisticated thinker, don't say things like "Although dx/d is a notation I've never seen before I assume it's an operator." What you should ask is "Is there a useful mathematical definition for the notation dx/d?".

Sorry for asking it in this way then, but I did say that what I think what I'm assuming is wrong.

I admit you're right, I like to think that there's more to mathematical notation than meets the eye. I expect whoever came up with the notation to have thought it out, so that it makes sense. The notation for the first derivative is quite straightforward, dy/dx can be understood simply as the infinitesimal change in y divided by the infinitesimal change in x, at the point at which it's evaluated. The logic behind the notation for the second derivative is not as apparent. Of course, it's just the derivative of the first derivative, but why did someone decide to write it in that way?

The first derivative notation is very handy with the way it can be inverted to get the derivative of the bottom variable with respect to the top one. The reciprocal of d²y/dx² is not equal to d²x/dy² and the notation certainly implies that. The purpose of this exercise was to see if the "meaning" ends there, i.e. if the notation does anything more than tell us the derivative can't be inverted in this way. For example, does the second derivative have anything similar to the chain rule for the first derivative, where the notation is used to represent a rather complicated operation as simple algebra.

 

[my2cts]

Oh please, I've studied calculus in both high school and college, and I've been applying it to problems in physics for the past 3 years. I think I know that much.

I used a different notation on purpose because I was wondering if it had any meaning.

This was not clear at the onset.
The only meaning of dy/d that I can come up with is (d/dy)^-1 in other words, integration.

 

[my2cts]

Yes, I have heard of fractional derivatives.

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

 

[DivergentSpectrum]

at first i thought this was an argument over semantics/notation, but i looked up
http://en.wikipedia.org/wiki/Fractional_calculus
very interesting.. seems like a sound idea but i can't imagine what it could possibly mean

lol my 2cts literally beat me to it by a second

 

[BHL 20]

at first i thought this was an argument over semantics/notation, but i looked up
http://en.wikipedia.org/wiki/Fractional_calculus
very interesting.. seems like a sound idea but i can't imagine what it could possibly mean

lol my 2cts literally beat me to it by a second

This argument isn't actually about fractional derivatives, that was just an example I gave of a strange notation which nevertheless has a meaning.

 

[DivergentSpectrum]

jeez if we can do fractional derivatives what's next?
d^a+biy/dx^a+bi
:p

 

[BHL 20]

jeez if we can do fractional derivatives what's next?
d^a+biy/dx^a+bi
:p

Old idea. Here's a paper on it: http://www.google.ie/url?sa=t&rct=j...wIGIBg&usg=AFQjCNGZDi2bV7JzaYye65eOOLOI6QlQsg

 

[my2cts]

Old idea. Here's a paper on it: http://www.google.ie/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCAQFjAA&url=http://jlms.oxfordjournals.org/content/s2-3/2/241.full.pdf&ei=rYZ_VOm1MImU7AanwIGIBg&usg=AFQjCNGZDi2bV7JzaYye65eOOLOI6QlQsg

 

[my2cts]

Setting the exponent to -1 is the answer to your original question .
I do not like the notation dy/dx, I prefer d_xy.
Thus you can write d_x^p, the partial derivative wrt x to the power p.

 

[Stephen Tashi]

Returning to the original post, if we seek to interpret $\frac {d^2 y}{d^2 x}$ as the application of an operator, there are some difficulties. It wouldn't be an operator that operated on a single function. It requires a pair of functions. if we denote the operator with the single symbol $W$ then $W$ is a mapping from a a pair of functions $(f,g)$ to the single function $\frac{f'}{g'}$. Trying to define what the operator $W$ does to a single function written as a fraction would required overcoming the fact that same function can be denoted by different fractions. For example, $f(u) = \frac{u^2}{u} = \frac{u^4}{u^3}$.

 

[DivergentSpectrum]

really i think it would be nice if someone put up a list of "acceptable manipulations of leibniz notation"

 

[Mark44]

really i think it would be nice if someone put up a list of "acceptable manipulations of leibniz notation"

There's really not much to say, at least if we limit the discussion to positive integer indexes. The motivation for second- and higher-order derivatives in Liebniz notation is this, I believe:
$$ \frac{d}{dx} \frac{dy}{dx} = \frac{d^2y}{dx^2}$$
We're applying the $\frac{d}{dx}$ operator to the derivative $\frac{dy}{dx}$. As a notational convenience to be able to write the derivative in a more compact form, someone thought that d²y should be what appears in the "numerator". The notation dx² that appears in the "denominator" is suggestive of dx * dx, or shorthand for (dx)².

In any case, it's just notation.

 

[BHL 20]

There's really not much to say, at least if we limit the discussion to positive integer indexes. The motivation for second- and higher-order derivatives in Liebniz notation is this, I believe:
$$ \frac{d}{dx} \frac{dy}{dx} = \frac{d^2y}{dx^2}$$
We're applying the $\frac{d}{dx}$ operator to the derivative $\frac{dy}{dx}$. As a notational convenience to be able to write the derivative in a more compact form, someone thought that d²y should be what appears in the "numerator". The notation dx² that appears in the "denominator" is suggestive of dx * dx, or shorthand for (dx)².

In any case, it's just notation.

At this stage, I agree with you. I suppose it can't get any more profound than this because people prefer to take derivatives and integrals one at a time. Maybe if we had a set of rules for taking second derivatives, derived from:
$$lim_{h→0}\ \dfrac{f(x+2h)-2f(x+h)+f(x)}{h^{2}}$$
it would be possible to invent a notation that incorporates at least one of those rules in a clever way. But yes, this approach would be impractical so the official second derivative notation is nothing more than a notation.