29.5. EXERCISES 593

16. Suppose k (x) = k, a constant and f = 0. Then in one dimension, the heat equation isof the form ut = αuxx. Show that u(x, t) = e−αn2t sin(nx) satisfies the heat equation3

17. Let U be a three dimensional region for which the divergence theorem holds. Showthat

∫U ∇×F dx =

∫∂U n×F dS where n is the unit outer normal.

18. In a linear, viscous, incompressible fluid, the Cauchy stress is of the form

Ti j (t,y) = λ

(vi, j (t,y)+ v j,i (t,y)

2

)− pδ i j

where p is the pressure, δ i j equals 0 if i ̸= j and 1 if i = j, and the comma followedby an index indicates the partial derivative with respect to that variable and v is thevelocity. Thus vi, j =

∂vi∂y j

. Also, p denotes the pressure. Show, using the balance ofmass equation that incompressible implies divv = 0. Next show that the balance ofmomentum equation requires

ρ v̇− λ

2∆v = ρ

[∂v

∂ t+

∂v

∂yivi

]− λ

2∆v = b−∇p.

This is the famous Navier Stokes equation for incompressible viscous linear flu-ids. There are still open questions related to this equation, one of which is worth$1,000,000 at this time.

3Fourier, an officer in Napoleon’s army studied solutions to the heat equation back in 1813. He was interestedin heat flow in cannons. He sought to find solutions by adding up infinitely many multiples of solutions ofthis form, the multiples coming from a given initial condition occurring when t = 0. Fourier thought that theresulting series always converged to this function. Lagrange and Laplace were not so sure. This topic of Fourierseries, especially the question of convergence, fascinated mathematicians for the next 150 years and motivated thedevelopment of analysis. The first proof of convergence was given by Dirichlet. As mentioned earlier, Dirichlet,Riemann, and later Lebesgue and Fejer were all interested in the convergence of Fourier series and the last bigresult on convergence did not take place till the middle 1960’s and was due to Carleson and more generally byHunt. It was a surprise because of a negative result of Kolmogorov from 1923. Actually these ideas were used bymany others before Fourier, but the idea became associated with him.

Fourier was with Napoleon in Egypt when the Rosetta Stone was discovered and wrote about Egypt in Descrip-tion de l’Égypte. He was a teacher of Champollion who eventually made it possible to read Egyptian by usingthe Rosetta Stone. This expedition of Napoleon caused great interest in all things Egyptian in the first part of thenineteenth century.