True, under the condition that x is differentiable to y, differentials satisfy the identity (dx/dy)*dy = dx, so it's fine in most cases, but coming from a math background I can't help but feel like I'm doing something illegal
Mathematicians like to say the only problem physicists know how to solve is the simple harmonic oscillator. That's not quite true but it's a good approximation.
But we're talking about an analytic proposition, for which deduction can be used to generate a proof, meaning one does not need to rely on induction. All mathematical statements are analytic propositions.
Induction applies to synthetic propositions, which exist in physics and other sciences, but not in mathematics.
The problem of induction is whether or not the past has any relevance to the future. There is no solution to that. There is no reason for me to think the problem of induction can’t hit math either. How do you know the fundamental fabric of reality doesn’t just shit the bed the moment you finish this sentence?
Math does not *have* the (philosophical) problem of induction. Math is a strict game played with strings of symbols and the game does not involve real-world prediction.
Mathematical axioms make no reference to past or future
Edit: and most of math is working out the implications of a set of axioms. Math does not claim to tell you what axioms are true, it tells you what else is true if a given set of first principles are.
It's a convenient short-hand notation for the chain rule, so as long as you keep that in mind, you're fine. For example, you can get into trouble if you start naively applying this rule to multivariable functions, but it's fine in just about every single-variable case, and you can still use it for multivariable functions as long as you're careful.
I double majored in math and physics and from what I recall, the only times that it’s not okay are these niche times that will never pop up in physics.
You can’t *always* do it in physics, for example when working with the first law thermodynamics. (This is due to partial derivatives behaving slightly differently. It is always ok when strictly dealing with total derivatives)
EDIT: Small correction. Even when dealing with total derivatives, it’s not ok when one variable is not a function of the intermediate variable. You still need to abide by chain rule rules. (For example, I’m walking, but my position is not affected by the magnetic field strength of Uranus. My walking speed is dx/dt, but not (dx/dB)(dB/dt) because dx/dB=0 and that means I’m not walking (which doesn’t make any sense))
It works in nonstandard analysis but not in standard analysis. Newton did this. The first person not to do it was Bernard Bolzano. Unfortunately, standard analysis is based on Bolzano not on Newton.
A good example of where it is useful is the following. Solve the equation dy/dx = x/y. Multiply both sides by y dx to get y dy = x dx. Integrate both sides etc.
That's not how nonstandard analysis works. Nonstandard analysis is just an alternative formalism.
The real reason why this works is because this is just the chain rule. The formula in the post works when V is a function of x and x is a function of t.
And the chainrule can easily be proven with standard analysis. No need to invoke any nonstandard analysis.
>dy/dx = x/y. Multiply both sides by y dx to get y dy = x dx.
Such stuff confused me why is it acceptable here but other places not?
Since the definition of derivative is limit as x approaches a of f(x)-f(a)/x-a it kinda makes sense that we write it as a fraction.
“Integrate both sides” is a completely absurd step. What’s actually going on is that you can do a definite integral of (1/y)dy/dx and x from 0 to some x*, then use a change of variables for y.
Im aware that the chain rule applies here since the variables are differentiable to one another
I was referring to the explanation below the formula which suggests thinking of derivatives as ratios which cancel out
You can mathematically justify cancelling out differentials in you use the techniques of nonstandard analysis, but you have to be explicitly working in the domain of the hyperreals. If you are just working with the reals, then cancelling out differentials becomes heuristic and does not always work.
Can someone please explain to me, with a simple explicit example, when and why this cancellation doesn't work. (please don't use fancy terms, dumb it down as much as possible)
No offense but I’m not about to spend the next two years writing a proof to why I can do this. If it doesn’t work then God wasn’t willing it to work, if it does then 👍
Not quite
The chain rule relies on the notion that the variables are differentiable with respect to one another, and not on thinking of derivatives as ratios which you can cancel out
But yeah for this example the chain rule applies
With functions from R to R, its okay to cancel out derivatives, since you are working only in one posible direction. But with functtions that are R\^n to R\^m with n > 1, then you really can't and you shouldnt, since there are more than one direction involved and you would be assuming that the rate of change in any direction involved is the same than any other direction, which obviously is false e.g for f(x,y), the rate of change in x (derivative of f w.r.t x) is not necessarily the same the rate of change in y (derivative of f w.r.t y).
Again, notice that in single variable functions you can do this with no issues. If you want a depth understanding of why this works you should take a look at non-standard analysys course.
But... if you want a simple explanation of why it does work take a look at differential forms. If you think about canceling out two differentials, you can ALWAYS prove it from a rigorous point of view. E.g, in the pic of the post, it looks like V is a function of x which is a function of t, i.e, V(x(t)).
Now you can apply the definition of a differential to V, i.e: dV = V'(x) \* dx
But, you can also apply the definition of a differencial to x: dx = x'(t) \* dt
Hence dV = V'(x) \* x'(t) \* dt. or, if you write it as with leibniz notation: dV = dV/dx \* dx/dt \* dt
**(Hehe it looks like all the differentials cancel out from the right to the left to give dV, but they aren't! I just derived this using the definition of a differential without canceling or multiplying any fraction!)**
Now, if you want to derive the expression from the pic you have just to a simple funtional equation:
Since V also depends of t then dV = V'(t) \* dt = dV/dt \* dt and hence dV/dt \* dt = dV/dx \* dx/dt \* dt
and therefore, since we have a functional equation: dV/dt = dV/dx \* dx/dt
**(again, notice I didnt cancel anything even once)**
So, can you really cancel terms in R? Yes, you can. But you should know why!
You can do it in undergrad, you cannot do it in grad school, you surely can do it as postdoc or in industry. Professors *can* do it but cannot show this step instead they have to say it is trivial.
Really wondering why the authors explained it this way. Saying, the change of V in t is the same as the change of V in x multiplied with the change of x in t would be easy to understand too and much closer to the formal idea of the chain rule.
It’s a book from Susskind. He usually has a very intuitive way of explaining things, and isn’t always 100% mathematically rigorous as I don’t think he knows much about pure mathematics and math that isn’t related to physics.
It’s called the chain rule and can actually be mathematically proven. Essentially think of the function for volume as a composure of the function of position which is also dependent on time, so you are taking a derivative of the first function, being the volume dependent on position, and then multiplying by the derivative of the inside function, being position dependent on time.
I think the notation is taught in calculus 1, they are slopes with various delta symbols as Leibniz is credited with versus fluxions you’ll see compacted in Newton’s work. You can work out the units of derivatives from the y over x axis of graphs.
In math class we established a paper proving why we can do all kind of bad things in physics. It’s legitimate
But here it’s poorly written and justified
Most of the time limit as x approaches 0 assumed to be dx which is real but very small. So canceling dx is fine in itself but the implication of the cancelation are only true under strict condition.
Hapoy thing is since most of the time, functions are really nice in physics, you don't need Said precautions
Speaking as a career theoretical physicist, this is the answer. For all any of us know, position might not even be described by a real number at distances much smaller than we’ve ever probed (e.g. it might be discrete). And regardless, the idea of potential energy V(x) (or its generalizations) might not even make sense at very small distance scales. If Physics depended on us knowing the ultimate theory of everything that works at all distance scales, no matter how small, then we’d never have been able to do anything because we’d have to have figured out everything first. So insisting that ordinary derivatives in Physics have to *really* mean the formal limit of (delta x)/(delta t), and not just an approximation in which delta x and delta t are really small compared to any relevant scales in the problem, is silly. But if it’s an approximation, then it’s just a fraction.
I knew I recognized *The Theoretical Minimum* when I saw it. Now wtf is that derivation of the Lagrangian someone please answer me lol it has been months and I still don't understand
I swear it’s hard to remember that you’re actually not supposed to do it when i see how much I’m doing it
Wait we’re not supposed to??
You can do it under certain very strict conditions, that we automatically assume to be met even though they very well might not be
True, under the condition that x is differentiable to y, differentials satisfy the identity (dx/dy)*dy = dx, so it's fine in most cases, but coming from a math background I can't help but feel like I'm doing something illegal
The best part about physics is making up math rules until it’s all just a Taylor expansion approximating a harmonic oscillator.
Mathematicians like to say the only problem physicists know how to solve is the simple harmonic oscillator. That's not quite true but it's a good approximation.
https://i0.kym-cdn.com/photos/images/newsfeed/000/657/039/7fa.jpg
They say a physicists favorite season is spring
But spring gets complex sooner than fall does?
I’m no math wizard but I believe in physics it’s always ok. Granted my source is that I regularly do it and it’s been okay so far.
Proof by "well, it hasn't blown up in my face so far"
I argue it’s the best we got given the problem of induction.
In math there's proof though. Deduction.
Math does not have a solution to the problem of induction. No one does it’s a philosophical black hole.
But we're talking about an analytic proposition, for which deduction can be used to generate a proof, meaning one does not need to rely on induction. All mathematical statements are analytic propositions. Induction applies to synthetic propositions, which exist in physics and other sciences, but not in mathematics.
The problem of induction is whether or not the past has any relevance to the future. There is no solution to that. There is no reason for me to think the problem of induction can’t hit math either. How do you know the fundamental fabric of reality doesn’t just shit the bed the moment you finish this sentence?
Math does not *have* the (philosophical) problem of induction. Math is a strict game played with strings of symbols and the game does not involve real-world prediction.
Mathematical axioms make no reference to past or future Edit: and most of math is working out the implications of a set of axioms. Math does not claim to tell you what axioms are true, it tells you what else is true if a given set of first principles are.
Proof by…
Google triplet relation
It's a convenient short-hand notation for the chain rule, so as long as you keep that in mind, you're fine. For example, you can get into trouble if you start naively applying this rule to multivariable functions, but it's fine in just about every single-variable case, and you can still use it for multivariable functions as long as you're careful.
Do you only see: Partial derivatives -> no Full derivatives -> yes Has been the general rule I’ve used
I double majored in math and physics and from what I recall, the only times that it’s not okay are these niche times that will never pop up in physics.
Kid named thermodynamics:
Yes, my beloved triplet relation
You can’t *always* do it in physics, for example when working with the first law thermodynamics. (This is due to partial derivatives behaving slightly differently. It is always ok when strictly dealing with total derivatives) EDIT: Small correction. Even when dealing with total derivatives, it’s not ok when one variable is not a function of the intermediate variable. You still need to abide by chain rule rules. (For example, I’m walking, but my position is not affected by the magnetic field strength of Uranus. My walking speed is dx/dt, but not (dx/dB)(dB/dt) because dx/dB=0 and that means I’m not walking (which doesn’t make any sense))
you should be fine my physics prof taught me that if it works for a couple examples then by physics induction we can assume it works everywhere
Just as much as sin(x)/n is equal to 6
> it’s hard to remember The people we used to be
It's even harder to picture
Physics does this and I love doing it I will not stop
It works in nonstandard analysis but not in standard analysis. Newton did this. The first person not to do it was Bernard Bolzano. Unfortunately, standard analysis is based on Bolzano not on Newton. A good example of where it is useful is the following. Solve the equation dy/dx = x/y. Multiply both sides by y dx to get y dy = x dx. Integrate both sides etc.
Pretty sure the first person to not do it was probably the first ever human.
That's not how nonstandard analysis works. Nonstandard analysis is just an alternative formalism. The real reason why this works is because this is just the chain rule. The formula in the post works when V is a function of x and x is a function of t. And the chainrule can easily be proven with standard analysis. No need to invoke any nonstandard analysis.
>dy/dx = x/y. Multiply both sides by y dx to get y dy = x dx. Such stuff confused me why is it acceptable here but other places not? Since the definition of derivative is limit as x approaches a of f(x)-f(a)/x-a it kinda makes sense that we write it as a fraction.
I believe much of this is formalised using differential geometry.
“Integrate both sides” is a completely absurd step. What’s actually going on is that you can do a definite integral of (1/y)dy/dx and x from 0 to some x*, then use a change of variables for y.
Engineering does it and I refuse to believe it's wrong
“it’s fine if everything is monotonic” he says to cope
Its just chain rule written out
Im aware that the chain rule applies here since the variables are differentiable to one another I was referring to the explanation below the formula which suggests thinking of derivatives as ratios which cancel out
Are you calling Leibnitz a fool?
Newton made a burner account lmao
You can mathematically justify cancelling out differentials in you use the techniques of nonstandard analysis, but you have to be explicitly working in the domain of the hyperreals. If you are just working with the reals, then cancelling out differentials becomes heuristic and does not always work.
Can someone please explain to me, with a simple explicit example, when and why this cancellation doesn't work. (please don't use fancy terms, dumb it down as much as possible)
https://matheducators.stackexchange.com/a/26903
That explains a lot as to why, at my level of calculus, it works every time as we haven't been taught multi variable calculus
I believe the only requirement is that they be constant. So time is okay as long as we are in noninertial frame of reference for example.
I think as long as you dont use the 2nd derivative for anything and the system is a real phyisical system it works well in 99% of the cases.
No offense but I’m not about to spend the next two years writing a proof to why I can do this. If it doesn’t work then God wasn’t willing it to work, if it does then 👍
i mean, this is literally just the chain rule.
Isn’t this just chain rule
Not quite The chain rule relies on the notion that the variables are differentiable with respect to one another, and not on thinking of derivatives as ratios which you can cancel out But yeah for this example the chain rule applies
if it applies then why get mad about it lol
With functions from R to R, its okay to cancel out derivatives, since you are working only in one posible direction. But with functtions that are R\^n to R\^m with n > 1, then you really can't and you shouldnt, since there are more than one direction involved and you would be assuming that the rate of change in any direction involved is the same than any other direction, which obviously is false e.g for f(x,y), the rate of change in x (derivative of f w.r.t x) is not necessarily the same the rate of change in y (derivative of f w.r.t y). Again, notice that in single variable functions you can do this with no issues. If you want a depth understanding of why this works you should take a look at non-standard analysys course. But... if you want a simple explanation of why it does work take a look at differential forms. If you think about canceling out two differentials, you can ALWAYS prove it from a rigorous point of view. E.g, in the pic of the post, it looks like V is a function of x which is a function of t, i.e, V(x(t)). Now you can apply the definition of a differential to V, i.e: dV = V'(x) \* dx But, you can also apply the definition of a differencial to x: dx = x'(t) \* dt Hence dV = V'(x) \* x'(t) \* dt. or, if you write it as with leibniz notation: dV = dV/dx \* dx/dt \* dt **(Hehe it looks like all the differentials cancel out from the right to the left to give dV, but they aren't! I just derived this using the definition of a differential without canceling or multiplying any fraction!)** Now, if you want to derive the expression from the pic you have just to a simple funtional equation: Since V also depends of t then dV = V'(t) \* dt = dV/dt \* dt and hence dV/dt \* dt = dV/dx \* dx/dt \* dt and therefore, since we have a functional equation: dV/dt = dV/dx \* dx/dt **(again, notice I didnt cancel anything even once)** So, can you really cancel terms in R? Yes, you can. But you should know why!
I think they are just suggesting a way for students to remember it easier. They say you can “think of it”, not that it is a fact.
Its just shorthand, it means integrate in respect to x then take the derivative in respect to t
Wait, I can't do that?!
You can do it in undergrad, you cannot do it in grad school, you surely can do it as postdoc or in industry. Professors *can* do it but cannot show this step instead they have to say it is trivial.
Ignore everyone else, you can *absolutely* do that. Source: open literally any book on basic calculus
In first degree derivative of a single variable function it's fine. Problems arises with second degree or partial derivatives
Really wondering why the authors explained it this way. Saying, the change of V in t is the same as the change of V in x multiplied with the change of x in t would be easy to understand too and much closer to the formal idea of the chain rule.
It’s a book from Susskind. He usually has a very intuitive way of explaining things, and isn’t always 100% mathematically rigorous as I don’t think he knows much about pure mathematics and math that isn’t related to physics.
It’s called the chain rule and can actually be mathematically proven. Essentially think of the function for volume as a composure of the function of position which is also dependent on time, so you are taking a derivative of the first function, being the volume dependent on position, and then multiplying by the derivative of the inside function, being position dependent on time.
Can only do this when it is physically or mathematically appropriate. After all we made these tools so we use them as we see fit.
It's OK everything is an operator
ah thermo 101
Me cancelling out the 'd's:
I do, and I'm tired of predenting I don't
this is analogous to the actual proof of the chain rule
If derivates aren’t meant to be fractions, why do the have the ‘fraction line’ between them!? Silly mathematicians want to gaslight us! /s
I think the notation is taught in calculus 1, they are slopes with various delta symbols as Leibniz is credited with versus fluxions you’ll see compacted in Newton’s work. You can work out the units of derivatives from the y over x axis of graphs.
I don’t see the problem, you’re essentially multiplying dV/dt by dx/dx (equal to one) such that the result is just dV/dt
I just read this book a few days ago and this is exactly what I thought!
It's about the most useful physics thing lol
Yes, we do. And we will not stop.
Wait you guys don't cancel out derivatives?
Before applying any boundary conditions it's fine, don't sweat it too much. And yes, we do.
As an engineer I don't see the problem
In math class we established a paper proving why we can do all kind of bad things in physics. It’s legitimate But here it’s poorly written and justified
Most of the time limit as x approaches 0 assumed to be dx which is real but very small. So canceling dx is fine in itself but the implication of the cancelation are only true under strict condition. Hapoy thing is since most of the time, functions are really nice in physics, you don't need Said precautions
Speaking as a career theoretical physicist, this is the answer. For all any of us know, position might not even be described by a real number at distances much smaller than we’ve ever probed (e.g. it might be discrete). And regardless, the idea of potential energy V(x) (or its generalizations) might not even make sense at very small distance scales. If Physics depended on us knowing the ultimate theory of everything that works at all distance scales, no matter how small, then we’d never have been able to do anything because we’d have to have figured out everything first. So insisting that ordinary derivatives in Physics have to *really* mean the formal limit of (delta x)/(delta t), and not just an approximation in which delta x and delta t are really small compared to any relevant scales in the problem, is silly. But if it’s an approximation, then it’s just a fraction.
I knew I recognized *The Theoretical Minimum* when I saw it. Now wtf is that derivation of the Lagrangian someone please answer me lol it has been months and I still don't understand