Evaluate || (x + y + z)dS, where S is the surface defined parametrically by S ", v) (2u + v)i+ (u – 2v)j+ (u+ 3v)k for 0 < u < 1, and 0 < v < 2. -
Evaluate || (x + y + z)dS, where S is the surface defined parametrically by S ", v) (2u + v)i+ (u – 2v)j+ (u+ 3v)k for 0 < u < 1, and 0 < v < 2. -
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem 292: Surface Integral Evaluation
Evaluate the surface integral
\[
\iint_S (x + y + z) \, dS
\]
where \( S \) is the surface defined parametrically by
\[
\mathbf{R}(u, v) = (2u + v)\mathbf{i} + (u - 2v)\mathbf{j} + (u + 3v)\mathbf{k}
\]
for \( 0 \leq u \leq 1 \), and \( 0 \leq v \leq 2 \).
### Key Concepts
1. **Surface Integral:** Surface integrals extend the concept of integrals to functions over surfaces. In this problem, we need to integrate the function \( x + y + z \) over the surface \( S \).
2. **Parametric Surface:** The surface \( S \) is given parametrically with the vector function \(\mathbf{R}(u, v)\). The parameters \( u \) and \( v \) range over the intervals \( 0 \leq u \leq 1 \) and \( 0 \leq v \leq 2 \).
3. **Vector Notation:** \(\mathbf{R}(u, v) = (2u + v)\mathbf{i} + (u - 2v)\mathbf{j} + (u + 3v)\mathbf{k}\) describes a surface in 3D space where \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are the standard unit vectors in the \( x \)-, \( y \)-, and \( z \)- directions, respectively.
### Approach to Solution
To solve this problem, follow these general steps:
1. **Express \( x \), \( y \), and \( z \) in terms of \( u \) and \( v \).**
- \( x = 2u + v \)
- \( y = u - 2v \)
- \( z = u + 3v \)
2. **Compute the appropriate partial derivatives \( \partial \mathbf{R} / \partial u \) and \( \partial \mathbf{R} / \partial v \).**
- \(\frac{\partial \mathbf{R}}{\partial u} = 2\mathbf{i} + \mathbf](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F444dbf3b-270a-42aa-8883-ea946de0a843%2Fe345c4f3-e39f-4851-a422-803a36aff206%2Fh9t6g7.png&w=3840&q=75)
Transcribed Image Text:### Problem 292: Surface Integral Evaluation
Evaluate the surface integral
\[
\iint_S (x + y + z) \, dS
\]
where \( S \) is the surface defined parametrically by
\[
\mathbf{R}(u, v) = (2u + v)\mathbf{i} + (u - 2v)\mathbf{j} + (u + 3v)\mathbf{k}
\]
for \( 0 \leq u \leq 1 \), and \( 0 \leq v \leq 2 \).
### Key Concepts
1. **Surface Integral:** Surface integrals extend the concept of integrals to functions over surfaces. In this problem, we need to integrate the function \( x + y + z \) over the surface \( S \).
2. **Parametric Surface:** The surface \( S \) is given parametrically with the vector function \(\mathbf{R}(u, v)\). The parameters \( u \) and \( v \) range over the intervals \( 0 \leq u \leq 1 \) and \( 0 \leq v \leq 2 \).
3. **Vector Notation:** \(\mathbf{R}(u, v) = (2u + v)\mathbf{i} + (u - 2v)\mathbf{j} + (u + 3v)\mathbf{k}\) describes a surface in 3D space where \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are the standard unit vectors in the \( x \)-, \( y \)-, and \( z \)- directions, respectively.
### Approach to Solution
To solve this problem, follow these general steps:
1. **Express \( x \), \( y \), and \( z \) in terms of \( u \) and \( v \).**
- \( x = 2u + v \)
- \( y = u - 2v \)
- \( z = u + 3v \)
2. **Compute the appropriate partial derivatives \( \partial \mathbf{R} / \partial u \) and \( \partial \mathbf{R} / \partial v \).**
- \(\frac{\partial \mathbf{R}}{\partial u} = 2\mathbf{i} + \mathbf
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