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Chapter 14: Q. 25 (page 1119)

Find the areas of the given surfaces in Exercises 21鈥�26.

S is the portion of the surface parametrized by r(u,v)=(3u-v,v+u,v-u) whose preimage (the domain in the uv-plane) is the unit square [0,1]脳[0,1]

Short Answer

Expert verified

The area of the surface determined by r(u,v)=(3u-v,v+u,v-u)is 26.

Step by step solution

01

Step 1. Given Information.

The function:

r(u,v)=(3u-v,v+u,v-u)

02

Step 2. Formula for finding the area of the surface.

The area of the smooth surface is given by,

鈭�SdS=鈭�Sru脳rvdA=鈭�鈭�Dru脳rvdudv

03

Step 3. Find ru,rv.

r(u,v)=(3u-v,v+u,v-u)

ru=鈭�r鈭�u=(3,1,-1)rv=鈭�r鈭�v=(-1,1,1)

04

Step 4. Find ru×rv.

ru脳rv=(3,1,-1)脳(-1,1,,1)=2i-2j+4kru脳rv=2i-2j+4k=22+(-2)2+42=26

05

Step 5. Find the area.

鈭�SdS=鈭�鈭�Dru脳rvdudv=鈭�01鈭�0126dudv=鈭�01[鈭�0126du]dv=鈭�0126[u]01dv=26鈭�01dv=26[v]01=26

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