346 DIFFERENTIAL CALCULUS⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
(a)∂k
∂T=AH
RT^2eTS−S
RT(b)∂A
∂T=−kH
RT^2eH−TS
RT(c)∂(S)
∂T=−H
T^2(d)∂(H)
∂T=S−Rln(
k
A)⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦34.3 Second order partial derivatives
As with ordinary differentiation, where a differen-
tial coefficient may be differentiated again, a partial
derivative may be differentiated partially again to
give higher order partial derivatives.
(i) Differentiating∂V
∂rof Section 34.2 with respecttor, keepinghconstant, gives∂
∂r(
∂V
∂r)
whichis written as∂^2 V
∂r^2
Thus if V=πr^2 h,then∂^2 V
∂r^2=∂
∂r(2πrh)= 2 πh.(ii) Differentiating∂V
∂hwith respect toh, keepingrconstant, gives∂
∂h(
∂V
∂h)
which is written as∂^2 V
∂h^2Thus∂^2 V
∂h^2=∂
∂h(πr^2 )= 0.(iii) Differentiating
∂V
∂hwith respect tor, keepinghconstant, gives∂
∂r(
∂V
∂h)
which is written as∂^2 V
∂r∂h. Thus,
∂^2 V
∂r∂h=∂
∂r(
∂V
∂h)
=∂
∂r(πr^2 )= 2 πr.(iv) Differentiating∂V
∂rwith respect toh, keepingrconstant, gives∂
∂h(
∂V
∂r)
, which is written as∂^2 V
∂h∂r. Thus,
∂^2 V
∂h∂r=∂
∂h(
∂V
∂r)
=∂
∂h(2πrh)= 2 πr.(v)∂^2 V
∂r^2,∂^2 V
∂h^2,∂^2 V
∂r∂hand∂^2 V
∂h∂rare examples ofsecond order partial derivatives.(vi) It is seen from (iii) and (iv) that∂^2 V
∂r∂h=∂^2 V
∂h∂r
and such a result is always true for continuous
functions (i.e. a graph of the function which has
no sudden jumps or breaks).Second order partial derivatives are used in the
solution of partial differential equations, in waveg-
uide theory, in such areas of thermodynamics cov-
ering entropy and the continuity theorem, and when
finding maxima, minima and saddle points for func-
tions of two variables (see Chapter 36).Problem 7. Givenz= 4 x^2 y^3 − 2 x^3 + 7 y^2 find(a)∂^2 z
∂x^2(b)∂^2 z
∂y^2(c)∂^2 z
∂x∂y(d)∂^2 z
∂y∂x(a)∂z
∂x= 8 xy^3 − 6 x^2∂^2 z
∂x^2=∂
∂x(
∂z
∂x)
=∂
∂x(8xy^3 − 6 x^2 )= 8 y^3 − 12 x(b)∂z
∂y= 12 x^2 y^2 + 14 y∂^2 z
∂y^2=∂
∂y(
∂z
∂y)
=∂
∂y(12x^2 y^2 + 14 y)= 24 x^2 y+ 14(c)∂^2 z
∂x∂y=∂
∂x(
∂z
∂y)
=∂
∂x(12x^2 y^2 + 14 y)= 24 xy^2(d)∂^2 z
∂y∂x=∂
∂y(
∂z
∂x)
=∂
∂y(8xy^3 − 6 x^2 )= 24 xy^2