Compute the surface integral over the given oriented surface: F=y³i+8j −æk, 81 SfF.ds = 21 portion of the plane x + y + z = 1 in the octant x, y, z ≥ 0, downward-pointing n
Compute the surface integral over the given oriented surface: F=y³i+8j −æk, 81 SfF.ds = 21 portion of the plane x + y + z = 1 in the octant x, y, z ≥ 0, downward-pointing n
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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how do i solve the attached calculus problem?
![### Surface Integral Calculation
**Problem Statement:**
Compute the surface integral over the given oriented surface:
\[ \mathbf{F} = y^5 \mathbf{i} + 8y \mathbf{j} - x \mathbf{k}, \]
where the surface is the portion of the plane \( x + y + z = 1 \) in the octant \( x, y, z \geq 0 \), with a downward-pointing normal.
**Solution:**
\[
\iint_S \mathbf{F} \cdot d\mathbf{S} = \frac{81}{21}
\]
This calculation demonstrates the surface integral of the vector field \(\mathbf{F}\) over the specified plane in the first octant. The result is given as a fraction \(\frac{81}{21}\).
### Explanation:
- **Vector Field \(\mathbf{F}\):** The vector field \(\mathbf{F}\) is composed of three components \((y^5, 8y, -x)\). The calculation involves integrating these components over the surface defined by the plane equation.
- **Surface:** The portion of the plane \( x + y + z = 1 \) considered for the calculation is restricted to the first octant, where all coordinate values \(x\), \(y\), and \(z\) are non-negative (\(x, y, z \geq 0\)).
- **Orientation:** The surface is oriented with a downward-pointing normal, affecting the direction over which the surface integral is calculated. This affects how the vector field interacts with the surface at each point.
This mathematical expression provides insight into the fields that intersect this plane, useful in physics and engineering contexts such as fluid flow or electromagnetic fields.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3cfeff4-8ba6-46a8-98d4-804b4f4f620a%2F7a4a7e0c-cd5c-43bd-b69e-bcce97dfb4ae%2Fnqfmedm_processed.png&w=3840&q=75)
Transcribed Image Text:### Surface Integral Calculation
**Problem Statement:**
Compute the surface integral over the given oriented surface:
\[ \mathbf{F} = y^5 \mathbf{i} + 8y \mathbf{j} - x \mathbf{k}, \]
where the surface is the portion of the plane \( x + y + z = 1 \) in the octant \( x, y, z \geq 0 \), with a downward-pointing normal.
**Solution:**
\[
\iint_S \mathbf{F} \cdot d\mathbf{S} = \frac{81}{21}
\]
This calculation demonstrates the surface integral of the vector field \(\mathbf{F}\) over the specified plane in the first octant. The result is given as a fraction \(\frac{81}{21}\).
### Explanation:
- **Vector Field \(\mathbf{F}\):** The vector field \(\mathbf{F}\) is composed of three components \((y^5, 8y, -x)\). The calculation involves integrating these components over the surface defined by the plane equation.
- **Surface:** The portion of the plane \( x + y + z = 1 \) considered for the calculation is restricted to the first octant, where all coordinate values \(x\), \(y\), and \(z\) are non-negative (\(x, y, z \geq 0\)).
- **Orientation:** The surface is oriented with a downward-pointing normal, affecting the direction over which the surface integral is calculated. This affects how the vector field interacts with the surface at each point.
This mathematical expression provides insight into the fields that intersect this plane, useful in physics and engineering contexts such as fluid flow or electromagnetic fields.
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