Oxford Calculus: Heat Equation Derivation
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- Опубліковано 24 чер 2022
- University of Oxford mathematician Dr Tom Crawford derives the Heat Equation from physical principles. The Heat Equation is one of the first PDEs studied as an undergraduate maths student.
Test yourself with some exercises on the Heat Equation with this FREE worksheet in Maple Learn: learn.maplesoft.com/doc/yllfw...
Investigate separable solutions to the Heat Equation here: learn.maplesoft.com/d/PQOSOUL...
We start with a one-dimensional long and thin cylindrical rod which has heat flowing through it from left to right. We calculate the change in the internal energy for a short section between x=a and a+h (for small h) using Liebniz Rule. This must be equal to the heat flux in minus the heat flux out of the segment.
By dividing both sides of the equation by h and taking the limit as h tends to zero we arrive at the first equation relating the temperature to the heat flux. We then use Fourier’s Law which relates the heat flux to the temperature gradient to rewrite the equation in terms of only the temperature. Finally, the extension of the equation to 2D and 3D is discussed.
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Other videos in the Oxford Calculus series can be found here: • Oxford Calculus
Finding critical points for functions of several variables: • Oxford Calculus: Findi...
Classifying critical points using the method of the discriminant: • Oxford Calculus: Class...
Partial differentiation explained: • Oxford Calculus: Parti...
Second order linear differential equations: • Oxford Mathematics Ope...
Integrating factors explained: • Oxford Calculus: Integ...
Solving simple PDEs: • Oxford Calculus: Solvi...
Jacobians explained: • Oxford Calculus: Jacob...
Separation of variables integration technique explained: • Oxford Calculus: Separ...
Solving homogeneous first order differential equations: • Oxford Calculus: Solvi...
Taylor’s Theorem explained with examples and derivation: • Oxford Calculus: Taylo...
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I am a mechanical engineer who studied about the various forms of the heat equation in the Cartesian, Cylindrical and Spherical coordinates but I never actually understood the derivation of it so thank you for clearing that up for me Dr. Crawford 💯
glad I could help :)
Universities are full of awful “professors”.
Im a lawyer (zero relation to maths) but i love your videos! I will make sure my future kids watch your videos 🙏🏼
I love how you teach! Very inviting and makes topics like this unintimidating!
Excellent method of explaining complex derivations and solutions, thank you!
Glad it was helpful!
Thank you Sir! Well explained! English is not my native language but I got you at 100%
I'M OUT OF WORDS. SIMPLY THANK YOU!
easy to follow and extremely thorough, Thanks!
Test yourself with some exercises on the Heat Equation with this FREE worksheet in Maple Learn: learn.maplesoft.com/d/FJNFDPCMIOLMHKOLCFPHBMCUPLCOJOALBKFNNHBTARLRGFLPDIIUHOPMLKGOFLJPPGEKGQDHKLNNLPLUPRNKFTOLLNELMLNNOQLP
Thx 👍
I never learn this kind of math but only in you😂🤣no Question for an Oxford math professor keep it up sir tom🙂
I really hope this is what my physics degree will be like, amazing vid as always.
18:02 That is just a simple L'Hospital rule. The denominator h tends to 0, and the nominator, the integral, also tends to 0. Differentiating both yields the derivative of the integral, over 1. Taking the limit as h goes to 0 we simply get the the argument of the previous integral evaluated at a.
That's redundant, the quotient is already a derivative of an integral
I think that, for the LHS, it would have been clearer if you had relied on the Fundamental Theorem of Calculus:
int_a^b f(x) dx = F(b) - F(a)
with dF(x)/dx = f(x) so that:
1/h ( int_a^{a+h} f(x) dx = 1/h (F(a+h) - F(a))
whose limit is more obviously the required derivative.
For those wondering how do you rigorously proof that the limit of that integral divided by h is just the integrand, here's the proof:
The limit is for h tending to 0, the integral from x to x+h and I call the integrand "A" (a function of x):
lim((Integral(A)/h)= lim[(F[A(x+h)] - F[A(x)])/h] = dF/dx=A(x)
Where F is the primitive and therefore, by definition, dF/dx is the integrand. QED
I love how you teach mathematics. I wish that I'm a student in your classes
Well, that was fun! Bit of a handwaving argument as to how you can differentiate an integral, as you acknowledged, but good enough for an engineer!
Eqs for both gr and qm are heat equations. Einstein's field equations describe the flow of heat. Schrödinger's equation is also a diffusion equation.
Thanks for the clear explanation.
Glad it was helpful!
I enjoyed, and mostly understood, that a lot!
you're welcome :)
Looking forwards to the video
Amazing video! Thanks!
Glad you liked it!
Couple of points when you are mentioning specific heat capacity you need to mention that there are two- the specific heat capacity at constant volume and the specific heat capacity at constant pressure. For solids they are identical but for gases they are different. The integral for internal heat is not required if you assume that the material is homogeneous and the temperature is evenly distributed along the y axis. k is seen as old hat for thermal conductivity and new symbols used by the heat transfer community . You will find most new books use lambda for thermal conductivity and alpha for heat transfer coefficients. A good introduction.
@17:17 since it is similar to 0/0 form, you can think of applying the L'Hôpital's rule which basically means differentiating both the numerator and the denominator.
We can just use the definition of partial differentiation
Fun fact after I worked through this further. If you make the time derivative a second derivative and replace kappa with the speed of light squared, you get the wave equation. By setting it up in 3D with the Laplacian and dividing the speed of light squared on both sides of the equation, you can pull the spacial derivatives over, negating them from the temporal derivative with a factor of 1/c^2. This yields the d'Alembert operator acting on a perturbation. Generalizing the perturbation function as a tensor such that space and time act on equal footing yields the gravitational wave equation (e.g the d'Alembert acting on the perturbation tensor = 0).
Now let's do this derivation for heat dispersion in fluids! heat equation for fluids!!
Hey Tom, when does one use the Lagrange interpolation Formula? I was working with functions and then derived the formula, but I am not very sure of its application.
Awesome.
Brilliant!
Would be class having this guy as my lecturer
Thanks!
شكرا جزيلا
So A is mall enough that heat is the same everywhere in the area, but large enough that a change in A results in a change in the mass of the rod, and so of the change in tempt.
At the same time, 'h' is small enough that the temperature at A and A+h is the same.
could you give an insight as to why the temperature curve slope @21:25 is positive, if going by the assumption that x axis our parameter t- time, shouldn't it be a descending curve and hence a negative slope?
great video indeed, thanks :)
Glad you liked it!
That was the nabla or del or grad operator rather than the symbol for the laplacian (which looks like a delta).
I've heard that the Black-Scholes formula for valuing options was derived based on this process. Any insight?
thank you
You're welcome
As you substituted the term for the thermal diffusivity my brain went like: 😌
What you could have done, which would have been neater (IMHO) is to write q(a+h,t)-q(a,t)=\int_{a}^{a+h}\frac{\partial q}{\partial x}dx. Then you would have taken everything to one side and just looked at the integrated and your equation would have popped out.
More derivation videos please 😁
wave equation to come next :)
Hey Dr.Tom, is it possible we could use the heat equation and boundary conditions as the Nuemann condition to model tumor growth. I am right now in the 10th standard and I am trying to model tumor growth with partial differential equations like the kolmogrov fischer equation, and stuff with nondimensional variables. If so, is it possible you could list a short explanation or so?
Thanks for video, but what is the difference between the Heat Equation and the Diffusion Equation, they seem to be the same, am I wrong ?
is funtion 'q' parametric?
funny, density times specific heat capacity is the volumetric heat capacity isn't it? Since density and specific heat capacity are both physical properties, you should get the volumetric heat capacity from the physicists directly because some materials might have had their heat capacity measured by volume rather than or in addition to by mass and they will be able to give you the most accurate number. As it is, you also have to know how to incorporate the uncertainty of two numbers together to derive the uncertainty in your answer but combining those two uncertainties is part of the physicists specialism even if they didn't measure the volumetric heat capacity.
Specific heat capacity at constant pressure...
I feel back again
That shirt is very badass
I've never seen a teacher in France calling for intuition
Is this kind of maths expected to be done by your students? What year students? And how easily?
I teach this to my first year students as part of the PDEs course. This would be around 5 marks out of 20 on an exam question (total 30 minutes per question).
@@TomRocksMaths Hmmm.. speaking from personal experience as an Engineering Science undergrad at Tom's college 40+ years ago, I'd expect this to be one of those derivations that you learnt as part of the course, rather than seeing it for the first time in an exam. You'd "rattle it off" in about five minutes in the exam then go on to apply it in the rest of the question. I seem to remember learning about 50 such things for Engineering Science finals. Maths might be different. I'm actually surprised this is part of the maths course as I always assumed maths at university level was much more abstract. Nice.
How old are you and what age were you when you got undergrad / phd degrees?
He is 32 so you can calculate the age when he got his undergrad and phd degrees
@@vihaannair5165 oh so it was normal times lmaoooo on google it said when he was 13 he started undergrad. I was very confused thank you for clarifying.
@@defaultd-6979 no problem. By the way, I think he completed his PhD in 2012
@@vihaannair5165 interesting I’m assuming he had a good environment to allow him to do so, but I don’t think your year is correct cuz wouldn’t that put him like year older than when terrence tao did and he doesn’t seem like he comes close to terrence
@@defaultd-6979 oh sorry I made a mistake; that is when he started it
Is it the same way Fourier came across the equation.....
What is “haich”?
Weird huh? don't know the hour they no longer honored the silent aspect of the letter H. Honestly, the what, where, and why of it seems like a whale for rhyme over reason.
Please, Tom, make a video deriving Laplace Transform… I beg you. There is no satisfactory explanation on the internet 😭
I will get there eventually in the 'Oxford Calculus' series
problem no free
Prelims are over… it’s not Fourier season
It will be again soon :)
So what's "Oxford" about this
This is based on the content I teach to my first year undergraduate students at Oxford.
@@TomRocksMaths Fair enough, the video title was just kind of clickbait, as in "look, it's Oxford math, look how hard and impressive this is"
about 5 days too late but thanks i guess
Is much easier to learn math with someone nice than with a 3000yo homophobic, xenophobic and misogynist teacher, thank you so much!!!
I'd rather have liked to see the rigorous derivation instead of the intuitive.
The creation of the known 1D-9D contingent universe:
(Anu = extension of God/Tathamet = extension of Ayin)
"Before the beginning there was void. Nothing. No flesh. No rock. No air. No heat. No light. No dark. Nothing, save a single, perfect pearl.
Within that pearl dreamed a mighty, unfathomable spirit-the One- Anu. Made of shining diamond.
Anu was the sum of all things: good and evil, light and dark, physical and mystical, joy and sadness-all reflected across the crystalline facets of its form. And, within its eternal dream-state, Anu considered itself-all of its myriad facets.
Seeking a state of total purity and perfection, Anu cast all evil from itself. All dissonance was gone.
But what of the cast-off aspect of its being? The dark parts, the sharp, searing aspects of hate and pridefulness?
Those could not remain in a state of separation, for all things are drawn to all things. All parts are drawn to the whole.
Those discordant parts assembled into the Beast-the Dragon.
Tathamet was his name-and he breathed unending death and darkness from his seven devouring heads.
The Dragon was solely composed of Anu's cast-off aspects. The end sum of the whole became a singular Evil- the Prime Evil, from which all the vileness would eventually spread throughout existence.
Though separate beings, Anu and the Dragon were bound together within the Pearl's shadowed womb. There they warred against each other in an unending clash of light and shadow for ages uncounted.
The diamond warrior and the seven-headed dragon proved to be the equal of the other, neither ever gaining the upper hand in their fierce and unending combat-till at last, their energies nearly spent after countless millennia of battle, the two combatants delivered their final blows.
The energies unleashed by their impossible fury ignited an explosion of light and matter so vast and terrible that it birthed the very universe all around us.
All of the stars above and the darkness that binds them. All that we touch. All that we feel. All that we know. All that is unknown.
All of it continues through the night and the day in the ebbing and flowing of the ocean tides and in the destruction of fire and the creation of the seed.
Everything of which we are aware, and that of which we are utterly unaware, was created with the deaths of Anu and the Dragon, Tathamet.
In the epicenter of reality lies Pandemonium, the scar of the universe's violent birth.
At its chaotic center lay the Heart of Creation, a massive jewel unlike any other: the Eye of Anu- the Worldstone.
It is the foundation stone of all places and times, a nexus of realities and vast, untold possibility.
Anu and Tathamet are no more, yet their distinct essences permeated the nascent universe-and eventually became the bedrock of what we know to be the High Heavens and the Burning Hells.
Anu's shining spine spun out into the primordial darkness, where it slowed and cooled. Over countless ages it formed into the Crystal Arch, around which the High Heavens took shape and form.
Though Anu was gone, some resonance of it remained in the holy Arch. Spirits bled forth from it-shining angels of light and sound who embodied the virtuous aspects of what the One had been.
Yet, despite the grace and beauty of this shining realm, it lacked the perfection of Anu's spirit.
Anu had passed into a benevolent place beyond this broken universe- a paradise of which nothing is known and yet represents perhaps the greatest-kept secret of Creation.
Longed for, but unimaginable."
-THE DAWN, BOOK OF CAIN.
Galileo proved the earth is not the center of the solar system.
Earth being the center of the universe is still on the table.
Great video until you mis pronounced “h”. I had to turn it off in protest!
Howdy.
The Theory of Everything, as to not be contradictory, would mean Theory of the Fundamentals of Everything.
Could you imagine the Theory of the Specifics of Everything? hahah.
I contend Gottfried Leibniz was correct about the fundamentals of our contingent universe and he just lacked 2022 verbiage/common knowledge.
More importantly is that humanity chose Isaac Newton's "real" universe, calculus, gravity, etc. This was a mistake. We need to correct this problem.
I've done my couch 🛋 warrior self's best impression of finishing what Leibniz started (with the intention of destroying what Newton started):
0D = (point); exact location only; non-composite substance.
(How ironic the symbol for Physics is the symbol for Metaphysics)
1D = line, straight; two points; composite substances
《0D (point) is exact location only; zero size; not a 'thing', not a 'part'; Monad》
"He is the invisible Spirit, of whom it is not right to think of him as a god, or something similar. For he is more than a god, since there is nothing above him, for no one lords it over him. For he does not exist in something inferior to him, since everything exists in him. For it is he who establishes himself. He is eternal, since he does not need anything. For he is total perfection. A being can have a relationship with a God but not the Monad as that would be a contradiction."
- The Apocryphon of John, 180 AD.
Monad (from Greek μονάς monas, "singularity" in turn from μόνος monos, "alone") refers, in cosmogony, to the Supreme Being, divinity or the totality of all things.
The concept was reportedly conceived by the Pythagoreans and may refer variously to a single source acting alone, or to an indivisible origin, or to both.
The concept was later adopted by other philosophers, such as Gottfried Wilhelm Leibniz, who referred to the monad as an elementary particle.
It had a geometric counterpart, which was debated and discussed contemporaneously by the same groups of people.
1st four dimensions are 0D, 1D, 2D, 3D ✅.
1st four dimensions are not 1D, 2D, 3D, 4D 🚫.
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 "imaginary/contingent" (1D xi, 2D yj, 3D zk).
1D-9D 'contingent' universe has "conscious lifeforms" (1D xi, 2D yj, 3D zk)..."turning" 'time'. We're "turners", "to turn". Humanity 3D zk.
"Turn" to what, you might ask. 5D is the center. All things and parts are drawn to the center, the whole.
[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 metaphysically create this middle set similar to a dimensional Venn Diagram with polarized lenses that we "turn" with our consciousness (which requires energy that we must consume i.e. calories to continue "to turn").
1D-3D set/7D-9D set creating the temporal illusion of 4D-6D set.
1D, 2D, 3D = spatial composite
4D, 5D, 6D = temporal illusory
7D, 8D, 9D = spectra energies
1D, 2D, 3D line, width, height
4D, 5D, 6D length, breadth, depth
7D, 8D, 9D continuous, emission, absorption
Symmetry:
1D, 4D, 7D line, length, continuous
2D, 5D, 8D width, breadth, emission
3D, 6D, 9D height, depth, absorption
Time-relative food for thought:
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).
4D = architecture (structure)
5D = design (solution)
Our universal constants have convoluted answers. Leibniz's Law of Sufficient Reason fixes this.
FUNDAMENTALS > specifics, basically.
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.
0D = (point)/not a 1D point:
[Math; Geometry]
A point is a 0-dimensional mathematical object which can be specified in -dimensional space using an n-tuple ( , , ..., ) consisting of. coordinates. In dimensions greater than or equal to two, points are sometimes considered synonymous with vectors and so points in n-dimensional space are sometimes called n-vectors.
[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.
What do we mean by tuple?
In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer. There is only one 0-tuple, referred to as the empty tuple. An n-tuple is defined inductively using the construction of an ordered pair.
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.
How do you make a quaternion?
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.
[Biology]
Points, conjugate. (Science; Microscopy) The pair of points on the principal axis of a mirror or lens so located that light emitted from either point will be focused at the other. Related points in the object and image are located optically so that one is the image of the other.
(See: polarizing element)
Where is this information useful when our planet is on fire and we lose habitat for humans in the near term future?
Why am I watching his?! I hated, hate and will always HATE math, the most boring subject I was forced to take.
as long as you're having fun now :)
Tom. a nice go at getting into the zone of some applied mathematics. Althoug you have only done the 1-d case the d dimensional Laplacian is of intrinsic interest. Your viewer should be aware aware that Luis Cafarelli ( who is a well known PDE expert) has given an interesting averaging explanation of the Laplaian. I came across this reading this guy's paper on Fourier theory : www.gotohaggstrom.com/Basic%20Fourier%20integrals.pdf It seems a bit hard core but I guess that comes with the territory.
3blue1brown's video on the heat equation is also excellent, especially for understanding the averaging property of the Laplacian.