Of course the issue stems from mathematicians and physicists each consolidating on different conventions!! Mathematicians d/dt Ordinary Derivative ∂/∂t (Total) Partial Derivative Physicists d/dx Total Derivative ∂/∂x (Explicit) Partial Derivative Mathematicians stuck around using d/dt for one variable case and ∂/∂t for just a multivariable version of it. So ∂/∂t means the "total" partial derivative with respect to an independent variable. Since 1800s, mathematics was all about about functions, not equations, so they got around the issue by defining explicit functions. Physicists on the other hand use d/dt as total derivative to mean every chain leading to that terminal independent variable, and use ∂/∂t to mean the "explicit" partial derivative leading directly to that variable. I left a link in the video description detailing the difference in convention.

"This is why we cant have nice things" 😂

I have a phd in math, and i still hadn't understood some partial derivative notations in thermodynamics... until i watched this video. Thank you!

I knew something was wrong with the notation!! For years I have been working my way around this pesky partial symbols without thinking about the root problem. Excellent video!!

i knew there was a reason i didnt like this notation but i could never quite put my finger on it

This was so good it deserved money

Something I don't understand about your f(x, y, z, t) example at 2:32 is that if we take the total derivative with respect to t then we add a df/dz dz/dt term (with partial ds). dz/dt is 0 so it evaluates to the same thing as before. This seems to be working as intended in my opinion.

Physicist here, this is the way I learnt it. Say we have u(x(t), y(t), z, t) where u is a function of x, y, z and t and where x and y both depend on t but z doesn't. Then the total derivative of u with respect to t would be written as du/dt = ∂u/∂x * dx/dt + ∂u/∂y * dy/dt + ∂u/∂z * dz/dt + ∂u/∂t Here ∂u/∂[.] refers to the explicit partial derivative where all variables except [.] are interpreted as constants. In physics we say dz/dt = 0 which then gives the correct formula. This removes any ambiguity that mathematics has.

As a math students, I would say it is just a bad habit not to think about what the actual function is. It’s actually f:U⊂R^3->R, f(x1,x2,x3), g:R->R^3, g(t)=(x(t),y(t),t), and u(t)=f(g(t)), and the problem is solved automatically. du/dt is well defined, ∂g/∂xi is well defined, not ambiguity. Often people use the notion without care, this is just the consequence of such carelessness, not of the notation.

This used to confuse the hell out of me when I studied physics. From a pure mathematical perspective, we are really talking about two different (but related) functions. The first one is the function (x, y, t)->u(x, y, t), where x, y and t are free variables, while the second one is the composition t -> u(x(t), y(t), t), where x and y are now functions of their own. The latter function may be called g := u \circ (x, y, id) . The confusion comes from the fact that we are treating x and y both as functions and as free variables depending on the context, and that context is what you have to keep in mind when doing it. Now (\del u / \del t) is just d_3(u) (i.e. the differentiation of u along the 3rd variable), while what is called (du/dt) is actually d_1(g). Of course, it would be too tiresome to write this out each time, so it is fine to use non-rigorous shorthands as long as the context is implicitly understood, but I still wish I learned to think about it this way earlier. Edit: I see somebody already pointed this out in another comment.

wow, thank you for finally making me understand the parenthesis+subscript in thermodynamics classes... and why one fluid dynamics book was very serious (but very unclear) about calling it material derivative

Cool video. I remember how the total and partial derivative w.r.t. time used to confuse the hell out of me in fluid mechanics.

This has cleared up so many things in just 10 minutes! Ive always felt the notation was slightly off but I did not know why

A new upload from epsilon delta, was waiting for this! Yay! Love your work ❤

THANK YOU FOR ADRESSING THESE FRUSTRATING AMBIGUITIES!

Instantly subscribed. This is god tier, and not just the same regurgitated lesson I've seen a million times! I've had partials explained like 3 times now and it's never been so clear how to truly utilize this notation. I'd become aware of functionals when learning Lagrangian mechanics a couple months ago and this effortlessly cleared up a serious tension i hadn't even been able to understand yet. God tier video, i thank god and you that you made it

I’ve always had an issue with the apparent ambiguity of partial derivatives. You explained it beautifully.

Your visualizations honestly helped cement these kinds of concepts more than whole classes that I’ve taken… great job! I’ve always been left wanting to know more about the math intuition behind things like wavefunctions or heat transfer 😊

great video, always happy for a new upload

I think you explained the difference between partial and total derivatives better than my rational mechanics professor. I think now I actually visualise the difference correctly, so really cool video!

These are the most useful yet easy to understand math explainers I have seen. So straightforward. 3b1b videos are too clever for me 😅

Clear and engaging, well done.

Thanks for the video. This has always confused me *a lot*. Now, at least, I learned that I wasn't totally stupid.

That's amazing. Always found it so confusing. As a mathematician studying physics, the most confusing step is Legendre transformation.

Your visuals are top notch

I’m not quite into material science, however, I’m a fan of machine learning in which partial derivatives come up often. I never considered the ambiguity of partial derivatives, but this video did a great job laying them out. I’ll try to follow the scheme you shared for whatever work I share, so thanks for awesome content.

Another fantastic video!!👍👍

Great video man!

sweet rant dude. I totally agree 👍

What a great channel!

Maybe I am not getting the main point you are trying to convey, but it cleared some of my other doubts and confusions ❤

This reminds me of the introductory chapter of Sussman and Wisdom's “Functional Differential Geometry”. Nicely done.

Lol! Solid ending.

I bet you don’t know how much this video has saved me just in time 😎 🕶️👌🏻😭

oh hey now i kinda see why we were using the parentheses and subscripts for partials in thermodynamics and nowhere else. that was always weird to me. cool!

I don't see the problem, if z doesn't depend on t, then you can take the total derivative and dz/dt is just 0. Whether or not all paths lead to t is irrelevant.

we need a type system for math that you can "import" from to give you some set of expressions or functions but its not an actual program its just like virtual. But the type checker could still work

Great video. In Spivak's Calculus on Manifolds, he uses the notations \partial_1 f, \partial_2 f, etc to indicate the partial derivative of f with respect to the 1st argument, 2nd argument, etc. This sort of notation is similar to how when writing computer code you can call a function with the first argument and second argument specified however you like but what really matters is the first argument is the first argument regardless of name.

4:45 So THAT'S what it means... I've been seeing that notation in my thermodynamics course all over and was wondering what the point of it was. It's why whenever I do an exercise I'm always confused whether, when differentiation with respect to T for example, I should take P to be constant, or replace it with its expression as a function of T. This really was very helpful. We're not taking multi-variable calculus as a module in maths until next year, so we've had to navigate our way through it using butchered physics math. Thank you for this video.

Ahh I knew there was something about partials that was tripping me up and this video hit it right on the mark! Also I genuinely wish my p-chem prof explained the equational approach to me. I had no idea what the difference the subscripts made before watching this video.

Not necessarily a correction, but to add some context to the temperature function: the "original" temperature function u(x,y,t) is dependent on 3 independent variables and there is no meaning to the idea of (non-zero) dx/dt or dy/dt since x and y are independent variables. A different scenario you can consider is taking a _path_ (x(t),y(t)) through the 2d space and consider the temperature along that path through time. This temperature function through time would just be u(t) (or use a different name for the function like @alexatg1820 said, let's call it U(t)). We can then say U(t) = u(x(t),y(t),t). And we can also make sense of total vs partial derivatives as d/dt U = (δ/δt u)|_(x(t),y(t),t) + (δ/δx u)|_(x(t),y(t),t) dx(t)/dt + (δ/δy u)|_(x(t),y(t),t) dy(t)/dt. Note that the |_ means that you plug in the underscored variables after calculating the partial derivative. This is an important idea for something like Liouville's theorem, which says that d/dt ρ = 0, where ρ is the phase-space distribution function. It seems strange at first that something that clearly has time dependence like ρ would have zero time derivative! But what it's really saying is that for any trajectory (q(t),p(t)) that satisfies Hamilton's equations, the partial time derivative of ρ will satisfy (δ/δt ρ)|_(q(t),p(t),t) = -(δ/δq ρ)|_(q(t),p(t),t) dq(t)/dt - (δ/δp ρ)|_(q(t),p(t),t) dp(t)/dt. So the d/dt ρ = 0 is really talking about the possible "valid" trajectories through time, not ρ by itself. In fact, the underlying idea is very simple: the weighting of a trajectory to the probability distribution shouldn't change through time, thus d/dt ρ = 0.

I think the most general and reliable solution is to always consider the computational graph. Algebraic notations can only take you so far before it becomes overencumbered. It's precisely how modern backpropagation algorithm is implemented.

As a mathematician this annoyed me so much in the physics and engineering lectures I had. People overcomplicate things so much for no good reason - just write your shit down properly and suddenly all the problems resolve themselves

Good choice of notation; giving Cyrillic some love. If I ever need to crunch or reference a PDE with such a dependency arrangements, I'm gunna use this notation. And when my colleagues are like WTF? I'm going to send them to this video, and they'll likely adopt this, too.

Thank you for using examples in physics! I learnt about the material derivative in the context of fluid dynamics, maybe it would be easier to explain the operator if we imagine a particle flowing in a velocity field

Very interesting, tough a little hard for my level. However, I can understand the ambiguity in notation, thanks to the dependency graphs, and despite the vocabulary. But my most complete incomprehension is of a logical nature and is as follows: said very simply, how is possible that someone decides that x(t) is fixed when t varies ? My question may sound silly to specialists, but this is what is really blocking me. To me, fixing something that is bound to vary cannot be decided, because there is no choice. I hope I'm clear.

I use ∂ for "single-layer" derivatives (do not go into dependencies of dependencies) and d for "every-layer" derivatives (go into every possible sub-dependency, i.e. a total derivative) - regardless of the quantity of variables of a function (one or more). This solves the ambiguity presented in this video still in a third way (which seems cleaner, in my opinion). A consequence is that it no longer holds that d is for when there is only one variable and ∂ is for when there are more (which has always added more confusion than usefulness, in my opinion). Another consequence is that, for a function with a single variable, f(t), we have that ∂f/∂t (its derivative - no word "partial" needed) equals df/dt (its total derivative). Of course, mathematics is kept intact and this is just an alternative notation that changes the meaning of the symbols ∂ and d - also, one should always be cautious of what is a function and what is a variable.

Thanks a lot!!! Nice video!!! I would also like to watch something about the differentials in thermodynamics because I don't want get why using differentials instead of simply derivatives... Thanks!!!🤩🤗

The way U understood is, the partial derivative is the derivative of a function with respect to one argument, *while fixing all other arguments constant*. I see no problem there. Partial derivative of u is undefined if it isn't defined as a function.

For u(x(t), y(t), t, z) I would have said du/dt (total) still encompasses t, x(t), y(t), with z being irrelevant. And ∂u/∂t (partial) identifies the direct dependence on t, without x or y

This would also be very enlightening for students learning the Lagrangian formalism for the first time, with its quirky functional derivative symbols.

A few things that I would say: 1. There is never any ambiguity if you go back to differences, instead of differentials. With differences you can write out exactly what you intend to difference (which is basically what differentiation is shorthand for) 2. Partial derivatives are defined functionally. So if a function does not depend explicitly on a variable it's partial derivative with respect to that variable can not be taken. This is why in PDEs partial derivatives are usually written with a number subscript to indicate the position of the variable with which a partial derivative is being taken. 3. Yes, you will say but what about the chain rule for partial derivatives? This is actually an abuse of notation. The actual chain rule should read: Partial f ○ phi / partial x = partial f / partial y1 • partial phi^1 / partial x + ..... + partial f / partial yn • partial phi^n / partial x Where phi is a differomorphism/change of coordinates. Sorry for not knowing how to write the partial symbol on yt

Very helpful.

Please, no, do not use "Д" or "д" (or, while we are at it, "Ð" or "ð")! I am running out if symbols that I can use when I was to use "d" or something similar.

Sooooo much of mathematics and physics is context dependent. Yes, the way "functions" and their derivatives work in physics is one. Or the Einstein summation notation. Keep your eyes open and it's all over though, right down to things you'd think would be set in stone like order of operations

This seems to be more about the imprecise notation rather than a flaw rooted in mathematics. Mathematics cant to anything about physicist abusing various notations.

Something about this video makes me want to do multivariate calculus with my hands at my side and my feet a-flailing.

Thanks for the informative video. Quick question, at 4:46 shouldn't the direct partial be written with 3 subscripts, x, y, z? as the direct partial assumes x,y,z are constant

It’s all very clear once you learn differential geometry.

Is this why we use the big D derivative for the navier stokes equation?

"Spaghetti code of mathematics" exactly lol

7:44 where can I read more about this diagram? I'm interested of its use

Engineer here. This notation of using the partial derivative symbol for the "complete [partial] derivative" with respect to t (or any other variable, but let's stick to t) seems obfuscatory. For the example function u=f(t,x(t),y(t);z) (with dz/dt=0), the quantity [ ∂f/∂t+(∂f/∂x)(dz/dt)+(∂f/∂y)(dy/dt) ] *is* the total derivative of f wrt t --- that f coincidentally has other arguments (in this case, z) which are independent of t doesn't change that. (In fact, one might as well include the term (∂f/∂z)(dz/dt), except óila!, dz/dt=0 and the term vanishes.) Using ∂ for anything other than the explicit partial derivative just seems like a pedagogical nightmare.

I hope I would had this video before learning thermodynamics a decade ago

0:19s terrified me. I thought my computer had finally had it with me.

I learned how to solve a bunch of different types of dif eqs in various math and physics courses, but whenever I tried to deeply understand the notation (like I would for a new programming language) it felt like my brain was returning "incomplete and inconsistent notation" errors. No one told me that there are technically several extra types of "d"'s or that physics & math sometimes used the symbols in slightly different ways. Before I started finding videos like this I thought I'd have to either dedicate a year of my life to reverse engineering the subject or wait till AI got advanced enough to explain it to me.

Well you could always use the total derivative. Two unrelated variables are functions of each other, as constants.

Inverse notation, powers, and derivative notation video next?

9:12 : What if the graph is not three layers deep, but is instead four or more layers deep?

How is equation vs function being defined here?

I think part of the ambiguity is that the same glyph is often used a placeholder (to distinguish input position) *and* as an actual variable used as an input. This naming conflict arises even in one dimension, it just typically does not cause any issues. To illustrate, suppose we have a function f that takes a single variable as input. If I write df/dx(x^2), do I mean to evaluate the derivative of f (a map R -> R) at the point x^2, or do I mean the derivative of the function x ↦ f(x^2)? For this reason, I tend to use notation like D^α f. For a function f taking n inputs, the superscript α is a tuple of length n whose entries are nonnegative integers. The ith entry of α tells you how many times the ith input position to f is differentiated. The drawback here is that Clairaut's theorem does not always hold, but I don't encounter these too often in my work

See Johnathan Bartlett's minor change to differential notation. He is a computer graphics programmer, rather than a mathematician. There is a bug in standard calculation that causes people to say "dy/dx is not really a fraction". But they can be if you are careful. You can even solve for "dx/dy" without too much trouble after you fix the notation. It basically means to ONLY use implicit differentiation for everything. That causes dividing by the var we are respect to happens later, as does holding things constant. The second derivative is kind of surprising. It's a bug to assume that d^2[x]=0, though it usually is. Explicitly nest second derivative, and it's richer than what you normally get: d[ dy/dx ]/dx = d[ dy * dx^(-1) ]/dx = (dy * d[dx^(-1) + d[dy] * dx^(-1)])/dx = (dy * -dx^(-2)*d[dx] + d^2[y]/dx)/dx = (-dy * d^2[x]/(dx^2) + d^2[y]/dx)/dx = d^2[y]/(dx^2) - (dy/dx)*(d^2[x]/(dx^2)) wow that subtracted term looks strange. these get huge for higher derivatives too. "acceleration is d^2[y]/(dx^2)" is only true when d^2[x]=0. And when x is a line, it's true: d[c]=0 "c is constant" d[d[t]] = "t is a line" d[] is an operator with binary operations, and it is recursively evaluated. we treat everything as variable. declaring "a is constant" is done exlicitly like "d[a]=0". To simplify with infinitesimals you could say that "b is positive infinitesimal", we could use dual numbers like "d[b] > 0 and d[b]^2=0" ... though maybe "d[b]^2 -> 0" is a one-way relation (not sure). Anyways, the point is to make Calculus fit on one page by explicitly defining the implicit differentiation operator d[]: d[a+b] = d[a] + d[b] d[a-b] = d[a + -b] d[a*b] = d[a]*b + a*d[b] d[a/b] = d[a * b^(-1)] d[a^b] = b*a^(b-1)*d[a] + log_e[a] * a^b * d[b] d[log_a[b]] = -log_e[b] /(a * log_e[a]^2) * d[a] + 1/(b * log_e[a]) * d[b] S[d[f]] = f - f_0; d[f_0]=0 d[d[x]] = d^2[x] d[x] = dx The reason for doing this is that when you implicitly differentiate, you don't lose the information about what var it was respect to: f = x^2 + y^2 df = 2x dx + 2y dy assuming dy=0, df/dx = 2x + 2y(dy/dx) = 2x assuming dx=0, df/dy = 2y Doing this, the derivative notation can be tedious, but it is legitimately algebraic now.

Can't we just make z depend on t in a trivial way so that dz/dt = 0?

please do a video on cantor schroder theorem.

At (6:08) is that actually a PVT diagram of a general substance that isn't water? Water expands on freezing so there should be some region of solid with volume greater (less dense) than some region of liquid. But excellent video either way.

At 2:30, I don't really see the problem with just writing the total derivative du/dt. You already have the t there indicating that you should ignore Z and focus on the time-dependent variables, and I'm not really sure what the 'real' total derivative including Z is supposed to be. Obviously there are still cases where the notation gets ambiguous, but I don't think this is one of them.

Explaining the whole thing (for mathematicians) in few paragraphs: There are two setups for partial derivative notations. *1) Setup number one.* Have a function of several (say, two) variables f: (x,y)-->f(x,y). The notation ∂f/∂x means the usual partial derivative of f w.r.t. its first argument (the one named x) and evaluated at (x,y). *2) Setup number two.* Let's make the case of two variables. Have a constraint F(x,y)=0, and you want to define a rate of change ∂y/∂x between two variables. Under usual assumptions, F(x,y)=0 represents a 1-submanifold of the plane. The notation ∂y/∂x means the following: - utilize the Implicit Function Theorem (where you can) to locally write y=g(x) such that F(x,g(x))=0, - then compute the usual derivative dg(x)/dx. By the way, the IFT gives you an expression of dg(x)/dx in terms of F and its partial derivatives (in the sense of Setup n.1). Let's now define *Setup n.2* in the case of constraints of more than two variables. In this case, you need to specify further constraints. For example, suppose you want to compute the rate of change ∂y/∂x of the variable y w.r.t. to the variable x under the constraint F(x,y,z)=0. You need a curve to project into the (x,y) plane to which to apply the IFT and compute ∂y/∂x according to Setup n.2. But F(x,y,z)=0 only gives you a surface in 3-space. So you need to specify an _additional_ equation, say h(x,y,z)=0, and then the notation (∂y/∂x)_(h(x,y,z)=0) makes sense (under usual transversality/genericity assumptions).

1:20 I don't understand how does x and y depend on t in the example. Are coordinates not constant?

its my opinion that f(x,y,z,t) is itself flawed notation as im pretty sure its supposed to be taken that the independent variables are, well ... *independent* (i know its technically not flawed conventionally speaking since its used everywhere, but i personally like writing f(x(t), y(t), z, s(t)) instead, where s(t) = t, removing the dependence)

I fully support using дu/дt as the notation for an implicit derivative. Math only uses latin and greek letters, when there's all of cyrillic, hebrew, arabic, hindi, chinese, korean, japanese, and ethiopic scripts all for free.

Thank

I want to defend the “material derivative” you mention at the end in a joke. It actually is really important to distinguish between the material derivative and the total derivative, and here is why. Suppose you are standing above a fluid, looking down at it (formally, I mean an Eulerian description of a fluid). You are interested in the change in some property, say temperature or fluid velocity or something, in the fluid. The fluid has some velocity function u(x,y,z,t), (u should be a three vector but I can’t draw arrows lol) which depends on the position in the fluid in your coordinate system, and the time. Now I ask, what is the time dependence of my property, e.g. fluid velocity? The problem is, I could mean two different things here. If I’m looking from above at a fluid that is in some steady flow state, there might be no change in the velocity field at any time: the fluid is just moving in the same pattern it always has been, and u does not change with time. Sure, u might be different at different spatial coordinates, but from my perspective, asking about the change of the fluid properties at a particular x, y, z with time, I would only get \partial u/\partial t. Now I ask: how does each fluid parcel (each fluid element that moves with the fluid) experience a change in time? Well now you want to account for HOW THE PARCEL MOVES IN x, y, z, as it is moving to new points. Ok, so simple, you want something that looks like \partial u/\partial t + \partial u/\partial x dx/dt + same for y and z. Which LOOKS LIKE the total derivative. BUT IT ISNT! If we actually compute this, well we are unfortunately stuck because x,y,z is MY FIXED COORDINATE SYSTEM! None of the coordinates depend on time! The x that enters the thing I want is not the same as the x in my coordinate system. So in fact, the expression I wrote above is just \partial u/\partial t, and I still haven’t computed the change of the fluid from the perspective of the parcel. So the thing you actually compute is not the total derivative, because it doesn’t really make sense here, x,y,z don’t depend on t. Instead, you compute the MATERIAL DERIVATIVE, which is D/Dt = \partial/\partial t + velocity_vector\cdot gradient_vector It’s just that the velocity vector is not the three vector of dx/dr, because again, x,y,z are fixed coordinates. I think it’s unfortunate that you didn’t mention this, because I think the notation of the material derivative is one of the best examples of GOOD notation that is clear about what you’re actually computing. It’s just when you write it like the total derivative, you’re missing that the x that appears in the right hand side is a different x to the coordinate system x (namely, that x is the parcel position in the coordinates). The material derivative is one of the nice things we have.

Ambiguity with partial 👌 notation

Hm.... I thought that in the case of one extra variable z, we should not invent any additional notation, because if we do not suppose that z = z(t), then both partial or full derivative of z by t is zero. So, basically we just add zero when use du/dt. But I have never thought about the case, when we want just to omit dependency of z on t even if we know that this dependency exists. Interesting. Although I don't like the proposed notation. Maybe something like Du/D_xy t would be better. Imho

Some remarks on the equation at 0.58. This is not necessarily true. It depends on how you represent z. If you write z as the function z(x,y) then it has only two possible partial derivatives: the one with respect to x and the one with respect to y. No t partial derivative exists in this setting. The partial derivatives refers to positions in the function signature. If you then let x and y depend on s and t, you get a function f(t,s) = z(x(t,s), y(t,s)). The full derivative of f with respect to t is then the partial derivative with respect to t of the second expression using the chain rule, as in your expression at 0.58. My point is that partial derivatives should be thought of as derivatives of the argument positions in the function signature. Let us take an example: Take the function f(t, x) = t * x, where x = t^2. If we replace x by t^2 so that we get the function g(t) = t^3, the full (and partial) derivative of this with respect to t is 3t^2. But the partial derivative of f with respect to t is x, which is t^2.

One way i kinda managed to by-pass the ambiguity is by considering two different things, let's give an example let u(x(t), y(t), t, z), if i want to know how u changes with respect the t input i would write it as ∂u/∂t, but if i want the total partial i would think it as an operator (∂/∂t)(u). With this i feel like i am asking the total partial derivative, since it feels like i am asking all the changes u when i move t. But i recognize that this solution is kinda jank

"Complete partial" is a funny name. Is there really a difference between it and a total though? Just take every direct or nth-order-implicit parameter which does not depend on, say, t to be a constant wrt t. Those terms are reduced to 0.

Issue: What is a differential of an irrational argument? Let a= some rational approximation, and A be the irrational number itself (if that makes sense). Then A - a > dA and there is no way a + dA > A can there be a length commensurate with all lengths (differential of length)? then all numbers would be rational

Would it kill mathematicians to just write out something instead of using 83 different notations? I just finished calculus 3 and my biggest problem was honestly just trying to interpret e ridiculous and confusing notation

Can you define each derivative (name, common notation(s), your proposed notations, context or graph of the input function dependency, the intuitive meaning of the derivative, and the rigorous mathematical definition of the derivative) in a single summary video or chart? I want to make sure that I perfectly understand this topic, and the explanations in the middle, while helpful for an introduction, make comparisons difficult. A follow-up video with each proposal flipped through quickly and in a bare-bones manner would be good. In particular, the names are not completely clear to me. For "rigorous mathematical definition", I mean any surrounding context about how to define the functions and intermediate variables, any relevant short-hand expansions of the formula (such as at 2:10), and perhaps even the limit definition which underlies it all.

personally, I would just say that the partial z with respect to t is zero, then the notation holds.

2:44 why can't you just do du/dt = pu/pt + pu/px * dx/dt + pu/py * dy/dt + pu/pz * dz/dt? Since z doesn''t depend on t, dz/dt = 0 and that's it, I don't see the problem.

6:04 - I don't think the notation (∂P/∂T)_V/n could possibly unambiguously indicate a derivative. Why? Because the equation of state PV=nRT (assuming R is a constant) is one constraint F(P,V,n,T)=0 in R^4, therefore defining a 3-submanifold of R^4. Its projection onto the (P, T) plane is therefore not a curve. You need a _curve_ (1-dim. constraint) φ(P,T)=0 in the (P, T) plane R^2 in order to compute a partial derivative ∂P/∂T (along such constraint, via the implicit function theorem). So, you would need another relation (beside V/n=c) between the 4 variables P, V, n, T - that is another 3-submanifold of R^4. The notation (∂P/∂T)_V,n on the contrary is unambiguous: you have specified _three_ 3-submanifolds of R^4 (namely: PV=nRT, V=c1, n=c2 - for arbitrary given constants c1 and c2), thereby determining a curve (1-submanifold) of R^4 whose projection onto the (P, T) plane will be a curve. [This works in general assuming suitable "general position" assumptions, such as certain ranks be maximal etc, like you don't want the resulting curve to be included in a fiber of R^4-> R^2 or stuff like that] Notice that your example with the three alloys works because in that case you're in R^3, so you need only one additional constraint (you chose x1/x2=c) beside your "equation of state" x1+x2+x3=1.

TL;DR: The partial derivative symbol does not denote the level of partial indirection. For the thumbnail, u(t):=f(y(t),z(t),w(t),t) or u(t,w)=f(y(t,z(t),w,t) do clearly have different partial derivatives after t.

I hoped there'd be a new, "better" solution at the end..

Indeed the notation with (partial) derivatives is confusing, why has this not been adressed

The issue is not with the derivative, it’s with whether you are taking a derivative of a function OR a function composition. If u depends on x,y,z. Then you can ONLY differentiate with respect to x, y, or z. That’s it. Plain and simple. If you want to differentiate with respect to t, it makes no sense and I mean that. However, if you make a composite function where x,y,z all or some depend on t, then you now have a *completely different function*. You have an outside function and and an inside(s) function. It is *not* possible to differentiate u with respect to t. But the composite function (emphasis on composite function) can be differentiated with respect to t. And if all 3 x,y,z depend on t, then the composite function is now a **single variable** function, while the outside function was a **multi variable** function. Do you see how an outside function and the full composite function are 2 completely different functions? **Always and forever** a function can **only** be differentiated with respect to its variables. But if you **overload** a function to mean different things, that’s where the confusion comes in. (A function is just a name given to a mapping. If I call a certain mapping h, and some other mapping h, do you see how confusion comes? There nothing wrong with the derivative, it’s your notation of the function you’re differentiating. Writing u(x,y,z) is **completely different** than u(x,y(t), z). These are 2 different functions. The first u can be differentiated with respect to x, y, or z. Not b or t or s or k. The second is a composite function. It’s domain is not xyz, but xtz. The outside function can be differentiated with respect to y, but not the composite function. That’s all) There is no “total” derivative. Just derivatives and partial derivatives. Single or multivariable calculus. And if you form a composite function, now you need the chain rule for the outside and inside functions.

The most confusing part of calc 3 solved yay

Can you please share the source code ?

Just write down the limit definition before using a notation - no confusion arises. For example, what you call "total derivative" is just lim_{h to 0} 1/h[u(x(t+h), y(t+h), t+h) - u(x(t), y(t), t)], and what you call the "direct partial" or "the explicit partial" is just lim_{h to 0} 1/h[u(x(t), y(t), t+h) - u(x(t), y(t), t)].