PARTIAL DIFFERENTIATION
(a) Find all the first partial derivatives of the following functionsf(x, y):
(i)x^2 y, (ii)x^2 +y^2 + 4, (iii) sin(x/y), (iv) tan−^1 (y/x),
(v)r(x, y, z)=(x^2 +y^2 +z^2 )^1 /^2.
(b) For (i), (ii) and (v), find∂^2 f/∂x^2 ,∂^2 f/∂y^2 and∂^2 f/∂x∂y.
(c) For (iv) verify that∂^2 f/∂x∂y=∂^2 f/∂y∂x.5.2 Determine which of the following are exact differentials:
(a) (3x+2)ydx+x(x+1)dy;
(b)ytanxdx+xtanydy;
(c) y^2 (lnx+1)dx+2xylnxdy;
(d)y^2 (lnx+1)dy+2xylnxdx;
(e) [x/(x^2 +y^2 )]dy−[y/(x^2 +y^2 )]dx.5.3 Show that the differential
df=x^2 dy−(y^2 +xy)dx
is not exact, but thatdg=(xy^2 )−^1 dfis exact.
5.4 Show that
df=y(1 +x−x^2 )dx+x(x+1)dy
is not an exact differential.
Find the differential equation that a functiong(x)mustsatisfyifdφ=g(x)df
is to be an exact differential. Verify thatg(x)=e−xis a solution of this equation
and deduce the form ofφ(x, y).
5.5 The equation 3y=z^3 +3xzdefineszimplicitly as a function ofxandy. Evaluate
all three second partial derivatives ofzwith respect toxand/ory.Verifythatz
is a solution of
x∂^2 z
∂y^2+
∂^2 z
∂x^2=0.
5.6 A possible equation of state for a gas takes the form
PV=RTexp(
−
α
VRT)
,
in whichαandRare constants. Calculate expressions for
(
∂P
∂V)
T,
(
∂V
∂T
)
P,
(
∂T
∂P
)
V,
and show that their product is−1, as stated in section 5.4.
5.7 The functionG(t) is defined by
G(t)=F(x, y)=x^2 +y^2 +3xy ,
wherex(t)=at^2 andy(t)=2at. Use the chain rule to find the values of (x, y)at
whichG(t) has stationary values as a function oft. Do any of them correspond
to the stationary points ofF(x, y) as a function ofxandy?
5.8 In thexy-plane, new coordinatessandtare defined by
s=^12 (x+y),t=^12 (x−y).
Transform the equation
∂^2 φ
∂x^2−
∂^2 φ
∂y^2=0
into the new coordinates and deduce that its general solution can be written
φ(x, y)=f(x+y)+g(x−y),
wheref(u)andg(v) are arbitrary functions ofuandv, respectively.