Lévy’s
N
N
-parameter Brownian motion in
d
d
-dimensional space is denoted by
W
(
N
,
d
)
{W^{(N,d)}}
. Using uniform partitions and a Vitali-type variation, Berman recently extended to
W
(
N
,
1
)
{W^{(N,1)}}
a classical result of Lévy concerning the relation between
W
(
1
,
1
)
{W^{(1,1)}}
and
2
2
-variation. With this variation
W
(
N
,
d
)
{W^{(N,d)}}
has variation dimension
2
N
2N
with probability one. An appropriate definition of weak variation is given using powers of the diameters of the images of sets which satisfy a parameter of regularity. A previous result concerning the Hausdorff dimensions of the graph and image is used to show the weak variation dimension of
W
(
N
,
d
)
{W^{(N,d)}}
is
2
N
2N
with probability one, extending the result for
W
(
1
,
1
)
{W^{(1,1)}}
of Goffman and Loughlin. If unrestricted partitions of the domain are used, the weak variation dimension of a function turns out to be the same as the Hausdorff dimension of the image.