The arrows look good... and they make it easier to find the problem: There are variables with the same name... like in 3:24 It has a function u(x, y, z, s, t) When you call u(x(t), y(t,s), z, s, t) the new t and s are not the same as in the original function... There must be a t above x, and an s between y and z... and the ones in the right need a different name, like t1 and s1
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.
Crap, I've just seen your previous video and you already discussed this. I have another complaint, though: the notation using ð might not be enough to solve everything if there are even more layers of dependence than just the most superficial and the most simplified.
@@LucasSilva-ut7nm Well, physicists are expected to work with variables instead of functions because it is variables that represent actual measurable magnitudes, functions just represent how one of them depends on others in a fixed reference. The inconveniences of abusing notation are fewer than the ones of using mathematically precise notation, I guess.
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.
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
For me the worst one is the inner product, in math and physics they are typically defined with the variables swapped around, this caused me so many headaches translating theorems in math use them in bra-ket notation
@@mattias2576 I have a math background, and physics way is superior in this regard. thinking of bra = covector = dual as a function acting on regular vectors from left side just seems like the superior choice.
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.
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.
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 many students were upset by this teacher using a different convention but I liked the clarity.
With a more formalist mindset, i feel like it ultimately breaks down to physicists (i study physics myself) ignoring what functions actually depend on which then leads to the confusions which we physicists have to solve by introducing new types of derivatives instead of being precise with the arguments/domains of a function. You pointed in the direction in the video when talking about f(x,y) and then doing the change of variables and using the same letter f(r,theta). I think this is where the problem lies. Because already when you insert values for x,y or r, theta into the function, even though you would write the same in both cases the value from the physics perspective should change. This doesn't make much sense as an expression like f(2,3) is the ambiguous. Paying attention to the dependencies also solves the Euler Lagrange problem as the EL equations are, in the example you gave, dL/du°(u,u_x,u_y,p_x,p_y) + d/dx[dL/du_x°(u,u_x,u_y,p_x,p_y)] + d/dy[dL/du_y°(u,u_x,u_y,p_x,p_y)] Here ° refers to the composition of functions and i used p_x and p_y as the projections onto x and y Its important here that dL/du_x°(u,u_x,u_y,p_x,p_y) is not the same function as dL/du_x as the former depends on u,u_x,u_y,x,y and the latter only depends on x,y because of the composition The composition also nicely shows why we need the chain rule there I think instead of inventing new symbols for new meanings of derivatives, we should advocate being more aware about the actual arguments and domains of the functions we take derivatives of the function (x,y)->F(x,y) is not the same as t->F(x(t),y(t)) because the argument of this function is a different one. With this in mind there is no meed for new terminology for derivatives and we can just use the partial derivative (which turns out to be the ordinary derivative for functions f:R->R) Edit: after sending the comment i realised that it looks like im goung on a rant here. I still think this video was great at highlighting the differences, advantages and disadvantages of different approaches. Thanks for the good work :)
In reality instead of f(x,y) or f(r, theta) you should have f(v) where v is a vector, and then you have the coordinate descriptions del v/del x, del v/del y, del v/del r, del v/del theta
I am a physicist and I completely disagree with your first line. In fact, it doesn't make sense to me, or to the content of the video, in which it is explained that physicist use partial derivatives and total derivatives precisely because we do not ignore on the actual dependencies of a function. Or maybe you are talking about some weird physics convention of which I am not aware.
@@JackDespero By ignoring, I believe op means to explicitly state the independent variables whenever a function is introduced, which is a math thing that rarely any physicists do. For example, physicists do not distinguish between u(x,y) and u(r,\theta), physicists see them as quantities instead of functions. But in strict mathematical terms, these are two different functions with conversion function composed at the end of u.
@@viliml2763 yes that makes a lot of things clearer in my opinion. That also shows a difference whether a function depends on a point or on the coordinates of this point in a certain coordinate system
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.
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.
TBH, I think we just need more explicit "paths" or restriction on independent variables. By that, I mean for example the conversion from (x,y) to (r,\theta) should be explicitly written as a function f(x,y)=(r(x,y),\theta(x,y)), and for u(r,\theta) we write u\circ f instead. With this explicitness, we arrive at no ambiguity at all, though I have to admit this could be cumbersome.
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.
IMNSHO We should teach the partial derivatives as components of the general derivative (Jacobi matrix). But, yes, also start with differentials. A derivative as a linear map between differentials is undamental.
I would like the topic applied to exterior calculus. Partial and total derivatives being vectors and covectors. The last is a reciprocal representation of slope by the run x, given constant y. The geographical water marks on a lake, across unit rise water marks seen from above seeing steeper slopes at constricted water mark paths. The covector is then the parallel tangent lines touching 2 water mark paths. This is a scale invariant representation as any vector crossing such a covector will magnify with a dilution of the covector giving an invariant inner product. The usage seems motivated by general relativity and exterior calculus, though Poncarie's Lemma involving boundaries and coboundaries is properly part of algebraic topology giving null results for 2nd derivatives and 2nd integrals unlike the case for gravity.
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.
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 does this even mean? A function does not depend on coordinate charts, it just associates points of one space with points of another space, if f(3) = 7 that fact holds regardless of the coordinates we use. When we change coordinates so that “f(3)=9” we changed the function f to some other function, otherwise we’d have 7=9
@@cm5754 by abstracting to functions defined over manifolds, i would usually expect that the points of the manifold are not themselves numbers. so, say that we have two coordinate charts x and y for a 1D manifold, as in your example. the points p(x=3) and p(y=3) can certainly be different points of the manifold. so the function f can take different values for x=3 and y=3.
@@myca9322 I do see what you mean. From this viewpoint the confusion with the f(t,x(t),y(t)) example is also easier to settle, because the first 't' is being used as a point in the manifold rather than a value in a chart, wile the t in x(t) is a value in a chart. But apart from the manifold language, I'm not sure this is not much different from the standard calculus solution of looking at f(u,v,w) with the parameterizations u = u(t), v = v(t) and w = w(t) taking the role of the chart. It is a nice perspective.
@@cm5754 yes. this type of example is actually what i was referring to in the last paragraph. one way of understanding is as you do, with t playing multiple roles. this is correct, but can be confusing. another way of thinking about it is that we are now considering f's value on a submanifold defined by a constraint: in this chart, the constraint is that the values of x and y have to lie on a curve parameterized by t. at abstract level, this actually has a very nice and precise interpretation in terms of diagrams which look quite similar to the arrow notation used in the video.
... 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)
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
Sometimes you also see ∂_i used as an operator, to mean ∂/∂x_i when you have variables x_1, ..., x_n. Which does this represent? And it could probably be extended to have ∂_x, ∂_y, ∂_u and so on without much confusion.
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
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.
I would be in favor of using capital D and little d. but in engineering fluid dynamics, the most common convention was the exact same as convention P, with the special treatment of total derivative with respect to time (directly and indrectly through spatial variables) aka the material derivative with big D. What a waste of perfectly fine symbol for a meaningless distinction. imo material derivative is just a total derivative with respect to time.
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.
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.
Partial derivative is just taking a 1-d derivative with every other variable set to constant with respect to the variable you are taking derivative of. There is a (somewhat) ambiguity in this definition because in different field of study, the levels of the actual variables are different. For example, let F:Rn->Rm, G:Rm->Rk. It could be unclear by writing the i-th partial of G(F(x1,...,x_n)) because it is not explicit whether we are taking the derivative of G or G\circ F where \circ is the composition operation. The way I prefer to do is that label the variable of G as (y1,...,ym) and F as (x1,...,xn), then when taking partials, we explicitly speak of whether we are taking partial of yi or xi. With this, I see no ambiguity. To answer your "what is held constant", I believe it is related to the level of variable I have spoken above. Sometimes we encounter functions like f(x(t),y(t),t) and f(x,y,t), and it is hard to know whether x,y are dependent on t or not. This is why clear definition of functions are needed, because otherwise it is impossible to know what variables are independent and how the dependence is distributed.
In my mind, the justification for the partial derivative is to break a problem into pieces. For example, if you have "z=2x+y", one part of the value of "dz" is "2dx", and the other part of the value of "dz" is "dy". Doing two partial derivatives instead of one total derivative across the function gives you a chance to take a breather and do some easier sub-problems. But it comes with the obvious problem that you are _in fact_ working with "The derivative of 'z=2x+y' is 'dz=2dx+dy'" the whole time, and if you do an operation to one of the parts that doesn't fully distribute over addition then it all breaks.
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
I like the idea of differentiating between explicit and total partial derivatives, but I don't your particular choice of notation: Firstly, by keeping the old symbol (∂) for explicit partial derivatives, you haven't made those more clear, since people who see "∂" might not be able to tell whether or not you're using the new convention. I think both symbols need to replaced with new symbols. (Even if people don't recognize either symbol, they will at least know that they don't know what it means, and so look it up, rather than moving on with a false belief about what it means. This is same reason why I think "injective" and "surjective" are better terms than "one-to-one" and "onto", which can both be misinterpreted in some contexts.) Secondly, you used "ð" for total partial derivatives, despite the fact that it looks like a "d" with the stem modified to look like an "x" for "eXplicit". That being said, it could also be a modification to make the d look like a "t" for "Total", or maybe since Old English "ð" became Modern English "th" you think it's more related to "t". If so, then that means mnemonics could connect it to either "x" or "t", making them less useful, but that's just like if it were some random alteration with no such mnemonic usefulness, so this second point wouldn't really be valid at all. I should probably watch your other video to see how you came up with your new notation.
I see you actually didn't propose this new notation in your previous "Ambiguity With Partial ∂ Notation, and How to Resolve It" video, like I assumed. By the way, that video made it clear to me that easier than pushing yet more notations on people is to either use different letters for different functions or use parentheses if you're really dealing with dependent variables rather than functions.
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.
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.
For anyone wordering why this is true: x=π1(x,y, etc) where π1 is the projection onto the x axis (ie π1(x,y, etc)=(1,0,..) dotted with (x,y,...)). Because π1 is linear its differential is itself and since π1(x,y, etc)=(1,0,..) dotted with (x,y,...) we can see that the derivative/jacobian matrix is just (1,0,..). Similarly for the other variables.
This is good, but instead of writing (1,0) and (0,1) I would write dx and dy (or x' and y'), the way it's done in differential equations. Just me not being used to using matrixes, I suppose. (The upside is I don't have to know all the variables involved before I start calculating the derivative.)
@@Tzizenorec Its really the same, by putting dx etc you are computing the differential, by putting (1,0) you are computing the derivative/Jacobian matrix
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
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.
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?
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.
Differentials are covector field, and derivative is vector field, so this "cancellation" somehow working out is something very specific to 1 dimensional integral, where gradient and differential of a function largely looks the same. Even simpler, in 1 dimensional vector space (talking about function of 1 variables), scalars = vectors = covectors, so we can multiply and divide without too much proper care But for example, take a look at the statement of Green's theorem, and see if you can "prove" it by multiplying or dividing by differentials Fdx + Gdy = (Gx - Fy)dxdy
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
Likewise, all the notation up to 2:19 is wrong. You are defining u in all sorts or ways which is why you’re confused. You can’t write du/dt and then also write partial u/partial x. Is it a single variable function of t or a multivariable function with x being one of the inputs? The last line is bad too
Likewise 2:52. You write u(x,y,t) - but there is no dependence on t through x and y. You say so but where? There is none. Now if you wrote u(x(t), y(t), t) - this composite function, say z(t) for shorthand, is single variable in t. Now you are right to say something along the lines you said. In this case (for clarity), there is no partial z / partial t but only a dz/dt. Another example: If you wrote u(x(t), y, t) let this be z(y, t) for shorthand. The only derivatives that exist here are partial z/partial y or partial z/partial t. [write out u(x(t),y,t) in place of z if you need to]. Partial z/partial t and partial u/partial t are very different because you understand the difference. u is the normal outside function of 3 variables. z or u(x(t),y,t) is a composite function of 2 variables - which requires the chain rule if you would like
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
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).
Principal of Least Action is a physical law regarding time integral, action is minimized over a period of time. Minimizing potential energy with respect to spatial variables on the other hand is to find a steady state solution. Like ball sitting on bottom of the hill will stay there and objects at thermal equilibrium will not have change in distribution of temperature. And its a borrowed name from Lagrangian mechanics, but in the context of calculus of variations, any integrand of an integral that is to be optimized is called Lagrangian.
@@HaramGuys True. Still though the thing you minimize is called action, I do not think that the name for functional to minimize is reserved, we just called it "functional" in our differential equations course, but in physics in this case it is. And because L = T - V = -V, where V is potential when there is no kinetic terms involved, saying that you minimize energy is not true either... Better to say that you find an extrema.
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.
I am somewhat surprised you did not put any attention on "The Grand Unified Theory of CLASSICAL Physics," zero errors, 85 zeros, quite old itself, now. Would you dare to correct this last comment? I am convinced that is unlikely. Also, quantum mechanics does not exist, not for the classical. The universe is oscillatory. All of WEBB's 'older' galaxies R from a former iteration, again, easily verifiable. Time, gravity, and entropy R solved for, 1st principles only. That said, I love your work, Russell's Paradox solved or not.
His English is completely fine. The way he speaks is more intelligible than that of some native English speakers. Maybe you should practice YOUR English instead by listening to English? Giving you the benefit of the doubt, I can interpret it as a "don't forget your homework kids" or "brush your teeth kids" kind of comment, but the period at the end doesn't really belong then.
At least he bothered to learn your language. How many languages do you speak? I have no issue with his English by the way. Scottish, Irish and other accents are way harder to understand than this. Not sure why it matters.
I mean it all makes sense, but in all honesty the symbol you suggest for total partial at the end kinda sucks: not too legible juxtaposed to the explicit partial, awkward when handwritten, and can be mixed up with the Feynman slash notation. My first reaction was to simply usurp the symbol for functional derivative "𝛿", but this is of course with its own disadvantages.
![The Most Useful Curve in Mathematics [Logarithms]](http://i.ytimg.com/vi/OjIwCOevUew/mqdefault.jpg)
The arrows to breakdown the chain rule is something I’ve never seen in all my years
Same here 😑
The arrows look good...
and they make it easier to find the problem:
There are variables with the same name... like in 3:24
It has a function u(x, y, z, s, t)
When you call u(x(t), y(t,s), z, s, t)
the new t and s are not the same as in the original function...
There must be a t above x, and an s between y and z...
and the ones in the right need a different name,
like t1 and s1
I’ve seen it a few times done this way at uni, I’m in Asia.
Are you for serious 😂. I learned it that way. It’s cool to see how differently others have learned the same topic
I always imagine them cancelling each other so they equal the same thing
Time to just use an apostrophe (prime) but the apostrophe has a subscript with a QR code linking to a haskell program detailing what we mean.
"*** Exception: partial:2:1-12: Non-exhaustive patterns in case."
@@user-pe7gf9rv4m*no **_😊_*
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.
Crap, I've just seen your previous video and you already discussed this. I have another complaint, though: the notation using ð might not be enough to solve everything if there are even more layers of dependence than just the most superficial and the most simplified.
In the end, the problem is the physicists lol...
@@LucasSilva-ut7nm Well, physicists are expected to work with variables instead of functions because it is variables that represent actual measurable magnitudes, functions just represent how one of them depends on others in a fixed reference.
The inconveniences of abusing notation are fewer than the ones of using mathematically precise notation, I guess.
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.
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
ikr, drives me crazy that physicists switch theta with phi
@@FranciscoCunha2004 drives me nuts as well.
@@FranciscoCunha2004 We also sometimes use elevation rather than inclination :P
For me the worst one is the inner product, in math and physics they are typically defined with the variables swapped around, this caused me so many headaches translating theorems in math use them in bra-ket notation
@@mattias2576 I have a math background, and physics way is superior in this regard. thinking of bra = covector = dual as a function acting on regular vectors from left side just seems like the superior choice.
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.
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.
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 many students were upset by this teacher using a different convention but I liked the clarity.
With a more formalist mindset, i feel like it ultimately breaks down to physicists (i study physics myself) ignoring what functions actually depend on which then leads to the confusions which we physicists have to solve by introducing new types of derivatives instead of being precise with the arguments/domains of a function. You pointed in the direction in the video when talking about f(x,y) and then doing the change of variables and using the same letter f(r,theta). I think this is where the problem lies. Because already when you insert values for x,y or r, theta into the function, even though you would write the same in both cases the value from the physics perspective should change. This doesn't make much sense as an expression like f(2,3) is the ambiguous. Paying attention to the dependencies also solves the Euler Lagrange problem as the EL equations are, in the example you gave, dL/du°(u,u_x,u_y,p_x,p_y) + d/dx[dL/du_x°(u,u_x,u_y,p_x,p_y)] + d/dy[dL/du_y°(u,u_x,u_y,p_x,p_y)]
Here ° refers to the composition of functions and i used p_x and p_y as the projections onto x and y
Its important here that dL/du_x°(u,u_x,u_y,p_x,p_y) is not the same function as dL/du_x as the former depends on u,u_x,u_y,x,y and the latter only depends on x,y because of the composition
The composition also nicely shows why we need the chain rule there
I think instead of inventing new symbols for new meanings of derivatives, we should advocate being more aware about the actual arguments and domains of the functions we take derivatives of the function (x,y)->F(x,y) is not the same as t->F(x(t),y(t)) because the argument of this function is a different one. With this in mind there is no meed for new terminology for derivatives and we can just use the partial derivative (which turns out to be the ordinary derivative for functions f:R->R)
Edit: after sending the comment i realised that it looks like im goung on a rant here.
I still think this video was great at highlighting the differences, advantages and disadvantages of different approaches. Thanks for the good work :)
In reality instead of f(x,y) or f(r, theta) you should have f(v) where v is a vector, and then you have the coordinate descriptions del v/del x, del v/del y, del v/del r, del v/del theta
I am a physicist and I completely disagree with your first line. In fact, it doesn't make sense to me, or to the content of the video, in which it is explained that physicist use partial derivatives and total derivatives precisely because we do not ignore on the actual dependencies of a function.
Or maybe you are talking about some weird physics convention of which I am not aware.
@@JackDespero By ignoring, I believe op means to explicitly state the independent variables whenever a function is introduced, which is a math thing that rarely any physicists do. For example, physicists do not distinguish between u(x,y) and u(r,\theta), physicists see them as quantities instead of functions. But in strict mathematical terms, these are two different functions with conversion function composed at the end of u.
@@viliml2763 yes that makes a lot of things clearer in my opinion. That also shows a difference whether a function depends on a point or on the coordinates of this point in a certain coordinate system
@@MH-sf6jzexactly thanks for making it more clear :)
Outstanding. Well done. Discussion of the Lagrangian is particularly great.
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.
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.
Hah, I studied physics, and I'm doing the same 😅. To be fair, a lot of physicists do this
Hats off to your work 🫡
TBH, I think we just need more explicit "paths" or restriction on independent variables. By that, I mean for example the conversion from (x,y) to (r,\theta) should be explicitly written as a function f(x,y)=(r(x,y),\theta(x,y)), and for u(r,\theta) we write u\circ f instead. With this explicitness, we arrive at no ambiguity at all, though I have to admit this could be cumbersome.
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.
Now it works with stochastic calculus as well!!! Great point
IMNSHO We should teach the partial derivatives as components of the general derivative (Jacobi matrix). But, yes, also start with differentials. A derivative as a linear map between differentials is undamental.
I would like the topic applied to exterior calculus. Partial and total derivatives being vectors and covectors. The last is a reciprocal representation of slope by the run x, given constant y. The geographical water marks on a lake, across unit rise water marks seen from above seeing steeper slopes at constricted water mark paths. The covector is then the parallel tangent lines touching 2 water mark paths. This is a scale invariant representation as any vector crossing such a covector will magnify with a dilution of the covector giving an invariant inner product. The usage seems motivated by general relativity and exterior calculus, though Poncarie's Lemma involving boundaries and coboundaries is properly part of algebraic topology giving null results for 2nd derivatives and 2nd integrals unlike the case for gravity.
_this is physics, we _*_do not_*_ give a fuck!_
Who are we?
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.
Total derivative in math is just the Jacobian or it's generalized manifold version i.e. the best linear approximation.
i have seen the 3 symbol usage convention before in physics lectures
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 does this even mean? A function does not depend on coordinate charts, it just associates points of one space with points of another space, if f(3) = 7 that fact holds regardless of the coordinates we use. When we change coordinates so that “f(3)=9” we changed the function f to some other function, otherwise we’d have 7=9
@@cm5754 by abstracting to functions defined over manifolds, i would usually expect that the points of the manifold are not themselves numbers. so, say that we have two coordinate charts x and y for a 1D manifold, as in your example. the points p(x=3) and p(y=3) can certainly be different points of the manifold. so the function f can take different values for x=3 and y=3.
@@myca9322 I do see what you mean. From this viewpoint the confusion with the f(t,x(t),y(t)) example is also easier to settle, because the first 't' is being used as a point in the manifold rather than a value in a chart, wile the t in x(t) is a value in a chart. But apart from the manifold language, I'm not sure this is not much different from the standard calculus solution of looking at f(u,v,w) with the parameterizations u = u(t), v = v(t) and w = w(t) taking the role of the chart. It is a nice perspective.
@@cm5754 yes. this type of example is actually what i was referring to in the last paragraph. one way of understanding is as you do, with t playing multiple roles. this is correct, but can be confusing. another way of thinking about it is that we are now considering f's value on a submanifold defined by a constraint: in this chart, the constraint is that the values of x and y have to lie on a curve parameterized by t.
at abstract level, this actually has a very nice and precise interpretation in terms of diagrams which look quite similar to the arrow notation used in the video.
... 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)
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
The important thing is that we get a long... a long explanation of how every convention works
Sometimes you also see ∂_i used as an operator, to mean ∂/∂x_i when you have variables x_1, ..., x_n. Which does this represent? And it could probably be extended to have ∂_x, ∂_y, ∂_u and so on without much confusion.
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
Can someone please explain to me the difference between the total derivative and the total partial derivative as shown in the video?
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.
I would be in favor of using capital D and little d. but in engineering fluid dynamics, the most common convention was the exact same as convention P, with the special treatment of total derivative with respect to time (directly and indrectly through spatial variables) aka the material derivative with big D. What a waste of perfectly fine symbol for a meaningless distinction. imo material derivative is just a total derivative with respect to time.
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.
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.
Partial derivative is just taking a 1-d derivative with every other variable set to constant with respect to the variable you are taking derivative of. There is a (somewhat) ambiguity in this definition because in different field of study, the levels of the actual variables are different. For example, let F:Rn->Rm, G:Rm->Rk. It could be unclear by writing the i-th partial of G(F(x1,...,x_n)) because it is not explicit whether we are taking the derivative of G or G\circ F where \circ is the composition operation. The way I prefer to do is that label the variable of G as (y1,...,ym) and F as (x1,...,xn), then when taking partials, we explicitly speak of whether we are taking partial of yi or xi. With this, I see no ambiguity.
To answer your "what is held constant", I believe it is related to the level of variable I have spoken above. Sometimes we encounter functions like f(x(t),y(t),t) and f(x,y,t), and it is hard to know whether x,y are dependent on t or not. This is why clear definition of functions are needed, because otherwise it is impossible to know what variables are independent and how the dependence is distributed.
In my mind, the justification for the partial derivative is to break a problem into pieces. For example, if you have "z=2x+y", one part of the value of "dz" is "2dx", and the other part of the value of "dz" is "dy". Doing two partial derivatives instead of one total derivative across the function gives you a chance to take a breather and do some easier sub-problems. But it comes with the obvious problem that you are _in fact_ working with "The derivative of 'z=2x+y' is 'dz=2dx+dy'" the whole time, and if you do an operation to one of the parts that doesn't fully distribute over addition then it all breaks.
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
I like the idea of differentiating between explicit and total partial derivatives, but I don't your particular choice of notation:
Firstly, by keeping the old symbol (∂) for explicit partial derivatives, you haven't made those more clear, since people who see "∂" might not be able to tell whether or not you're using the new convention. I think both symbols need to replaced with new symbols. (Even if people don't recognize either symbol, they will at least know that they don't know what it means, and so look it up, rather than moving on with a false belief about what it means. This is same reason why I think "injective" and "surjective" are better terms than "one-to-one" and "onto", which can both be misinterpreted in some contexts.)
Secondly, you used "ð" for total partial derivatives, despite the fact that it looks like a "d" with the stem modified to look like an "x" for "eXplicit". That being said, it could also be a modification to make the d look like a "t" for "Total", or maybe since Old English "ð" became Modern English "th" you think it's more related to "t". If so, then that means mnemonics could connect it to either "x" or "t", making them less useful, but that's just like if it were some random alteration with no such mnemonic usefulness, so this second point wouldn't really be valid at all.
I should probably watch your other video to see how you came up with your new notation.
I see you actually didn't propose this new notation in your previous "Ambiguity With Partial ∂ Notation, and How to Resolve It" video, like I assumed. By the way, that video made it clear to me that easier than pushing yet more notations on people is to either use different letters for different functions or use parentheses if you're really dealing with dependent variables rather than functions.
Amazing video, thank you!
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.
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.
For anyone wordering why this is true: x=π1(x,y, etc) where π1 is the projection onto the x axis (ie π1(x,y, etc)=(1,0,..) dotted with (x,y,...)). Because π1 is linear its differential is itself and since π1(x,y, etc)=(1,0,..) dotted with (x,y,...) we can see that the derivative/jacobian matrix is just (1,0,..). Similarly for the other variables.
This is good, but instead of writing (1,0) and (0,1) I would write dx and dy (or x' and y'), the way it's done in differential equations. Just me not being used to using matrixes, I suppose. (The upside is I don't have to know all the variables involved before I start calculating the derivative.)
@@Tzizenorec Its really the same, by putting dx etc you are computing the differential, by putting (1,0) you are computing the derivative/Jacobian matrix
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
I wish I understood any of this. I did 10 years ago, but now I just lost the foundations.
This one video i would refer all throught my semester ... soo insightful
could you add on to this video btw?
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.
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?
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.
how about we use the cyrillic d with a T synbol for total partial derivative?
Distinction without a difference.
How is dividing by dt an abuse of notation? I thought it was an essential part of deriving calculus expressions
Differentials are covector field, and derivative is vector field, so this "cancellation" somehow working out is something very specific to 1 dimensional integral, where gradient and differential of a function largely looks the same.
Even simpler, in 1 dimensional vector space (talking about function of 1 variables), scalars = vectors = covectors, so we can multiply and divide without too much proper care
But for example, take a look at the statement of Green's theorem, and see if you can "prove" it by multiplying or dividing by differentials
Fdx + Gdy = (Gx - Fy)dxdy
Ok thank you. I'll have to take your word on a lot of that
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
Likewise, all the notation up to 2:19 is wrong. You are defining u in all sorts or ways which is why you’re confused. You can’t write du/dt and then also write partial u/partial x. Is it a single variable function of t or a multivariable function with x being one of the inputs? The last line is bad too
Likewise 2:52. You write u(x,y,t) - but there is no dependence on t through x and y. You say so but where? There is none. Now if you wrote u(x(t), y(t), t) - this composite function, say z(t) for shorthand, is single variable in t. Now you are right to say something along the lines you said. In this case (for clarity), there is no partial z / partial t but only a dz/dt.
Another example:
If you wrote u(x(t), y, t) let this be z(y, t) for shorthand. The only derivatives that exist here are partial z/partial y or partial z/partial t. [write out u(x(t),y,t) in place of z if you need to]. Partial z/partial t and partial u/partial t are very different because you understand the difference. u is the normal outside function of 3 variables. z or u(x(t),y,t) is a composite function of 2 variables - which requires the chain rule if you would like
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
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).
Or just use currying. Write f(x)(y) instead of f(x, y), then the partial with respect to x is f'(x)(y) and the partial with respect to y is f(x)'(y).
You might find sussman's functional differential geometry on that topic interesting
@@SVVV97 Thanks for the recommendation
If you use Lagrangian, you minimize not energy, but the "action".
Principal of Least Action is a physical law regarding time integral, action is minimized over a period of time.
Minimizing potential energy with respect to spatial variables on the other hand is to find a steady state solution. Like ball sitting on bottom of the hill will stay there and objects at thermal equilibrium will not have change in distribution of temperature.
And its a borrowed name from Lagrangian mechanics, but in the context of calculus of variations, any integrand of an integral that is to be optimized is called Lagrangian.
@@HaramGuys True. Still though the thing you minimize is called action, I do not think that the name for functional to minimize is reserved, we just called it "functional" in our differential equations course, but in physics in this case it is. And because L = T - V = -V, where V is potential when there is no kinetic terms involved, saying that you minimize energy is not true either... Better to say that you find an extrema.
who cuts the d's tail!
d😂/d❤=∆❤/∆👆=😂+ d❤/d👆= ∆👆/∆😂+ð👆/ð😂
Derivatives of emojis
Ah yes, those infinitesimal smiles ... :-)
We have the same books :D
crying in thermodynamics
my head hurts
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.
I am somewhat surprised you did not put any attention on "The Grand Unified Theory of CLASSICAL Physics," zero errors, 85 zeros, quite old itself, now.
Would you dare to correct this last comment? I am convinced that is unlikely. Also, quantum mechanics does not exist, not for the classical.
The universe is oscillatory. All of WEBB's 'older' galaxies R from a former iteration, again, easily verifiable. Time, gravity, and entropy R solved for, 1st principles only.
That said, I love your work, Russell's Paradox solved or not.
you should practice your english.
I can clearly understand what he is saying, how much more english does he need?
His English is completely fine. The way he speaks is more intelligible than that of some native English speakers. Maybe you should practice YOUR English instead by listening to English?
Giving you the benefit of the doubt, I can interpret it as a "don't forget your homework kids" or "brush your teeth kids" kind of comment, but the period at the end doesn't really belong then.
In all fairness I believe he'd be easier to understand if he switched off the distracting background music. Listeners have enough to pay attention to.
I agree, I can't even spell multivariable correctly 2:30
At least he bothered to learn your language. How many languages do you speak?
I have no issue with his English by the way. Scottish, Irish and other accents are way harder to understand than this. Not sure why it matters.
I mean it all makes sense, but in all honesty the symbol you suggest for total partial at the end kinda sucks: not too legible juxtaposed to the explicit partial, awkward when handwritten, and can be mixed up with the Feynman slash notation. My first reaction was to simply usurp the symbol for functional derivative "𝛿", but this is of course with its own disadvantages.