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Question 1. The curvature of a smooth vector-valued function (or space curve) r(t) is defined to be dT ds where s = s(t) is arclength parametrisation and T is the unit tangent vector of r. ||T' (t)||| ||r' (t)|| (a) with respect to t). K = (b) Use the Chain Rule to show that = (where' denotes differentiation Find the curvature K of the vector-valued function r(t) = (t², sint - tcost, cost + tsint), t>0.

Question 1. The curvature of a smooth vector-valued function (or space curve) r(t) is defined to be dT ds where s = s(t) is arclength parametrisation and T is the unit tangent vector of r. ||T' (t)||| ||r' (t)|| (a) with respect to t). K = (b) Use the Chain Rule to show that = (where' denotes differentiation Find the curvature K of the vector-valued function r(t) = (t², sint - tcost, cost + tsint), t>0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 1. The curvature of a smooth vector-valued function (or space curve) r(t) is defined to be dT ds where s = s(t) is arclength parametrisation and T is the unit tangent vector of r. ||T' (t)||| ||r' (t)|| (a) with respect to t). K = (b) Use the Chain Rule to show that = (where' denotes differentiation Find the curvature K of the vector-valued function r(t) = (t², sint - tcost, cost + tsint), t>0.
Transcribed Image Text:Question 1. The curvature of a smooth vector-valued function (or space curve) r(t) is defined to be dT ds where s = s(t) is arclength parametrisation and T is the unit tangent vector of r. ||T' (t)||| ||r' (t)|| (a) with respect to t). K = (b) Use the Chain Rule to show that = (where' denotes differentiation Find the curvature K of the vector-valued function r(t) = (t², sint - tcost, cost + tsint), t>0.
Q1 continued (c) (i) Give the definition of the binormal B(t) of a vvf r(t) in terms of the unit tangent T(t) and principal normal N(t). (ii) Hence write down a formula for B' (t). (iii) From the definition of curvature, T' = KN. Use this to prove that B' is perpendicular to T for all t. (You may want to consider properties of the scalar triple product given in the Formula Sheet.)
Transcribed Image Text:Q1 continued (c) (i) Give the definition of the binormal B(t) of a vvf r(t) in terms of the unit tangent T(t) and principal normal N(t). (ii) Hence write down a formula for B' (t). (iii) From the definition of curvature, T' = KN. Use this to prove that B' is perpendicular to T for all t. (You may want to consider properties of the scalar triple product given in the Formula Sheet.)
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