PARTIAL DIFFERENTIATION
the horizontal. The cost of construction isαperunitheightofwallandβper
unit (slope) length of roof. Show that, irrespective of the values ofαandβ,to
minimise costswshould be chosen to satisfy the equationw^4 =16A(A−wh),andθmade such that 2 tan 2θ=w/h.
5.20 Show that the envelope of all concentric ellipses that have their axes along the
x-andy-coordinate axes, and that have the sum of their semi-axes equal to a
constantL, is the same curve (an astroid) as that found in the worked example
in section 5.10.
5.21 Find the area of the region covered by points on the lines
x
a+
y
b=1,
where the sum of any line’s intercepts on the coordinate axes is fixed and equal
toc.
5.22 Prove that the envelope of the circles whose diameters are those chords of a
given circle that pass through a fixed point on its circumference, is the cardioid
r=a(1 + cosθ).Hereais the radius of the given circle and (r, θ) are the polar coordinates of the
envelope. Take as the system parameter the angleφbetween a chord and the
polar axis from whichθis measured.
5.23 A water feature contains a spray head at water level at the centre of a round
basin. The head is in the form of a small hemisphere perforated by many evenly
distributed small holes, through which water spurts out at the same speed,v 0 ,in
all directions.
(a) What is the shape of the ‘water bell’ so formed?
(b) What must be the minimum diameter of the bowl if no water is to be lost?5.24 In order to make a focussing mirror that concentrates parallel axial rays to one
spot (or conversely forms a parallel beam from a point source), a parabolic shape
should be adopted. If a mirror that is part of a circular cylinder or sphere were
used, the light would be spread out along a curve. This curve is known as a
causticand is the envelope of the rays reflected from the mirror. Denoting byθ
the angle which a typical incident axial ray makes with the normal to the mirror
at the place where it is reflected, the geometry of reflection (the angle of incidence
equals the angle of reflection) is shown in figure 5.5.
Show that a parametric specification of the caustic is
x=Rcosθ( 1
2 +sin(^2) θ),y=Rsin (^3) θ,
whereRis the radius of curvature of the mirror. The curve is, in fact, part of an
epicycloid.
5.25 By considering the differential
dG=d(U+PV−ST),
whereGis the Gibbs free energy,Pthe pressure,Vthe volume,Sthe entropy
andTthe temperature of a system, and given further that the internal energyU
satisfies
dU=TdS−PdV,
derive a Maxwell relation connecting (∂V /∂T)Pand (∂S/∂P)T.