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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Question
#19
![## 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:**
\[
-\pi
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f1c559b-3699-4238-b9ff-9c1bfdecf463%2F9c9be21e-946c-4eed-aebb-eedf45201929%2Fef83ap_processed.png&w=3840&q=75)
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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