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a) Compute the surface area of the portion of the plane x+y=z = 0 contained inside the cylinder x² + y² = 9. b) Compute the flux of the vector field F(x, y, z) = ( −y, x, z) across the portion of the plane x + y z = 0 contained inside the cylinder x² + y² = 9, oriented with upward pointing normals.

a) Compute the surface area of the portion of the plane x+y=z = 0 contained inside the cylinder x² + y² = 9. b) Compute the flux of the vector field F(x, y, z) = ( −y, x, z) across the portion of the plane x + y z = 0 contained inside the cylinder x² + y² = 9, oriented with upward pointing normals.

Advanced Engineering Mathematics
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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### Educational Exercise: Calculating Surface Area and Flux #### Problem Statement: **a) Compute the surface area of the portion of the plane \( x + y - z = 0 \) contained inside the cylinder \( x^2 + y^2 = 9 \).** To solve this, you need to find the surface area of the intersecting region between the plane and the circular cylinder in 3D space. Use the parameterization of the plane and the given cylindrical boundary to set up the integral for surface area. **b) Compute the flux of the vector field \( \mathbf{F}(x, y, z) = \langle -y, x, z \rangle \) across the portion of the plane \( x + y - z = 0 \) contained inside the cylinder \( x^2 + y^2 = 9 \), oriented with upward pointing normals.** For this part, determine the flux of the vector field through the surface. The orientation with upward pointing normals is crucial for setting up the correct surface integral using the divergence theorem or direct computation of the flux integral. #### Key Concepts: - **Surface Area**: Calculated using a surface integral over the region defined by the constraints. - **Flux**: A measure of how much of the vector field passes through the surface, computed using the surface integral of the dot product of the vector field and the unit normal to the surface. #### Diagram Explanation: While there isn't an explicit diagram, mentally visualize or sketch the following: - The plane \( x + y - z = 0 \) is a tilted sheet in 3-dimensional space. - The cylinder \( x^2 + y^2 = 9 \) is a vertical cylinder enclosing a circular region on the xy-plane. - The intersection creates an elliptical region on the plane. This is where the calculations focus. These visual aids help in setting up and solving the integrals required for each part of the problem.
Transcribed Image Text:### Educational Exercise: Calculating Surface Area and Flux #### Problem Statement: **a) Compute the surface area of the portion of the plane \( x + y - z = 0 \) contained inside the cylinder \( x^2 + y^2 = 9 \).** To solve this, you need to find the surface area of the intersecting region between the plane and the circular cylinder in 3D space. Use the parameterization of the plane and the given cylindrical boundary to set up the integral for surface area. **b) Compute the flux of the vector field \( \mathbf{F}(x, y, z) = \langle -y, x, z \rangle \) across the portion of the plane \( x + y - z = 0 \) contained inside the cylinder \( x^2 + y^2 = 9 \), oriented with upward pointing normals.** For this part, determine the flux of the vector field through the surface. The orientation with upward pointing normals is crucial for setting up the correct surface integral using the divergence theorem or direct computation of the flux integral. #### Key Concepts: - **Surface Area**: Calculated using a surface integral over the region defined by the constraints. - **Flux**: A measure of how much of the vector field passes through the surface, computed using the surface integral of the dot product of the vector field and the unit normal to the surface. #### Diagram Explanation: While there isn't an explicit diagram, mentally visualize or sketch the following: - The plane \( x + y - z = 0 \) is a tilted sheet in 3-dimensional space. - The cylinder \( x^2 + y^2 = 9 \) is a vertical cylinder enclosing a circular region on the xy-plane. - The intersection creates an elliptical region on the plane. This is where the calculations focus. These visual aids help in setting up and solving the integrals required for each part of the problem.
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