9.3 Partial differentiation 249
Each of these graphs is a planar ‘cut’ through the three-dimensional surface. In the
general case, a function of nvariables defines a ‘surface’ in an(n 1 + 1 1)-dimensional
space, and its graph as a function of one of the variables is obtained by taking a planar
cut through the representative(n 1 + 1 1)-dimensional surface.
9.3 Partial differentiation
We saw in Chapter 4 that the first derivative of a function of one variable is
interpreted graphically as the slope of a tangent line to its graph, and dynamically as
the rate of change of the function with respect to the variable. For a function of
two or more variables there exist as many independent first derivatives as there are
independent variables. For example, the function
z 1 = 1 f(x, y) 1 = 1 x
21 − 12 xy 1 − 13 y
2can be differentiated with respect to variable x, with ytreated as a constant, to give the
partial derivativeof the function with respect to x
(read as ‘partial dz by dx’),
2and with respect to yat constant xfor the partial
derivative with respect to y
The existence of partial derivatives and the validity of the operation of partial differ-
entiation are subject to the same conditions of continuity and smoothness as for the
ordinary (total) derivative. If these conditions are satisfied then the partial derivatives
of a function of two variables are defined by the limits (compare equation (4.8))
(9.2)
(9.3)
The geometric interpretation of these quantities is shown in Figure 9.3. The plane
ABC is parallel to the xz-plane, so that y 1 = 1 constant in the plane and the values of
∂z 2 ∂xfor this value of yare the slopes of the tangent lines to the curve APB. In the
same way, the plane DEF is parallel to the yz-plane, and the values of∂z 2 ∂yare
∂
∂
=
,+∆ − ,
∆
→
z
y
fxy y fxy
y y
lim
()()
∆ 0
∂
∂
=
+∆ , − ,
∆
→
z
x
fx xy fxy
x x
lim
()()
∆ 0
∂
∂
=− −
z
y
26 xy
∂
∂
=−
z
x
22 xy
2The notation∂z 2 ∂xwas first used by Legendre in 1788, but began to be accepted only after Jacobi used it in
his theory of determinants in 1841.