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(a) Let C be a smooth curve drawn on a sphere S1, which is centred at the origin and has the radius 1: S1 = {x : |x| = 1}. The curve C is defined parametrically by the equations x = c(s), a(s), y= b(s), z = where a, b, c are smooth functions of s E (0, 1). Consider the vector- valued function F(s) = a(s)i+ b(s)j+c(s)k, %3D where i, j, k are unit basis vectors of the Cartesian system of coordi- nates. Verify that the vector field dF s € (0, 1), ds is tangent to the sphere S1. (b) For an arbitrary smooth curve C defined parametrically by the equations x = x(s), y = y(s), z= verify that the position vector r of a point along C satisifes the relation dr - = r- dr where r = = |r]. r

(a) Let C be a smooth curve drawn on a sphere S1, which is centred at the origin and has the radius 1: S1 = {x : |x| = 1}. The curve C is defined parametrically by the equations x = c(s), a(s), y= b(s), z = where a, b, c are smooth functions of s E (0, 1). Consider the vector- valued function F(s) = a(s)i+ b(s)j+c(s)k, %3D where i, j, k are unit basis vectors of the Cartesian system of coordi- nates. Verify that the vector field dF s € (0, 1), ds is tangent to the sphere S1. (b) For an arbitrary smooth curve C defined parametrically by the equations x = x(s), y = y(s), z= verify that the position vector r of a point along C satisifes the relation dr - = r- dr where r = = |r]. r

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section: Chapter Questions
Problem 18T
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2. (a) Let C be a smooth curve drawn on a sphere S1, which is centred at the origin and has the radius 1: S1 = {x : |x| = 1}. The curve C is defined parametrically by the equations a(s), y= b(s), z = c(s), x = where a, b, c are smooth functions of s E (0, 1). Consider the vector- valued function F(s) = a(s)i+ b(s)j+c(s)k, where i,j, k are unit basis vectors of the Cartesian system of coordi- nates. Verify that the vector field dF s E (0, 1), ds' is tangent to the sphere S1. (b) For an arbitrary smooth curve C defined parametrically by the equations x = x(s), y= y(s), z= z(s), verify that the position vectorr of a point along C satisifes the relation dr dr r. - = r- ds' where r = ds
Transcribed Image Text:2. (a) Let C be a smooth curve drawn on a sphere S1, which is centred at the origin and has the radius 1: S1 = {x : |x| = 1}. The curve C is defined parametrically by the equations a(s), y= b(s), z = c(s), x = where a, b, c are smooth functions of s E (0, 1). Consider the vector- valued function F(s) = a(s)i+ b(s)j+c(s)k, where i,j, k are unit basis vectors of the Cartesian system of coordi- nates. Verify that the vector field dF s E (0, 1), ds' is tangent to the sphere S1. (b) For an arbitrary smooth curve C defined parametrically by the equations x = x(s), y= y(s), z= z(s), verify that the position vectorr of a point along C satisifes the relation dr dr r. - = r- ds' where r = ds
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