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## Main Question or Discussion Point

I've been working my way through an old scientific paper on inductance calculation, and there's a fundamental principle I'm having a problem with.

Suppose there are two filamentary conductors of any arbitrary shape. Now consider an infinitesimal element dl1 somewhere on the first conductor, and another element dl2 somewhere on the second conductor. My understanding is that the contribution to the mutual inductance of the two conductors due to dl1 and dl2 is equal to the dot product of dl1 and dl2 divided by the distance r which separates them. Then Integrating twice with respect to dl1 and then dl2 should give the mutual inductance of the two conductors (Neumann integral). Is this correct so far?

If so, then my understanding of the dot product is not very good. Logically, it seems to me that the magnitude of dl1 should be multiplied by the magnitude of dl2 and the result then multiplied by a reduction factor which accounts for the projection of dl1 onto dl2, and and vice versa. My understanding of dot product is that the x, y and z components of the first element are multiplied by the x, y and z components respectively of the second element. While this would account for the the fact that the elements dl1 and dl2 may not be parallel, it doesn't fully account for how the elements may be located in space and the projection of one upon the other, which I think must play a part in the calculation. Can someone shed some light on this?

Suppose there are two filamentary conductors of any arbitrary shape. Now consider an infinitesimal element dl1 somewhere on the first conductor, and another element dl2 somewhere on the second conductor. My understanding is that the contribution to the mutual inductance of the two conductors due to dl1 and dl2 is equal to the dot product of dl1 and dl2 divided by the distance r which separates them. Then Integrating twice with respect to dl1 and then dl2 should give the mutual inductance of the two conductors (Neumann integral). Is this correct so far?

If so, then my understanding of the dot product is not very good. Logically, it seems to me that the magnitude of dl1 should be multiplied by the magnitude of dl2 and the result then multiplied by a reduction factor which accounts for the projection of dl1 onto dl2, and and vice versa. My understanding of dot product is that the x, y and z components of the first element are multiplied by the x, y and z components respectively of the second element. While this would account for the the fact that the elements dl1 and dl2 may not be parallel, it doesn't fully account for how the elements may be located in space and the projection of one upon the other, which I think must play a part in the calculation. Can someone shed some light on this?