In three dimensions the force per unit length acting on a vortex filament is

Here, the local velocity V includes the component from the freestream and a component from the induced downwash. This latter component produces a component of force in the direction of the freestream: the induced drag.

The induced drag is related to the lift by:

From the results of lifting line theory for lift and downwash in terms of the Fourier coefficients of the lift distribution:

so we have:

The induced drag is often written as:

This may be written in coefficient form as:

with the same definition of e. Note that e simply depends on the shape of the lift distribution. It is called the span efficiency factor or Oswald's efficiency factor. Note also that the induced drag force depends principally on the lift per unit span, L/b.

We can determine quickly, from the expression for induced drag above that drag is a minimum for a given lift and span when all of the Fourier coefficients except the A

If the box is made large, contributions from certain sides vanish. In the limit as the box sides go to infinity we obtain the following expressions for lift and drag:

Here, u, v, and w are the perturbation velocities induced by the wing and its wake. Note that the drag only depends on the velocities induced in the "Trefftz Plane" -- a plane far behind the wing.

The drag can be expressed as the integral over the infinite plane of the perturbation velocities squared. But, using Gauss' theorem (derivation) it can be expressed as a line integral over the wake itself:

This simplifies the calculation of the drag.

The normalwash, Vn, is just the downwash if the wake is flat, but the downwash far behind the wing, not at the wing itself.

Thus we would obtain the same expression as from lifting line theory if the downwash due to the wake at the wing is half the downwash at infinity. This is indeed the case for unswept wings modeled with a lifting line.

Far-field velocities can also be used to compute
the lift. The results are subtle, but rather interesting.

Perhaps the most useful of these is called Munk's stagger theorem. It states that:

The theorem applies when the distribution of circulation on the surfaces is held constant by adjusting the surface incidences as the longitudinal position is varied.

This implies that the drag of an elliptically-loaded swept wing is the same as that of an unswept wing. It also is very useful in the study of canard airplanes for which the canard downwash on the wing is quite complicated. Moving the canard very far behind the wing does not change the drag, but makes its computation much easier. One may use the stagger theorem to prove several other useful results. One of these is the mutual induced drag theorem which states that: The interference drag caused by the downwash of one wing on another is equal to that produced by the second wing on the first, when the surfaces are unstaggered (at the same streamwise location).

These results are especially useful in analyzing multiple lifting surfaces.