Open, multimodal PDE learning with active workbook spaces.
Consider the transport equation
ut + cux = 0, where c = constant.
• We look for functions u = u(x, t) that satisfy the transport equation via directional derivatives.
• Remember from vector calculus that the directional derivative of u in the direction of a
vector ˆv is
Dvˆu = ˆv • ∇u
where ˆv is the unit vector in direction of ˆv and ∇u = (ux, ut).
• In particular, if Dvˆu = 0 at all points then u is constant along all lines that are tangential
• Learn standard PDE solution methods through guided, step-by-step examples
• Use the companion YouTube lessons to follow each chapter’s subsection
• Practice actively with workbook “active learning spaces” and annotations
• Compare approaches across transport, wave, heat, Laplace, and Fourier methods
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