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Chapter 11: Q. 15 (page 879)

Under what conditions does a twice-differentiable vector valued function r(t) not have a binormal vector at a point in the domain of r(t)?

Short Answer

Expert verified

The functionrtdoesn't exists whenr't=0.

Step by step solution

01

Step 1. Given information.

Consider the given question,

A twice-differentiable vector-valued function isr(t).

02

Step 2. Definition of binomial vector.

Assume t0to be any point in the domain of rt.

Assume rtto be a differential vector function on the interval I鈯�Rsuch that T't0鈮�0, where, t0鈭�I.

The binomial vector B at rt0is defined as,

Bt0=Tt0脳Nt0

But the unit vector Tt0exist only when r't0鈮�0and the principal unit vector normal , Nt0=T't0T't0exists only when T't0鈮�0.

So the basic condition for the existence of Bt0is r't0=0.

Thus, the binomial vector doesn't exists at any point t0at whichr't0=0.

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