Consider all space curves of the form & (t) = (cos(t), sin(t), f (t)), where F: R -> R is a smooth function. Find all functions f such that X(t) has torsion identically 0.
Added by Justin P.
Step 1
Step 1: Recall that the torsion of a curve X(t) = (x(t), y(t), z(t)) is given by the formula T(t) = (d/dt) * B(t), where B(t) is the binormal vector of the curve. Show more…
Show all steps
Close
Your feedback will help us improve your experience
Adi S and 79 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Let A,C be nonzero/noncollinear vectors. Let a curve be parametrized by: r(t) = f(t)A + g(t)C for t ∈ R, where f and g are smooth functions and r0 is a constant vector. What is the torsion of the curve?
Adi S.
Differentiable curves with zero torsion lie in planes. That a sufficiently differentiable curve with zero torsion lies in a plane is a special case of the fact that a particle whose velocity remains perpendicular to a fixed vector C moves in a plane perpendicular to C. This, in turn, can be viewed as the following result. Suppose r(t) = f(t)i + g(t)j + h(t)k is twice differentiable for all t in an interval [a, b], that r = 0 when t = a, and that v • k = 0 for all t in [a, b]. Show that h(t) = 0 for all t in [a, b]. (Hint: Start with a = d^2r/dt^2 and apply the initial conditions in reverse order.)
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD