§5. Differentiable functions of several variables
275
the image of a vector h E JR2 under the linear map39 Df(z): ]R2 ---+ ]R2; the
expression o(h) now clearly denotes a function of h which is negligible with
respect to the length Ihl ~ lui + Ivl of the vector h.
The expression df = f'(a)dx introduced in nO 14 extends immediately to
functions of several variables. It is quite clear that at any point of the plane
the differentials of the coordinate functions (x, y) f-+ X and (x, y) f-+ yare
dx(a, b) : (u, v) ~ u, dy(a, b) : (u, v) ~ v.
As in nO 14, one can then write (3) in the form
(19.3')
df(a, b) = Dd(a, b)dx(a, b) + D2!(a, b)dy(a, b),
a relation between linear functions of (u, v). In short,
(19.3")
df = f~dx + f~dy.
This applies, for example, to the functions z = x + iy and z = x - iy, whence
dz = dx + idy,
dz = dx - idy,
which means that the differential of (x, y) f-+ Z at any point is the linear
function (u,v) f-+ u - iv. As we recalled above, a linear map of]R2 into]R2
can be represented by a 2 x 2 matrix. To obtain that of (3), i.e. of D f(a, b),
one has to calculate the real coordinates of the result, i.e., since u and v
are real, replace the derivatives in (3) either by their real parts, or by their
imaginary parts, which one obtains on performing the same operation on f.
It is clear that on putting (a, b) = z the matrix of df(z) or Df(z) is simply
(19.4)
it is called the Jacobian matrix of fat z and often we shall make no distinction
between the linear map Df(z) and the matrix (4). This causes no problem
so long as one does not change the system of coordinates in ]R2, i.e. makes
no distinction between a "geometric" vector h with coordinates u, v and the
matrix
associated with it.
39 Recall that in linear algebra one frequently writes Ah, rather than A(h), for the
the value at the vector h under a linear map A (inheritance from of the age when
one spoke of matrices instead of linear maps). Failing this convention, one would
have to write D f(z)(h) for what we write as D f(z)h. The notation df(z; h) is
also used often; see Vol. III, Chap. IX, § 1.
275
the image of a vector h E JR2 under the linear map39 Df(z): ]R2 ---+ ]R2; the
expression o(h) now clearly denotes a function of h which is negligible with
respect to the length Ihl ~ lui + Ivl of the vector h.
The expression df = f'(a)dx introduced in nO 14 extends immediately to
functions of several variables. It is quite clear that at any point of the plane
the differentials of the coordinate functions (x, y) f-+ X and (x, y) f-+ yare
dx(a, b) : (u, v) ~ u, dy(a, b) : (u, v) ~ v.
As in nO 14, one can then write (3) in the form
(19.3')
df(a, b) = Dd(a, b)dx(a, b) + D2!(a, b)dy(a, b),
a relation between linear functions of (u, v). In short,
(19.3")
df = f~dx + f~dy.
This applies, for example, to the functions z = x + iy and z = x - iy, whence
dz = dx + idy,
dz = dx - idy,
which means that the differential of (x, y) f-+ Z at any point is the linear
function (u,v) f-+ u - iv. As we recalled above, a linear map of]R2 into]R2
can be represented by a 2 x 2 matrix. To obtain that of (3), i.e. of D f(a, b),
one has to calculate the real coordinates of the result, i.e., since u and v
are real, replace the derivatives in (3) either by their real parts, or by their
imaginary parts, which one obtains on performing the same operation on f.
It is clear that on putting (a, b) = z the matrix of df(z) or Df(z) is simply
(19.4)
it is called the Jacobian matrix of fat z and often we shall make no distinction
between the linear map Df(z) and the matrix (4). This causes no problem
so long as one does not change the system of coordinates in ]R2, i.e. makes
no distinction between a "geometric" vector h with coordinates u, v and the
matrix
associated with it.
39 Recall that in linear algebra one frequently writes Ah, rather than A(h), for the
the value at the vector h under a linear map A (inheritance from of the age when
one spoke of matrices instead of linear maps). Failing this convention, one would
have to write D f(z)(h) for what we write as D f(z)h. The notation df(z; h) is
also used often; see Vol. III, Chap. IX, § 1.