The mutual inductance by circuit i on circuit j is given by the double integral Neumann formula
M
i
j
=
μ
0
4
π
∮
C
i
∮
C
j
d
s
i
⋅
d
s
j

R
i
j

Φ
i
=
∫
S
i
B
⋅
d
a
=
∫
S
i
(
∇
×
A
)
⋅
d
a
=
∮
C
i
A
⋅
d
s
=
∮
C
i
(
∑
j
μ
0
I
j
4
π
∮
C
j
d
s
j

R

)
⋅
d
s
i
where
Φ
i
is the magnetic flux through the
ith surface by the electrical circuit outlined by
C_{j}
C_{i} is the enclosing curve of S
_{i}.
B is the magnetic field vector.
A is the vector potential.
Stokes' theorem has been used.
M
i
j
=
d
e
f
Φ
i
I
j
=
μ
0
4
π
∮
C
i
∮
C
j
d
s
i
⋅
d
s
j

R
i
j

so that the mutual inductance is a purely geometrical quantity independent of the current in the circuits.