Brard 
This is the Poincaré formula. 
Let us suppose that the fluid is incompressible. The triple 
integral in the second term is not necessarily null for sources with 
a density o = div V could be distributed through D;. The double 
integral in the second term represents a source distribution over S 
with the density o =(n. Ve . There is a jump of V through S. 
Its normal component n (V, - is -@V)y, . According to (2.2) 
its tangential jump from ie e to Ded is that due to a vortex sheet 
over S, with the vorticity =-1 (m7 A Via 
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Poincaré's formula solves the equation 
o= WAV 
with respect to V, when the vorticity @ and div V are known 
inside Dj, if, furthermore, V is known on De ts 
Biot's and Savart's formula - (Figure 3) 
Let us suppose that all the points of S are at infinity, that 
div V=0 andthat @ is null everywhere except in a very thin tube 
with a transverse area 5 vie . Let us suppose that the measure of 
6 ys goes to zero, while ai eso . The vorticity reduces to a vor- 
tex filament with the intensity [ tangent to ee. Let ds denote 
the element of arc of 
Then Poincaré's formula gives 
(Mm) = Leupp ML). sea hay r ds(M! 
V (M) = curl — ds(M') = curl re [os , (2. 4) 
This is the Biot and Savart formula which gives the velocity induced 
at M by the vortex filament Fhe intensity of whichis [ 
As: uw, Si. excepton oe there is a velocity potential @ 
except on pices > be an open surface whose edge coincides 
with and n the unit vector normal to ye , Oriented in the posi- 
tive direction with respect to the arc ds of . One has 
Vv =< VO wine = ff =o — sar du (Mt), Gee 
7 M'! 
x 
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