Відео

I (as a mathematician) am not sure about any "Ambiguity" here, maybe it's a problem for physicists? The problem is just sloppy notation. If you suppress the dependency of the functions (not variables!) x = x(t) and y = y(t) on the time t for convenience, it might be helpful to state it explicitly before any calculations are done. But of course there is difference in the partial derivative of a function with three variables u = u(x, y, t) and a function depending on time and two parametrised curves u = u(x(t), y(t), t). For the former taking the derivative in t is just taking a partial derivative of the function u in one variable, for the latter you actually try to differentiate a function with indirect dependencies u = u(x(t), y(t), t) in the variable t (hence t, x(t), y(t)) so that the chain rule leads to the total derivative of u(x(t), y(t), t).

I had a teacher who was using a lower case delta (instead of your striked round d) to make the difference. Probably easier for handwriting. I remember maybe students were upset by this teacher using a different convention but I liked the clarity.

For me the background music puts ring theory into a mindless elevator.

Distinction without a difference.

In thermodynamics, there's another convention: You subscript the partial derivatives with the variables you keep constant. for example (partial V/partial T)_p means the change of the volume with temperature when pressure is kept constant. I think using that notation universally would solve all ambiguities. For example, in your example with direct and indirect paths, (partial f/partial t)_x,y,z,s means x, y and z are left constant, thus only the explicit dependence on t is considered. On the other hand, (partial f/partial t)_s,z means that the dependence of x and y from t is considered, as it is only s and z which are kept constant. The non-partial derivative would then simply denote the case that there are no variables that are held constant. Now in the example, s and z don't have any dependence on t (ds/dt = dz/dt=0), therefore df/dt = (partial f/partial t)_s,z. Or in more general terms, variables that don't depend either directly or indirectly on the variable in respect to which differentiation occurs (and only those) can always be omitted in the list of constant terms, That is, (partial f/partial t)_x,y,z,s= (partial f/partial t)_x,y because neither z nor s does depend on t. The visual picture would be that the dependencies of the variables in the subscript are simply cut out of the dependency graph.

What do you think of $PRICK? It is ALREADY possible to withdraw money from it and play it at the same time!

Nice video, this broke down a few things I'd just taken for granted. As someone with a physics background, it was also great to see things from another perspective. One small correction for the triangular graph: the red arrow's head should be in the bottom left corner, where you have 100% silver. As it is, you end up with Ag:Au:Cu = 60:40:0, which doesn't satisfy the original premise. Those graphs can be pretty tricky to read, angled tick marks would definitely have helped!

I do like the idea of using Eth, that one extra stroke ð makes it distinct enough to not mix up with ∂ and d In the Lagrangian treatment of field theory, if your Lagrangian is not explicitly dependent on time or space then energy or momentum (respectively) are conserved (practically the most important cases of Noether's theorem) when you try to write these down, you run into this notational problem exactly, and you have to write funky equations, like (∂L/∂t)_explicit = 0. If only English kept the letter ð, it would be reasonably accessible to use here as well. Also, Thorn makes a much better alternative to :P so that could've been nice :Þ :þ

d😂/d❤=∆❤/∆👆=😂+ d❤/d👆= ∆👆/∆😂+ð👆/ð😂 Derivatives of emojis

Really nice illustration. When I learned the distribution theory, books usually just introduce the "test function" by showing the f(x) = { exp(-1/x) (x>0); 0 (x<=0) } as an example, but never talked about how they came up with this 🤣

How is dividing by dt an abuse of notation? I thought it was an essential part of deriving calculus expressions

during my grad in engineering and later PhD, the convention is Du/dt for total derivative. mostly in the context of fluids, heat-mass systems, variational calculus, etc. I think it helps a lot

Notationally, 1:37 is wrong. You defined y in 2 different ways. You can’t write dy/dt and dy/dx. It’s either one or the other. Each implies y is a single variable function of t or x. Which is it? You need to write dy(x(t)) /dt on the left hand side. Now you don’t need to use the chain rule. Just differentiate with respect to t. But if you do, then the chain rule states that the derivative of the composite function is equal to a special combination of derivatives of the other functions. Note that a composite function is a brand new, 3rd function

Can someone please explain to me the difference between the total derivative and the total partial derivative as shown in the video?

From my perspective teaching calculus, if we start with f(x,y,t) and the want to make x,y depend on t, we need to change of one the two ‘t’ variable with something else, eg. we have f(x,s,t1), x(t2), y(t2) and t1(t2). This is because t1 represents a coordinate of a copy of R3, the domain of f, and t2 does not, it represents a coordinate in some other space, the domain of x and y. Fundamentally every derivative is a derivative along a parameterized curve, but in the f(x,y,t) example the notation is obscuring which curve that is.

The solution is clearly maplet/lambda expressions, which can turn any multivariable function into a single-variable function. Instead of ∂f/∂x, we'd write (x, y) ↦ (z ↦ f(z, y))'(x), and instead of ∂f/∂y, we'd write (x, y) ↦ (z ↦ f(x, z))'(y). Much less unwieldy! Actually undecided on how much of a joke this is, cause it can be cleaned up significantly: ∂f/∂x(x, y) = f(-, y)'(x), ∂f/∂y(x, y) = f(x, -)'(y).

4:15 종속변수와 함수는 구분해야해

Is "abstractify" actually a proper REAL word, per language authorities? Or is it just another one of those words that's "creeping in"?

5:58 if abstracted to functions defined over manifolds, the usage of f here for "both" of these functions is completely correct-because they are not actually distinct functions, just the same function represented using distinct coordinate charts. more generally, all of these rules and dependencies (especially your quite nice arrow notation) are natural in the context of manifolds and explicit charts. this can be quite helpful, for instance, in understanding definitions of thermodynamic quantities, where it's often required to take derivatives with other specified variables held constant (together, all these variables, varying or constant, should define one specific chart on the manifold of thermodynamic states; if they do not, then the derivative is not well-defined). indeed, something which can be confusing, such as cases where it makes sense to consider parameters x and y as independent but then for some calculations also consider them as dependent on other parameter(s) [e.g. time, or even that they are confined to some constraint surface], can be understood nicely using pullbacks/equalizers in the category of manifolds (and charts).

what if instead of your dashed-partials we use instead: du=u_x partial t + u_y partial t to tell is the chain rule as interpreted as "in line" partial t == partial "previous argument" over partial t Does it leads to an equivalent interpretation for the dashed-differentials application?

i have seen the 3 symbol usage convention before in physics lectures

Outstanding. Well done. Discussion of the Lagrangian is particularly great.

When I was first introduced to the partial derivative my intuition was that it was kind of an unnecessary notation change and I'm not fully away from that. It seems like the way physicists use the normal differential is fairly similar to if not the same as the material derivative Df/Dt.

... and there's also the material derivative Df/Dt in fluid dynamic, and also (but not always) the total derivative Du/Dt in general relativity. But great vid' to highlight these conventions! (I've always feel a bit confused, but couldn't pinpoint why)

I'm an advocate for throwing out "partial differentials" entirely and using total differentials from start to finish. It works out perfectly, and actually makes the notation simpler.

In mathematics we would consider the t as a driving variable and then consider a relationship of tau at one level higher with tau = t then we allow for chains with direct dependence and so can ignore the distinction

We have the same books :D

It’s hard to get a point from your talk. I am confident that in proper mathematics there is no issue here. Only when you start using 19th century improper mathematics like physists still tend to do you are introducing unclearities.

Let me tell you a little secret fellows: Take any function from R^n to R^m, say F(x,y)=(f1(x,y),f2(x,y)). You can find the derivative of f1 by simply differentiating it like its a 1 variable function, just remember the derivative of x is not longer 1, its (1,0), similarly for y its (0,1), this way you can both the partials and the jacobian matrix in one step. Ex, f1(x,y)=e^xy, (e^xy)'=e^xy times (xy)' = e^xy times (xy'+x'y)= e^xy (x(0,1)+y(1,0))= (ye^xy,xe^xy). In this simple case it take more time to compute but in a more complex case (say sin(xye^z)lnxyz) it actually takes less time because you are two all three partials at the same time. Also the chain rule is exactly the same as the chain rule in calc 1. If we had a function that outputted a vector like the one at the start we simply compute the derivative of the first (f1) and then the second (f2) and stack their entries on top of each other. Ex (e^xy, lnxy), For the first we already show that its derivative (gradiant) is (ye^xy,xe^xy). For the second : (sinxy)'=cosxy * (xy)'=cosxy * (x(0,1)+y(1,0))=(ycosxy,xcosxy). This means that our derivative/jacobian matrix is (first row) (ye^xy,xe^xy), (second row) (ycosxy,xcosxy). Last remark, the derivative Xi (where Xi is the ith variable from a total of n) is always (0,0,...1,....0) where 1 is in the ith place and this is an n dimensional column vector. So if F is from R^k to R^m then n=k.

Lost me at 6:47: "so it doesn't make sense to use different symbols for partial derivatives with respect to each coordinate direction, despite using chain rule to translate between them." What different symbols do you mean? Do you mean using f and g for the different coordinate systems? Or possibly ∂ vs d, but I don't see how that would be relevant.

I have also used partial and the "eth" symbol to distinguish between holomorphic and anti-holomorphic Dolbeault operators. In fact, I did so in my latest video.

My heuristic is this: Physicists work with dependent variables, whereas mathematicians work with functions, particularly in the cases of coordinate changes like we saw in the video. I'd argue that Leibniz notation works better with the physics interpretation than the mathematics interpretation. You see this more obviously in implicit differentiation. You write an implicitly defined curve, like an elliptic curve, in terms of variables x and y, but neither is really a function of the other. Even so, the quantity dy/dx still makes perfect sense. It's as though you've introduced a constraint equation between the nominally free and independent variables x and y, and consequently dy/dx is nonzero.

For me, what's unclear is always the question "what is held constant". It is like in probability theory, you need to know what is the conditions before a conditional probability makes sense. All partial derivative symbols elide that, unluckily. When I see partial f over partial x, somehow I need to figure out that it means to hold y and t constant instead of u and v. That's always confusing for me.

we are just constrained by existing writing systems. we could invent new symbols but good luck with adoption. though the total partial being partway from explicit and total in terms of looks fits, but feels confusing when written

how about we use the cyrillic d with a T synbol for total partial derivative?

Now i learn that explicit derivative is defined for a formula, while total derivative is defined for a value. Partiality or not is really not going to matter when computing. When a formula is simple univariate function, its own explicit derivative and the total derivative of the value it defines coincide. Thx

who cuts the d's tail!

Total derivative in math is just the Jacobian or it's generalized manifold version i.e. the best linear approximation.

The difference comes from the fact that physicists use a lot of variables that are related to each other, and usually the same variables (v, F,E,U,m etc) While in math the study of calculus is much more abstract, and since definitions and theorems are easier to understand the simpler they are, math tends to eliminate redundancy whenever it makes the ideas simpler. In physics convoluted relations between variables are natural and convenient.

The important thing is that we get a long... a long explanation of how every convention works

If you use Lagrangian, you minimize not energy, but the "action".

I'm not a chef, but I feel like I can conquer any recipe after watching this.

In mathematics, the notation of "total partial derivatives" is innecessary, because they would not apply the differential operators to variables, but to actual functions. Thus, if a mathematician had a variable u depending on x, y, t, s, z, and then they had x, y depend on t and y depend on s following x = X(t), y = Y(t, s), they would define two different functions for that variable: u = F(x, y, t, s, z) = G(t, s, z), with G being defined through composition as G(x, y, z) = F(X(t), Y(t, s), t, s, z). In this case, there is no confusion at all regarding the meaning of partial derivative with respect to t, for example, because ∂F/∂t and ∂G/∂t are obviously different despite both being a partial derivative of the same variable with respect to the same other variable.

I guess I am not fully understanding the problem since both in my Maths and Physics lectures, I always got the feeling that the convention over partial derivatives was the same.

This is something I "felt" but didn't even understand until seeing your video. Great work! Still, this problem goes much deeper as far as I know. "Physics" is not a single community and Engineering even uses wilder conventions. In fluid dynamics, if I recall correctly, the total derivative is represented by capital D, with d representing "intermediate" total derivatives. I think the convention P is most "confusing" when you are learning with thermodynamics for the first time.

my head hurts

Partial derivative - f-ion with more than 1 independent variable, we want to know how much the function changes when only 1 of the independent variables changes while the others are kept constants. In fluid mechanics with Laplace gradient operator and with partial derivative definition is very hard to solve because Brownian motion mixes all atoms together. Statistical-mechanics is better approach.

Write a nice introductory physics text using your convention.

Having taken Engineering in undergrad, I just always default to partial derivatives, because I see it as being less wrong than using a "d" when i shouldn't.

Luckily, computer science community diverged from mathematics community only around 70s-80s, so their conventions are largely the same. physics and math, they cant even agree on theta vs phi for azimuth vs longitude for spherical coordinates