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19. F(x, y, z) = yî + xyz ĵ – 2zx k, where S is the upward-facing paraboloid z = x² + y² lying in cylinder x² + y² = 1 Answer 一元

19. F(x, y, z) = yî + xyz ĵ – 2zx k, where S is the upward-facing paraboloid z = x² + y² lying in cylinder x² + y² = 1 Answer 一元

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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## Use Stokes' Theorem to Evaluate \[ \iint_{S} \text{curl} \, \vec{F} \cdot d\vec{S} \] ### Problem 18 Given: \[ \vec{F}(x, y, z) = y \, \hat{i} - x \, \hat{j} + z \, \hat{k} \] where \( S \) is the upper half of the unit sphere. ### Problem 19 Given: \[ \vec{F}(x, y, z) = y \, \hat{i} + xyz \, \hat{j} - 2xz \, \hat{k} \] where \( S \) is the upward-facing paraboloid \( z = x^2 + y^2 \) lying in the cylinder \( x^2 + y^2 = 1 \). **Answer:** \[ -\pi \]
Transcribed Image Text:## Use Stokes' Theorem to Evaluate \[ \iint_{S} \text{curl} \, \vec{F} \cdot d\vec{S} \] ### Problem 18 Given: \[ \vec{F}(x, y, z) = y \, \hat{i} - x \, \hat{j} + z \, \hat{k} \] where \( S \) is the upper half of the unit sphere. ### Problem 19 Given: \[ \vec{F}(x, y, z) = y \, \hat{i} + xyz \, \hat{j} - 2xz \, \hat{k} \] where \( S \) is the upward-facing paraboloid \( z = x^2 + y^2 \) lying in the cylinder \( x^2 + y^2 = 1 \). **Answer:** \[ -\pi \]
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