We shall wrap up our study with on of those fundamental theorems, this time the one of space curves. I really don't know they call these theorems 'fundamental'.
For this one, we make use of the Frenet frame and the Frenet-Serret equations to find when are two curves congruent. It help should we need to match two curves, for example the equations of lines of force, together the easiest way possible.
NOTE: Due to the intricate calculations, this is a 3-part video lecture. I suggest viewing part 1 and part 3 for the important portions.
VECTOR DIFFERENTIAL CALCULUS >
THE FUNDAMENTAL THEOREM OF SPACE CURVES.3
May want to check out:
... THEOREM OF SPACE CURVES.1
... THEOREM OF SPACE CURVES.2
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VIDEO LECTURE part 3
MAIN CONCEPTS
If two curves,  and  , have the same functions of  and  , then they are congruent. Get the Printer Friendly Version COMMENTS
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CONCEPT
Let
 and  be two non-vanishing curves, both of which has the same values of the curvature, , and torsion, . The Fundamental theorem of space says that these two curves are congruent, meaning to say that one can be moved rigidly and then superimposed on the other so that they coincide exactly.
We shall prove this result. A must set the condition What we would do next is to define a scalar function
What we anticipate is whether the above function would change as s varies. To find out, we differentiate
Therefore, the function
Bearing in mind that each vector is of unit length. However, we recall the definition of the dot product
which tells us that each dot product cannot exceed 1. Therefore, each dot product must equal to 1 as when one product is less than 1, the other products can’t make the sum to 3. In particular
Integrating with respect to s and using the fact that both curves start from the same point when s = 0, we obtain
for all s, concluding that NOTE: In the video, I mentioned this to be a ‘half proof’ or that it is ‘not a full proof’. Just to clarify, I never meant to discount or discredit the work of the mathematician involve in this. Instead, I was addressing the point of how the function All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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as the vectors 
and 
are not defined when 
. We then define the vectors 
and 
as the unit tangent, normal and binormal for the curves 
such that
with respect to s using the Product Rule and the Frenet-Serret formulas. Long equation coming up. I have dropped the 
to save space but vectors still mean the same.
is constant. We now move 
rigidly so that its initial point coincides with the initial point of 
and in a way ‘twist’ it so that the Frenet frames of both curves coincide at that point. Since the frames coincide at 
, the constant 
must be 3 or
which implies that these vectors are equal for all s as this occurs when 
or 
meaning both vectors the same direction and magnitude.
and 
are congruent.
came about and should we know that, it would make lesson more complete. If anything, I am praising the great intuition in thinking of such a function. 
