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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the curvature and torsion of the curve x = sinh t. y = cosh t, z = t at the point (0, 1, 0).

To determine

To find: The curvature and torsion of the curve x=sinht,y=cosht,z=t at point (0,1,0) .

Explanation

Given data:

Parametric equation of curve are x=sinht,y=cosht,z=t .

Formula used:

Write the expression for curvature of curve r(t) (K) .

K=|r(t)×r(t)||r(t)|3 (1)

Here,

r(t) is first derivative of r(t) ,

r(t) is second derivative of r(t) , and

r(t) is third derivative of r(t) ,

Write the expression for torsion of curve r(t) (τ) .

τ=(r(t)×r(t))r(t)|r(t)×r(t)|2 (2)

Consider the two three-dimensional vector functions such as u(t)=u1(t),u2(t),u3(t) and v(t)=v1(t),v2(t),v3(t) .

Cross product of vectors:

Write the expression for cross product of vectors u(t) and v(t) (u(t)×v(t)) .

u(t)×v(t)=|ijku1(t)u2(t)u3(t)v1(t)v2(t)v3(t)|=[(u2(t)v3(t)v2(t)u3(t))],[(u1(t)v3(t)v1(t)u3(t))],[(u1(t)v2(t)v1(t)u2(t))]

Dot product of vectors:

Write the expression for dot product of vectors u(t) and v(t) (u(t)v(t)) .

u(t)v(t)=u1(t),u2(t),u3(t)v1(t),v2(t),v3(t)=u1(t)v1(t)+u2(t)v2(t)+u3(t)v3(t)

Write the expression for curve r(t) .

r(t)=x,y,z

Substitute sinht for x, cosht for y, and t for z,

r(t)=sinht,cosht,t

Equate the components of r(t) with components of point (0,1,0) .

sinht=0t=sinh1(0)t=0

cosht=1t=cosh1(1)t=0

t=0

Hence, t value is 0.

Find the value of r(t) .

r(t)=ddtsinht,cosht,t=ddt(sinht),ddt(cosht),ddt(t)=cosht,sinht,1{ddx(sinhx)=coshx,ddx(coshx)=sinhx,ddx(x)=1}

Apply differentiation with respect to t on both sides of equation.

r(t)=ddtcosht,sinht,1=ddt(cosht),ddt(sinht),ddt(1)=sinht,cosht,0 {ddx(sinhx)=coshx,ddx(coshx)=sinhx,ddx(k)=0}

Apply differentiation with respect to t on both sides of equation.

r(t)=ddtsinht,cosht,0=ddt(sinht),ddt(cosht),ddt(0)=cosht,sinht,0 {ddx(sinhx)=coshx,ddx(coshx)=sinhx,ddx(0)=0}

Find the value of r(t)×r(t)

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