- #1
MichaelT
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Homework Statement
If c is given in terms of some other parameter t and c'(t) is never zero, show that
k = ||c'(t) x c"(t)||/||c'(t)||3
The first two parts of this problem involved a path parametrized by arc length, but this part says nothing about that, so I assume that this path is not parametrized by arc length.
Homework Equations
I have found that
T'(t) = c"(t) = [||c'(t)||2c"(t) - c'(t)(c'(t) dot c"(t))]/||c'(t)||3
I am having trouble figuring out how to relate the curvature to the equation
k = ||c'(t) x c"(t)||/||c'(t)||3
The Attempt at a Solution
I can express the equation as the components so
F(x,y,z) = [(z"y'-y"z')2+(x"z'-x'z")2(y"x'-x'y")2]1/2/(x'2+y'2+z'2)3/2
Where should I go from here, and is the above equation useful at all? Do I need to find
||c"(t)|| as well (and is that even possible)?