By Frances Kirwan, Jonathan Woolf

A grad/research-level advent to the ability and sweetness of intersection homology concept. available to any mathematician with an curiosity within the topology of singular areas. The emphasis is on introducing and explaining the most principles. tough proofs of vital theorems are passed over or purely sketched. Covers algebraic topology, algebraic geometry, illustration idea and differential equations.

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4) P = (Y - 1)PE. 5) PY ( p p - ’ ) = - P E - div q. Y-1 - ~ We shall call dissipationless any flow of such a gas in which P * E + div q = 0. 21) but independent of it, for here we have not introduced q or anything at all from thermodynamics. 6), we see that aflow ofan ideal gas with constant specijic heats is dissipationless if and only if pp-? = const. 7) for each fluid point. This statement is one form of POISSON’S law of adiabatic an equivalent statechange. 1). 17), is a given function of E and p.

THE STOKES-KIRCHHOFF GAS Of course, an ideal gas may be also a perfect fluid, but it need not be. There are many different kinds of internal friction which an ideal gas might have. 7). 12)r equivalently, in all flows m = p. 13)r The resulting theory, that of an ideal, linearly viscous gas with constant specijic heats and null bulk viscosity, we shall call the Stokes-Kirchhoff Theory. 4) P = (Y - 1)PE. 5) PY ( p p - ’ ) = - P E - div q. Y-1 - ~ We shall call dissipationless any flow of such a gas in which P * E + div q = 0.

Const. 7) for each fluid point. This statement is one form of POISSON’S law of adiabatic an equivalent statechange. 1). 17), is a given function of E and p. (ii) DYNAMICAL SIMILARITY 21 The special case of the Stokes-Kirchhoff theory obtained by taking p and Mp as 0 we shall call the Euler-Hadamard Theory. It represents the fluid as being an ideal perfect gas that does not conduct heat, so all its flows are dissipationless. In that theory we may apply HUGONIOT’S theorem and conthe clude that the intrinsic speed of propagation of weak waves is ,/=; value of this scalar field at a place and time is the corresponding speed ofsound.