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3. F(x, y, z) = y² î + 2x ĵ + 5 Â; S is hemisphere z = (4 — x² - y²)¹/²¸

3. F(x, y, z) = y² î + 2x ĵ + 5 Â; S is hemisphere z = (4 — x² - y²)¹/²¸

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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**Exercises: Flux and Circulation Integrals** **Objective:** In exercises 1 - 6, calculate directly both the flux of \( \text{curl} \, \mathbf{F} \cdot \mathbf{N} \) over the given surface and the circulation integral around its boundary, without using Stokes' theorem. Assume all are oriented clockwise. 1. **Problem 1:** \(\mathbf{F}(x, y, z) = y^2 \, \mathbf{i} + z^2 \, \mathbf{j} + x^2 \, \mathbf{k}; \ S\) is the first-octant portion of plane \(x + y + z = 1\). 2. **Problem 2:** \(\mathbf{F}(x, y, z) = z \, \mathbf{i} + x \, \mathbf{j} + y \, \mathbf{k}; \ S\) is the hemisphere \(z = (a^2 - x^2 - y^2)^{1/2}\). 3. **Answer:** \(\mathbf{F}(x, y, z) = y^2 \, \mathbf{i} + 2x \, \mathbf{j} + 5 \, \mathbf{k}; \ S\) is the hemisphere \(z = (4 - x^2 - y^2)^{1/2}\). **Note:** Ensure to approach each exercise with the correct vector field and surface parameters for accurate calculations.
Transcribed Image Text:**Exercises: Flux and Circulation Integrals** **Objective:** In exercises 1 - 6, calculate directly both the flux of \( \text{curl} \, \mathbf{F} \cdot \mathbf{N} \) over the given surface and the circulation integral around its boundary, without using Stokes' theorem. Assume all are oriented clockwise. 1. **Problem 1:** \(\mathbf{F}(x, y, z) = y^2 \, \mathbf{i} + z^2 \, \mathbf{j} + x^2 \, \mathbf{k}; \ S\) is the first-octant portion of plane \(x + y + z = 1\). 2. **Problem 2:** \(\mathbf{F}(x, y, z) = z \, \mathbf{i} + x \, \mathbf{j} + y \, \mathbf{k}; \ S\) is the hemisphere \(z = (a^2 - x^2 - y^2)^{1/2}\). 3. **Answer:** \(\mathbf{F}(x, y, z) = y^2 \, \mathbf{i} + 2x \, \mathbf{j} + 5 \, \mathbf{k}; \ S\) is the hemisphere \(z = (4 - x^2 - y^2)^{1/2}\). **Note:** Ensure to approach each exercise with the correct vector field and surface parameters for accurate calculations.
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