PREFACE 
Any course in Advanced Calculus must deal with partial differen 
tiation and multiple integrals, with systematic integration, with im 
proper integrals, and to some extent with complex quantities. Be 
fore modern methods in analysis — “e-methods,”as they are sometimes 
called — had been developed, the race became aware of a wide range 
of applications in physics (including geometry, the noblest branch of 
physics) which, although not yet technical, nevertheless make exact 
ing demands on precise formulation and thus bring out both the phys 
ical hypotheses and the analytic means of working with them. The 
deduction of the partial differential equation which governs the flow 
of heat or electricity in conductors, the establishment of the equation 
of continuity in hydrodynamics and elasticity, and the setting up of 
the equations which describe the motion of the vibrating string or 
membrane, are cases in point. In these days when modern physics 
is primarily interested in the motion of discrete particles, it is par 
ticularly timely to emphasize continuous distributions of physical sub 
stances throughout regions of space, and contimious transformations 
of space. 
The demands which geometry makes on partial differentiation are 
relatively slight. In thermodynamics a thoroughgoing appreciation 
of what the independent variables are (in order that, when the letters 
expressing the variables of two classes overlap, the meaning of the 
partial derivatives may be clear) and the ability to think in terms of 
line integrals, are indispensable. 
Oscillatory motion is a basal conception in physics. Simple har 
monic motion ; next, the simplest case of damping ; and finally the 
case of an impressed periodic force — these physical pictures are im 
portant alike for the student of physics and the student of pure 
mathematics, for they help to give him perspective as he proceeds 
with the study of the chapter in differential equations which relates 
to the solution of boundary value problems by means of developments 
into series. Moreover, Fourier’s series and similar developments are 
illumined by the applications they here find.