Let
F
n
{F_n}
be a free group of rank
n
n
with free basis
x
1
,
⋯
,
x
n
{x_1}, \cdots ,{x_n}
. Let
{
y
1
,
⋯
,
y
k
}
\{ {y_1}, \cdots ,{y_k}\}
be a set of
k
≦
n
k \leqq n
elements of
F
n
{F_n}
, where each
y
i
{y_i}
is represented by a word
Y
i
(
x
1
,
⋯
,
x
n
)
{Y_i}({x_1}, \cdots ,{x_n})
in the generators
x
j
{x_j}
. Let
∂
y
i
/
∂
x
j
\partial {y_i}/\partial {x_j}
denote the free derivative of
y
i
{y_i}
with respect to
x
j
{x_j}
, and let
J
k
n
=
|
|
∂
y
i
/
∂
x
j
|
|
{J_{kn}} = ||\partial {y_i}/\partial {x_j}||
denote the
k
×
n
k \times n
Jacobian matrix. Theorem. If
k
=
n
k = n
, the set
{
y
1
,
⋯
,
y
n
}
\{ {y_1}, \cdots ,{y_n}\}
generates
F
n
{F_n}
if and only if
J
n
n
{J_{nn}}
has a right inverse. If
k
>
n
k > n
, the set
{
y
1
,
⋯
,
y
k
}
\{ {y_1}, \cdots ,{y_k}\}
may be extended to a set of elements which generate
F
n
{F_n}
only if
J
k
n
{J_{kn}}
has a right inverse. Several applications are given.