   Chapter 16.6, Problem 36E

Chapter
Section
Textbook Problem

Find an equation of the tangent plane to the given parametric surface at the specified point.36. r(u, v) = sin u i + cos u sin v j + sin v k; u = π/6, v = π/6

To determine

To find: An equation of the tangent plane to parametric surface r(u,v)=sinui+cosusinvj+sinvk at the point u=π6,v=π6.

Explanation

Given data:

The vector function is given as follows.

r(u,v)=sinui+cosusinvj+sinvk;u=π6,v=π6

Formula used:

The expression to find tangent plane to the parametric surface with the normal vector n=a,b,c at the specified point (x0,y0,z0) is,

a(xx0)+b(yy0)+c(zz0)=0 (1)

The expression to find normal vector from the tangent vectors of the parametric surface is,

n=|ijka1b1c1a2b2c2| (2)

Where, the vector a1,b1,c1 is a tangent vector ru of the parametric surface and the vector a2,b2,c2 is a tangent vector rv of the parametric surface.

The expression to find the tangent vector ru of the parametric surface is,

ru=xui+yuj+zuk (3)

The expression to find the tangent vector rv of the parametric surface is,

rv=xvi+yvj+zvk (4)

Calculation:

The parametric surface r(u,v)=sinui+cosusinvj+sinvk at the point u=π6,v=π6.

Let x=sinu,y=cosusinv,z=sinv.

Write the expression to find point (x0,y0,z0) in the surface.

(x0,y0,z0)=(sinu,cosusinv,sinv)

Substitute π6 for u and π6 for v,

(x0,y0,z0)=(sin(π6),cos(π6)sin(π6),sin(π6))=(12,(32)(12),12)=(12,34,12)

Thus, (x0,y0,z0)=(12,34,12).

Calculation of tangent vector ru is as follows.

Substitute sinu for x, cosusinv for y, and sinv for z in equation (3),

ru=(sinu)ui+(cosusinv)uj+(sinv)uk=[u(sinu)]i+[u(cosusinv)]j+[u(sinv)]k=(cosu)i+[sinvu(cosu)]j+(0)k=(cosu)i+(sinusinv)j+(0)k

Modify the equation as follows.

ru=cosu,sinusinv,0

Substitute π6 for u and π6 for v,

ru=cos(π6),sin(π6)sin(π6),0=32,(12)(12),0=32,14,0

Thus, ru=32,14,0.

Calculation of tangent vector rv is as follows.

Substitute sinu for x, cosusinv for y, and sinv for z in equation (4),

rv=(sinu)vi+(cosusinv)vj+(sinv)vk=[v(sinu)]i+[v(cosusinv)]j+[v(sinv)]k=(0)i+[cosuv(sinv)]j+(cosv)k=(0)i+(cosucosv)j+(cosv)k

Modify the equation as follows

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