1. Stored Energy in Magnetically Coupled Inductors
Integration over multidimensional current space

2. Mutual Flux
Weakly Coupled Circuits

3. Two Coupled Inductors

4. General Integration Path
Conservative magnetic field
Energy independent of path of integration

5. Simple Path of Integration

6. Other Simple Path of Integration

7. Equivalence
In a conservative field, Lkm = Lmk
In general:

8. Aiding and Opposing Flux
When both currents enter the polarity marks or when both currents exit from the polarity marks, the mutual inductance, Lkm is considered as positive.
Otherwise, Lkm is negative.

9. Aiding and Opposing Flux
Here, L12 = L21 would be considered negative.