Deriving the Heat Equation: A Parabolic Partial Differential Equation for Heat Energy Conservation
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- Опубликовано: 1 авг 2024
- In this video we will derive the heat equation, which is a canonical partial differential equation (PDE) in mathematical physics.
@eigensteve on Twitter
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This video was produced at the University of Washington
%%% CHAPTERS %%%
0:00 Overview
3:02 Statement in Words
6:52 Statement in Math
10:30 Heat Flux
16:00 Fourier's Law of Heat Conduction
18:54 The Heat Equation 
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Building advanced concepts from simple principles.. excellent. This channel is such a gem
I am lucky to live in this time, having access to such a great teacher for free and just before the exams. Thank you sir!
That's a professor! Explains clearly because he knows deeply the topic. Thanks!
You and your channel is the BEST online science education so far! What a wonderful lecture! Thank you so much for your knowledge, the way you teach and your greatness and kindness of sharing !!!
No-no-no, thank YOU! Thank you for making this stuff so compelling and accessible.
always great to see a lecture push for common sense and knowhow 😄
it's actually reassuring, because it means that it's not always obvious how to map real world interactions that we have an intuitive understanding of into a formal system. It definitely intimidates me in a "not sure how to get started" sort of way.
my takeaways:
- identify the property you're modelling (heat over time and space)
- use a visual aid, draw out the physical setting
- set constraints on what you are & are not including in the model (e.g., not accounting for radiation)
- write down a laymen explanations of how the property changes in the dimensions (time and space)
- keep breaking those explanations down into smaller statements until you've got something isolated enough to model on its own (heat flux, sources/sinks)
- combine models
- know vector calculus
- ???
- win
Looking forward to future videos!
Steve... you are so great I have no words. I can only hope to have a fraction of your understanding and didactics. Thank God you exist, you've saved me so many times with your classes, you have no idea!
Thank you! I'm glad I could help :)
Excellent lecture. Thank you from Nebraska!
Excellent refresher. Thank you.
Really enjoy your lessons in this You tube channel . Very nice explanation, pretty neat, very comprehensable Professor Steve. Among all scientific books that I have there is a really good one from Walter A. Strauss from MIT in Partial Differential Equations PDE´s. In another [Elementary Differential Equations] from Lyman M Kells , Mc Graw Hill, fith edition [1960], the derivation of Second Order Partial Differential Equation applied to NUCLEAR FISSION , which defines the NEUTRON FLUX and MATERIAL BUCKLING.
If you have a negative heat gradient from left to right, more heat is entering on the left than is leaving on the right. Therefore the heat will accumulate in the body. You could break it down as little cars bring heat people in and leaving with less heat people. That means the heat ppl are accumulating. 😀 Hope that helps with understanding the sign convention.
Wonderful! Thanks.
Thank you for making such a video
That was beautiful how he explained the heat equation.
Thank you!
Hi! This is very helpful. My friend and I were wondering how you write on that screen and it shows up the right way from our perspective. That’s so cool.
A little bit confusing to me because both -(partial q / partial x) and q(x, t) are called 'heat flux'.
Your videos are very good. I like them. I have seen your entire series on vector calculus and partial differential equations.
brilliant thanks!
neat to see the equation being derived from the physics. i consider this several degrees more difficult than modelling a system when an equation is known
the hard thing was finding principles of physical theories indeed (Newton's principles of dynamics for classical mechanics, principles of thermodynamics, Maxwell equations and Lorentz force for electromagnetism). Once you (better, someone else back in the years) have found physical principles, what comes next is "only Math", where "only Math" hides quite a large number of mathematical tools and opportunities to make lots of mistakes
q(x + ∆x) is q(x) but with the additional change +(dq/dx)∆x. So written algebraically: q(x+∆x)=q(x) + (dq/dx)∆x you can quickly see that substituting this into the equation of transport for q(x+∆x) gives you the negative derivative
Excellent intro, just a couple of little points to add from the Wiki!
Although *q(x,t)* is, in general a vector function, _u(x,t)_ a scalar functions, in this 1D case both may be treated as simply scalars.
_K/c℘_ is called the _thermal diffusivity_ of the medium, Si units [m^2/s].
A good way to remember the signs is by making the normal unit vectors on the surfaces facing outwards and if the flow is opposite to the unit vector then - otherwise +
Ofc not reinventing the wheel , just a way on how I like to do analyze it but Nevertheless excellent video as always !
Thank you for these very clear explanations. Do you plan to make videos on the finite elements/volumes methods ?
Thanks! Yes, I’d like to make those videos too, but probably not for a while
thank you sir
In case the origin of the heat energy term around 8:30 is confusing, the terms can be obtained by unit analysis:
c(x) = [Heat energy] / [Mass x Temp] (*this is interpreted as the amount of heat energy per unit mass the system gains as a result of a change in the system's temperature, hence "specific" heat capacity)
p(x) = [Mass] / [Length] (*in 1D)
u(x,t) = [Temp]
So heat energy distribution function is in units of [Heat energy] / [Length].
very good
15:12
I felt if a
The sign is really difficult to think, so I always remember that flow out is positive, flow in is negtive
@22:30 If zero is replaced by Q than the konstant K has to be considered (again).
Excellent! Thanx! 😂
Explanations like in the book of Haberman :)
I notice that if thermal conductivity κ stays a function of x -> κ(x), the PDE will involve both first and second order derivative of thermal energy with respect to x. complicated, but surely someone out there has done it before
for some of the "most common" numerical methods, like Finite Element and Finite Volume, this is not a problem at all. These formulations naturally deal with diffusion coefficient not uniform in space with almost no additional cost
I would like to know what is the title of Fourier’s book about heat conduction.
Can I get the reference?
I believe it is called "The analytical theory of heat" by J. Fourier
@@Eigensteve indeed it is and out of copyright. here is a translation www.google.ca/books/edition/The_Analytical_Theory_of_Heat/No8IAAAAMAAJ?hl=en&gbpv=1&printsec=frontcover
I agree with another commenter, since q(x,t) is the heat flux, shouldn't dq/dx be written as something other than just heat flux? Perhaps flux gradient would be a better term?
I think I like flux gradient better too
It would be sensible for heat flux to be energy/time and heat flux density - or heat flux gradient - to be energy/time x area (or energy/time x length for 1D). But I don’t think there is such strict definition. In my readings heat flux generally refers to the unit normalized over area, but for a specific problem with a setup featuring a fixed boundary (as in this one), it is common to refer to the flux simply in terms of energy/time. Flux is somewhat of a loose term according to Wikipedia, in this aspect as well as others.
Does anyone know why at 8:40 the heat flux is written down as dq/dx when q seems to be the heat flux itself? Why the derivative?
Because there actually are two heat fluxes in the small segment, heat flux in through the left face and out through the right face. The difference between these heat fluxes is the accumulation of heat in the system. The key concept here is not to confuse heat flux with Temperature. Temperature is the driving force for heat flow. Hope that helps.
Hi Phillip! I actually was wondering the same thing. I think the idea here is that the presenter is explaining how this particular equation is derived.
It may not be the best equation to model heat. Describing a source term as q(x,t) is perhaps not the best, because you are assigning a heat value to every point on the line, rather than at the ends .
Yes, you would put those details in the initial conditions, but now that equation reads something like
“the heat change per second at location x and time t is equal to the heat change per metre at location x and time t”
Multiplicative constants aside, the units are not the same, and since you presumably want a 2d graph of temperature/heat with distance along the line, this equation puts unnecessary constraints on the behaviour of the temperature/heat with distance.
Another approach might be to write x as a parametric equation of t(time), and then model the behaviour entirely in terms of heat and metres, with time included implicitly in x(t).
That way, you are not putting extra constraints on the behaviour of u(x,t).
Essentially, you rewrite u(x,t) as u(x(t)), then derive everything again accordingly.
Any thoughts anyone?
because it's the total heat flux (by conduction, here) through the boundaries of the infinitesimal volume of the domain, of length dx with the left boundary at x and the right boundary at x+dx say.
In 3d space, it comes from the heat flux over the boundary of the volume, that equals the volume integral of the divergence (by divergence theorem).
Here, in 1D problem, the divergence equals the derivative along the only space dimension (x) of the only component of the heat flux (called here q)
To recap, dq/dx here because that term comes from the heat transfer by conduction along the 1D domain. If you have some heat source in the domain, no derivative is required. As an example, if you have a fluid in a long pipe where a chemical reaction is occurring in some region of the pipe, releasing or absorbing heat, that could be model with a source term. Even, if you have heat flux through the lateral walls of the long pipe, still modelled as a 1D equation, you can add that contribution as a source term
I like these lectures. Thanks for sharing. The common sense used to derive the heat equation is also referred to as the first law of thermodynamics.
better call paul
*2nd Year Uni Group Assignment flashbacks oh god*
Theory of Everything solution (short version):
Swap from Newton "real/necessary" universe over to Leibniz "contingent/not-necessary" universe as our fundamental blueprint of the universe.
This includes Leibniz calculus vs Newton calculus. Anywhere Leibniz and Newton thought different. All of it. Full swap.
Gottfried Leibniz "contingent/not-necessary" universe just lacked 2022 quantum physics verbiage (just match up definitions i.e. quark and Monad) and Hamilton's 4D quaternion algebra (created 200 years after Leibniz died).
Lastly, our first number is NOT 1.
It's 0.
Our ten numbers are 0, 1, 2, 3,...9 ✅.
Our ten dimensions are 0D, 1D, 2D, 3D,...9D ✅.
Ask someone to begin counting. I bet they begin with 1.
1 is not the beginning.
0 is the beginning.
1D is a Line; two points; physical; matter; beginning and ending; contingent.
0D is a (point); exact location only; no spatial extension; zero size; Monad (Leibniz) Quark (strong nuclear force); necessary.
Examples:
What is another word for quark?
fundamental particle, elementary particle.
Do quarks take up space?
Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up space.
How fast do quarks move?
the speed of light (see Leibniz's Law of Sufficient Reason).
What is an elementary particle example?
(0, 1, 2, 3)
Elementary particles include
quarks (the constituents of protons and neutrons),
leptons (electrons, muons, taus, and neutrinos),
gauge bosons (photons, gluons, and W and Z bosons) and the Higgs boson.
What is the size of an elementary particle?
The elementary particles are not believed to have any size at all. As currently understood they are zero size points. Protons and neutrons (and all hadrons) are about 10−15m.
Match Leibniz definitions to quantum physics definitions. Different word, same definition.
Not a coincidence.
Human consciousness, mathematically, is identical to 4D quaternion algebra with w, x, y, z being "real/necessary"
0D, 1D, 2D, 3D
and i, j, k being "contingent/not-necessary"
0D, 1D xi, 2D yj,
3D zk.
0D is always w (real/necessary)
1D-9D contingent/not-necessary universe has "conscious lifeforms" (1D xi, 2D yj, 3D zk)..."turning" 'time'.
[In mathematics, a versor is a quaternion of norm one (a unit quaternion). The word is derived from Latin versare = "to turn" with the suffix -or forming a noun from the verb (i.e. versor = "the turner"). It was introduced by William Rowan Hamilton in the context of his quaternion theory.]
[Math; 4D quaternion algebra]
A quaternion is a 4-tuple, which is a more concise representation than a rotation matrix. Its geo- metric meaning is also more obvious as the rotation axis and angle can be trivially recovered.
How do you make a quaternion? (Nobody is starting with 0)
You can create an N-by-1 quaternion array by specifying an N-by-3 array of Euler angles in radians or degrees. Use the euler syntax to create a scalar quaternion using a 1-by-3 vector of Euler angles in radians.
"Turn" to what, you might ask. 5D is the center of 1D-9D. The breadth (space-time). All things are drawn to the center, the whole. (Gravity means Nothing compared to the Strong Nuclear Force)
[Contingent Universe]:
3 sets of 3 dimensions:
(1D-3D/4D-6D/7D-9D)
The illusory middle set (4D, 5D, 6D) is temporal. Id imagine we create this middle temporal set similar to a dimensional Venn Diagram with polarized lenses that we "turn" by being conscious.
Which requires energy. 3D height symmetry/entanglement with 6D depth and
9D absorption is why we are "consumers", we must consume/absorb calories, and sleep, to continue "to turn" 'time' (be alive).
1D-3D spatial set/7D-9D spectral set overlap creating the temporal illusion of 4D-6D set. Transcending one another.
1D, 2D, 3D = spatial composite (line, width, height)
4D, 5D, 6D = temporal illusory (length, breadth, depth)
7D, 8D, 9D = spectra energies (continuous, emission, absorption)
Symmetry/entanglement:
1D, 4D, 7D line, length, continuous
2D, 5D, 8D width, breadth, emission
3D, 6D, 9D height, depth, absorption
[Time]
According to theoretical physicist Carlo Rovelli, time is an illusion: our naive perception of its flow doesn't correspond to physical reality. Indeed, as Rovelli argues in The Order of Time, much more is illusory, including Isaac Newton's picture of a universally ticking clock.
Does time exist without space?
Time 'is' as space 'is' - part of a reference frame in which in ordered sequence you can touch, throw and eat apples.
Time cannot exist without space and the existence of time does require energy.
Time, then, has three levels, according to Leibniz:
(i) the atemporality or eternality of God;
(ii) the continuous immanent becoming-itself of the monad as entelechy;
(iii) time as the external framework of a chronology of “nows”
The difference between (ii) and (iii) is made clear by the account of the internal principle of change.
The real difference between the necessary being of God and the contingent, created finitude of a human being is the difference between (i) and (ii).
Conclusion: Humanity needs to immediately swap from "Newton" to "Leibniz". Also from Edison to Tesla. Also the Aether guy.
Our calculus is incorrect (Leibniz > Newton):
What is the difference between Newton and Leibniz calculus?
Newton's calculus is about functions.
Leibniz's calculus is about relations defined by constraints.
In Newton's calculus, there is (what would now be called) a limit built into every operation.
In Leibniz's calculus, the limit is a separate operation.
The known multiverse:
1) real/
necessary
w, x, y, z being
0D, 1D, 2D, 3D
2) contingent/not-necessary
w, xi, yj, zk
(YOU ARE HERE) being
0D, 1D xi, 2D yj, 3D zk
3) simulated/imaginary
(working on it)
Our ten numbers are
0, 1, 2, 3,...9 ✅.
"10" is not one of our ten numbers.
If you think 10 comes before 0...you might be duped by Newton's logic/calculus/physics.
What an absolute fraud. Newton used political power to "win" against Leibniz.
And Humanity foots the bill.
The scumbag, trying to "win", always wins.
Edison was the scumbag vs Tesla.
The M and M "Moron" scumbags vs the Aether guy.
The list goes on and on. Read about Newton. Even his most significant achievement was disputed (Hooke had beef with Newton "stealing" HIS original idea).
We need to reassess who we're even studying 📖.
If humanity all had the correct fundamentals then we'd advance like Tesla was talking about.
Who's doing this...
It's like a Twilight Zone episode where Globalists are pushing fundamentally incorrect gobbledygook to keep the Plebians stupid.
fundamentals = rock
specifics = sand
Everyone is a genius at the fundamentals.
CERN so concerned with 1D, 2D, 3D, 4D 🚫 instead of
0D, 1D, 2D, 3D ✅.
Derp.
Naughty little signs...
Is he actually writing backward, or is there some technological magic here?
Monad (Greek for "singularity", or "alone") has the same definition as Quark (strong nuclear force):
No spatial extension. Zero size.
Exact location only.
6 thousand trillion trillion trillion (39 zeroes after 6k) times stronger than the force of gravity.
Leibniz is our Universal Genius, not Newton. Read 📚 Monadology.
Match up definitions from Leibniz's works verbiage 300 years ago with quantum physics verbiage definitions today. Different words, same definitions.
Our 0D (point)/Quark (strong nuclear force)/Monad (singularity):
just read what Christ and John (and Leibniz) said about Monad (also the Abrahamic Gnostics [Greek, Syrian and Persian]).
Not a coincidence.
Our ten numbers are 0, 1, 2, 3,...9.
Our ten dimensions are 0D, 1D, 2D, 3D,...9D.
1D-9D is contingent universe.
Hamilton's 4D quaternion algebra can prove Leibniz. Update first four dimensions w, x, y, z to 0D, 1D, 2D, 3D though (as opposed to the current 1D, 2D, 3D, 4D nonsense).
Newton drinks his own pee.
1 is not the beginning.
0 is the beginning.
1D is two points.
0D is a representative dot in a theoretical circle (point).
1 is contingent.
0 is necessary.
There is no 1 without 0.
A piece of 1, no matter how small, will NEVER be 0 (without becoming 0, thus no longer being 1).
What number is before 1? 0.
0D (point) is "One with Everything", literally
". with 0".
0D (point) is
Monad (singularity) is
Quark (strong nuclear force) is
Soul (6 thousand trillion trillion trillion times stronger than the force of gravity)
No spatial extension. Zero size.
Exact location only.
As Leibniz put it: “If an ontological theory implies the existence of two scenarios that are empirically indistinguishable in principle but ontologically distinct ... then the ontological theory should be rejected and replaced with one relative to which the two scenarios are ontologically identical.”
In other words, if a theory describes two situations as being distinct, and yet also implies that there is no conceivable way, empirically, to tell them apart, then that theory contains some superfluous and arbitrary elements that ought to be removed.
Leibniz’s prescription is, of course, widely accepted by most physicists today. The idea exerted a powerful influence over later thinkers, including Poincaré and Einstein, and helped lead to the theories of special and general relativity. And this idea, Spekkens suggests, may still hold further value for questions at the frontiers of today’s physics.
Leibniz’s correspondent
Clarke objected to his view, suggesting an exception. A man riding inside a boat, he argued, may not detect its motion, yet that motion is obviously real enough. Leibniz countered that such motion is real because it can be detected by someone, even if it isn’t actually detected in some particular case. “Motion does not indeed depend upon being observed,” he wrote, “but it does depend upon being possible to be observed ... when there is no change that can be observed, there is no change at all.”
In this, Leibniz was arguing against prevailing ideas of the time, and against Newton, who conceived of space and time in absolute terms. “I have said more than once,” Leibniz wrote, “that I hold space to be something merely relative.”
Einstein, of course, followed Leibniz’s principle when he noticed that the equations of electricity and magnetism make no reference to any absolute sense of motion, but only to relative motion. A conducting wire moving through the field of a magnet seems like a distinct situation from a magnet moving past a stationary wire. Yet the two situations are in fact empirically identical, and should, Einstein concluded, be considered as such. Demanding as much leads to the Lorentz transformation as the proper way to link descriptions in reference frames in relative motion. From this, one finds a host of highly counter-intuitive effects, including time dilation.
Einstein again followed Leibniz on his way to general relativity. In this case, the indistinguishability of two distinct situations - a body at rest in the absence of a gravitational field, or in free fall within a field - implied the impossibility of referring to any concept of absolute acceleration. In a 1922
lecture, Einstein recalled the moment of his discovery: “The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly the thought struck me: If a man falls freely, he would not feel his own weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity.”
Leibniz now mostly inhabits scientific history books, his ideas receiving scant attention in actual research. And yet, Spekkens argues, Leibniz’s principle concerning indistinguishability may be as useful as ever, especially when confronting foundational issues in physics. Consider the interpretation of quantum theory, where theorists remain separated into two opposing groups, loosely associated with the terms realism and empiricism. Although Leibniz’s principle can’t offer any way to unify the two groups, Spekkens argues, it might help them focus their attention on the most important issues dividing them, where progress might be made.
For example, one particular interpretation comes in the form of so-called pilot-wave theories, in which electrons and other particles follow precise but highly non-classical trajectories under the influence of a quantum potential, which produces the wave-like nature of quantum dynamics. These theories demonstrate by explicit example that nothing in quantum physics prohibits thinking about particles moving along well-defined trajectories. But the theory does require the existence of some absolute rest frame, while also implying that this frame can never be detected. Many other aspects of such theories also remain unconstrained by empirical data. Hence, one might take Leibniz’s principle as coming down against such theories.
On the other hand, Spekkens points out, Leibniz’s principle demands that distinct states be, in Leibniz’s own words, “empirically indistinguishable in principle,” and achieving such certainty is not easy. If several states appear indistinguishable now, future experiments might turn up measurable differences between them. So a proponent of the pilot-wave approach might agree with Leibniz’s principle, but still reject its application just yet. The aim of research, from this point of view, ought to be to seek out such evidence, or at least envision the conditions under which it might be obtained.
And in this sense, Spekkens notes, Leibniz’s principle also offers some criticism of
theorists from the empirical school, who object to pilot-wave or other realist interpretations of quantum theory for containing unmeasurable quantities. It implies, as he puts it, that the empiricists’ “set of mental tools is too impoverished.”
After all, progress in physics often requires imagination, and creative exploration of possible distinguishing features that have not yet been measured, or even thought to exist. Progress requires scientists to “entertain ontological hypotheses, expressed with concepts that are not defined purely in terms of empirical phenomena.”
Science thrives on the essential tension existing at the boundary between empirical observation and unconstrained imagination. Incredibly, Leibniz perceived that more than 300 years ago.