Deriving the Heat Equation in 2D & 3D (& in N Dimensions!) with Control Volumes and Vector Calculus

Here we derive the heat equation in higher dimensions using Gauss's theorem.

@eigensteve on Twitter

This video was produced at the University of Washington

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0:00 Overview
5:27 Heat Equation Derivation
11:45 Surface Integral to Volume Integral
15:04 Volume Integrals to PDEs

[shakennotstired]

Thanks professor. This is great staff. I am learning a lot. I am forever grateful for your effort

[ilpreterosso]

A great review for me of contents from last semester!

[abrarsalekinraiyan]

Hearing that Steve properly learned all these after becoming a grad student, makes me feel a bit better that I didn't understand all these in undergrad 😅. And now as a grad student its coming together slowly for me

[adityawagh]

Love your videos Steve, they really help me understand the physics to a basic level. However seeing you in real life would be mind-bending, as the Steve we know is your mirrored persona.

[erickleuro]

@13:40 my ¢5 cents to this amazing video, the partial derivative (with respect to time) can be moved inside the triple volume integral because the system is (or assumed to be) lineal and therefore obeys the principle of superposition; otherwise, this wouldn't be allowed

[tomctutor]

Excellent derivation, I no doubt expect some examples will follow, thankyou!♒

[ramkitty]

i apply your teaching of vcalc to self physiotherapy and joint manipulation with the goal of equalizing tesions through muscle bundles by forcing applications at muscle attachment or roots.

[ajitkumar-iphc]

Dear Steve, thank you for uploading your lectures it will be very helpful if you can make a video on Rayleigh-Benard convection simulation (DNS)

[Helmutandmoshe]

I sure hope this series continues. Is there a plan for more videos?

[maflojustin]

21:30: you almost divided by 'zero'😄

[mingusbingus]

"I didn't learn about PDEs until I was a graduate student"- Same, I feel like I was cheated out of such a deep way of understanding the world.

[benanderson]

Towards the end once you've written the equation in its simplified form, why is alpha squared? Is this just a simple way of denoting that the coefficient of the laplacian of u is always positive?

[oliviajulia]

Hi! Great content as always! 
Could you please do an example of this 3D heat equation including conduction, radiation and convection? 
It seems I can't find any example in the scientific literature... If you could refer to a source in the mean time where I could find this, it would be great!

[CyrusTabery]

You forgot the thermal conductivity in Fourier’slaw. Also it maybe good to be clear when constant heat capacity and density assumption is needed. Finally nit pick : u is for vectors, use a different letter for temperature, T maybe? It maybe fun to show the dimensionless versions of the equations…

[mathjitsuteacher]

Hi Steve, what do I need to do a post doc with you?

[santoshpathak]

(Time: 10: 50) Wow everybody is like me. I am gonna teach the same thing tomorrow and not sure how to explain that negative sign infront of the surface integral. I wanted to hear his explanation about it, he was so unhappy about his own explanation.

[MrFazeFaze]

great review! Would you be going into numerical methods too? i.e. FVM

[batripleO]

I think you forgot to add it to the playlist

[JonasAstrom-zvtu]

Sorry for a lightweight comment: you are writing in reverse, apparently without difficulty. I couldn't do that!

[sheraz]

Could you please write a little bigger?