Chapter 14: Q. 25 (page 1119)

Find the areas of the given surfaces in Exercises 21–26.
S is the portion of the surface parametrized by $r (u, v) = (3 u - v, v + u, v - u)$ whose preimage (the domain in the uv-plane) is the unit square $[0, 1] \times [0, 1]$

Short Answer

Expert verified

The area of the surface determined by $r (u, v) = (3 u - v, v + u, v - u)$ is $2 \sqrt{6}$ .

Step by step solution

Step 1. Given Information.

The function:

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

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

The area of the smooth surface is given by,

$\int_{S} d S = \int_{S} ||r_{u} \times r_{v}|| d A = \int \int_{D} ||r_{u} \times r_{v}|| d u d v$

Step 3. Find ru,rv.

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

$r_{u} = \frac{\partial r}{\partial u} = (3, 1, - 1) r_{v} = \frac{\partial r}{\partial v} = (- 1, 1, 1)$

Step 4. Find ru×rv.

$r_{u} \times r_{v} = (3, 1, - 1) \times (- 1, 1,, 1) = 2 i - 2 j + 4 k ||r_{u} \times r_{v}|| = ||2 i - 2 j + 4 k|| = \sqrt{2^{2} + {(- 2)}^{2} + 4^{2}} = 2 \sqrt{6}$

Step 5. Find the area.

$\int_{S} d S = \int \int_{D} ||r_{u} \times r_{v}|| d u d v = \int_{0}^{1} \int_{0}^{1} 2 \sqrt{6} d u d v = \int_{0}^{1} [\int_{0}^{1} 2 \sqrt{6} d u] d v = \int_{0}^{1} 2 \sqrt{6} {[u]}_{0}^{1} d v = 2 \sqrt{6} \int_{0}^{1} d v = 2 \sqrt{6} {[v]}_{0}^{1} = 2 \sqrt{6}$

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Find the areas of the given surfaces in Exercises 21–26.
S is the portion of the surface parametrized by $r (u, v) = (3 u - v, v + u, v - u)$ whose preimage (the domain in the uv-plane) is the unit square $[0, 1] \times [0, 1]$

Short Answer

Step by step solution

Step 1. Given Information.

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

Step 3. Find ru,rv.

Step 4. Find ru×rv.

Step 5. Find the area.

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Find the areas of the given surfaces in Exercises 21–26.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]×[0,1]

Short Answer

Step by step solution

Step 1. Given Information.

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

Step 3. Find ru,rv.

Step 4. Find ru×rv.

Step 5. Find the area.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Geometry

Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find the areas of the given surfaces in Exercises 21–26.
S is the portion of the surface parametrized by $r (u, v) = (3 u - v, v + u, v - u)$ whose preimage (the domain in the uv-plane) is the unit square $[0, 1] \times [0, 1]$