Evaluate JF- ds F(x,y,z) = zi + xj + yk, and S is the part of the sphere z+1=V5-x2 - y² with z>0 and in n directed upward. This is a hemishere above the xy plane. а. 2п b. 4 m с. п/2 d. 3п e. 3 π/2
Evaluate JF- ds F(x,y,z) = zi + xj + yk, and S is the part of the sphere z+1=V5-x2 - y² with z>0 and in n directed upward. This is a hemishere above the xy plane. а. 2п b. 4 m с. п/2 d. 3п e. 3 π/2
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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![### Surface Integral Evaluation
In this exercise, we aim to evaluate the surface integral:
\[
\iint_S \mathbf{F} \cdot d\mathbf{S}
\]
Given the vector field:
\[
\mathbf{F}(x,y,z) = zi + xj + yk
\]
And the surface \(S\), which is the part of the sphere defined by:
\[
z + 1 = \sqrt{5 - x^2 - y^2}
\]
where \( z > 0 \) and the normal vector \(\mathbf{n}\) is directed upward. This sphere represents a hemisphere above the \(xy\)-plane.
### Problem Choices
The possible answers to the integral are:
a. \( 2\pi \)
b. \( 4\pi \)
c. \( \pi/2 \)
d. \( 3\pi \)
e. \( 3\pi/2 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee1f5d00-2b06-4d9b-996b-2e4c16434e95%2Fcdb6e141-251f-48ed-93d7-b829197540cd%2Fprfodvm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Surface Integral Evaluation
In this exercise, we aim to evaluate the surface integral:
\[
\iint_S \mathbf{F} \cdot d\mathbf{S}
\]
Given the vector field:
\[
\mathbf{F}(x,y,z) = zi + xj + yk
\]
And the surface \(S\), which is the part of the sphere defined by:
\[
z + 1 = \sqrt{5 - x^2 - y^2}
\]
where \( z > 0 \) and the normal vector \(\mathbf{n}\) is directed upward. This sphere represents a hemisphere above the \(xy\)-plane.
### Problem Choices
The possible answers to the integral are:
a. \( 2\pi \)
b. \( 4\pi \)
c. \( \pi/2 \)
d. \( 3\pi \)
e. \( 3\pi/2 \)
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