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. Test yourself with some exercises on the Heat Equation with this FREE worksheet in Maple Learn: https://learn.maplesoft.com/doc/yllfw9gbg7/trm-heat-equation-worksheet Investigate separable solutions to the Heat Equation here: https://learn.maplesoft.com/d/PQOSOULGKPFHEREGBKJSJRHRKPCPJPHLGRCIFSHOOKFUERPPEGLUDILQNTLIAUJUBTJMLGKPMJJRHMJFIHKJKNLIKOLHNKFNONKU 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. Check your working using the Maple Calculator App – available for free on Google Play and the App Store. Android: https://play.google.com/store/apps/details?id=com.maplesoft.companion&hl=en Apple: https://apps.apple.com/us/app/maple-companion/id1466659419 Other videos in the Oxford Calculus series can be found here: https://www.youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S Finding critical points for functions of several variables: https://youtu.be/Leomuu82-u8 Classifying critical points using the method of the discriminant: https://www.youtube.com/watch?v=5M_ts8Q2LEM Partial differentiation explained: https://youtu.be/RVwcBGzQcT8 Second order linear differential equations: https://youtu.be/F54yhRB9qDI Integrating factors explained: https://www.youtube.com/watch?v=ftqKuOfOX3E Solving simple PDEs: https://youtu.be/uztjxrGY6Jw Jacobians explained: https://www.youtube.com/watch?v=YqMelRryG8U Separation of variables integration technique explained: https://youtu.be/zk41c0vs9XQ Solving homogeneous first order differential equations: https://youtu.be/uqvqjbAcbL8 Taylor’s Theorem explained with examples and derivation: https://www.youtube.com/watch?v=DULzJmUHN5g Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: https://www.youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website https://tomrocksmaths.com/ You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. https://www.facebook.com/tomrocksmaths/ https://twitter.com/tomrocksmaths https://www.instagram.com/tomrocksmaths/ Get your Tom Rocks Maths merchandise here: https://beautifulequations.net/collections/tom-rocks-maths