Oxford Calculus: Heat Equation Derivation

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.

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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[TheAnimesNanda]

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 💯

[msmsa]

Im a lawyer (zero relation to maths) but i love your videos! I will make sure my future kids watch your videos 🙏🏼

[ashafaghi]

Excellent method of explaining complex derivations and solutions, thank you!

[adershvarshnei]

I love how you teach! Very inviting and makes topics like this unintimidating!

[hameedmusa-basheer]

I'M OUT OF WORDS. SIMPLY THANK YOU!

[xaviergonzalez]

Thank you Sir! Well explained! English is not my native language but I got you at 100%

[AtricosHU]

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.

[samicalvo]

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

[scollyer.tuition]

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.

[evansherman]

easy to follow and extremely thorough, Thanks!

[TheLethalDomain]

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).

[tomgargan]

I really hope this is what my physics degree will be like, amazing vid as always.

[God_For_A_Reason]

@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.

[kramlyn]

I never learn this kind of math but only in you😂🤣no Question for an Oxford math professor keep it up sir tom🙂

[frun]

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.

[الصوتالرخيم]

I love how you teach mathematics. I wish that I'm a student in your classes

[mastershooter]

Now let's do this derivation for heat dispersion in fluids! heat equation for fluids!!

[zimzimal]

Would be class having this guy as my lecturer

[knowitall]

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.

[marouaniAymen]

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 ?