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Chapter 11: Q. 6 (page 889)

Let C be the graph of the vector-valued function r(t). Define the curvature at a point on C.

Short Answer

Expert verified

The curvature is defined as the rate of change of the tangent vector with respect to arc length.

Step by step solution

01

Step 1. Given information.

We have to define the curvature at a point on C where C be the graph of the vector-valued function r(t).

02

Step 2. Definition of the curvature

Let C be the graph of a vector function r(s) defined on an interval I, parametrized by arc length s, and with unit tangent vector T.

The curvature k of c at a point on the curve is the scalar given by

k=dTds

That is, the curvature is defined as the rate of change of the tangent vector with respect to arc length.

03

Step 3. Curvature k of c.

The Curvature k of c at a point on the curve is given by :

(a) k=T鈥�(t)r鈥�(t)where T(t) is the unit tangent vector to the r(t).

(b)k=r鈥�(t)脳r鈥测赌�(t)r鈥�(t)3

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Most popular questions from this chapter

Let r(t)be a vector-valued function whose graph is a curve C, and let a(t)be the acceleration vector. Prove that if a(t)is always zero, then C is a straight line.

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