Compute T, N, and B for a parameterized curve, Multivariable Calculus
In this exercise, we compute the unit tangent vector, unit normal vector, and unit binormal vector--the Frenet Frame vectors--for the parameterized curve r(t)=(cos(t),sin(t),sin(2t)) at the point where t=0. MISTAKE: I got very careless at 9:29 and did not differentiate correctly--luckily that term vanishes and does not affect the end result.
First, we calculate the velocity vector r′(t) and its magnitude to normalize it and find the unit tangent vector T(t). Then, we evaluate T(t) at t=0. Next, we differentiate T(t) to obtain T′(t) and normalize it to find the unit normal vector N(t), evaluated at t=0. Finally, we calculate the unit binormal vector B(0) by taking the cross product of T(0) and N(0).
My focus throughout is on computing these vectors efficiently, keeping the variable t as needed but bringing in t=0 for simplification when possible.
#mathematics #math #calculus3 #multivariablecalculus #TangentVector #NormalVector #BinormalVector #ParameterizedCurves #vectorcalculus
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