348 DIFFERENTIAL CALCULUS
- z=2lnxy
⎡⎣(a)− 2
x^2(b)− 2
y^2
(c) 0 (d) 0⎤⎦- z=
(x−y)
(x+y)⎡⎢
⎢
⎢
⎣(a)− 4 y
(x+y)^3(b)4 x
(x+y)^3(c)2(x−y)
(x+y)^3(d)2(x−y)
(x+y)^3⎤⎥
⎥
⎥
⎦- z=sinhxcosh 2y
⎡⎢
⎢
⎢
⎣(a) sinhxcosh 2y
(b) 4 sinhxcosh 2y
(c) 2 coshxsinh 2y
(d) 2 coshxsinh 2y⎤⎥
⎥
⎥
⎦- Given z=x^2 sin (x− 2 y) find (a)
∂^2 z
∂x^2and(b)∂^2 z
∂y^2Show also that∂^2 z
∂x∂y=∂^2 z
∂y∂x
= 2 x^2 sin(x− 2 y)− 4 xcos(x− 2 y).
⎡⎢
⎢
⎣(a) (2−x^2 ) sin (x− 2 y)
+ 4 xcos (x− 2 y)(b)− 4 x^2 sin (x− 2 y)⎤⎥
⎥
⎦- Find
∂^2 z
∂x^2,∂^2 z
∂y^2and show that∂^2 z
∂x∂y=∂^2 z
∂y∂x
whenz=arccosx
y⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
(a)∂^2 z
∂x^2=−x
√
(y^2 −x^2 )^3,(b)∂^2 z
∂y^2=−x
√
(y^2 −x^2 ){
1
y^2+1
(y^2 −x^2 )}(c)∂^2 z
∂x∂y=∂^2 z
∂y∂x=y
√
(y^2 −x^2 )^3⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦- Givenz=
√(
3 x
y)
show that∂^2 z
∂x∂y=∂^2 z
∂y∂xand evaluate∂^2 z
∂x^2whenx=1
2andy=3.[
−1
√
2]- An equation used in thermodynamics is the
Benedict-Webb-Rubine equation of state for
the expansion of a gas. The equation is:
p=RT
V+(
B 0 RT−A 0 −C 0
T^2)
1
V^2+(bRT−a)1
V^3+Aα
V^6+C(
1 +γ
V^2)T^2(
1
V^3)
e−γ
V^2Show that∂^2 p
∂T^2=6
V^2 T^4{
C
V(
1 +γ
V^2)
e−Vγ 2
−C 0}