Use Stokes' theorem to evaluate Scurl(F) · dS. F(x, y, z) = exy cos(z) i + x2z j + xy k, S is the hemisphere x = sqrt 49 − y2 − z2, oriented in the direction of the positive x-axis
Use Stokes' theorem to evaluate Scurl(F) · dS. F(x, y, z) = exy cos(z) i + x2z j + xy k, S is the hemisphere x = sqrt 49 − y2 − z2, oriented in the direction of the positive x-axis
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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16.8 #5.
Use Stokes' theorem to evaluate
curl(F) · dS.
| S |
F(x, y, z) = exy cos(z) i + x2z j + xy k,
S is the hemisphere
x =
,
oriented in the direction of the positive x-axis| sqrt | 49 − y2 − z2 |
![**Problem Statement:**
Use Stokes' theorem to evaluate the surface integral:
\[
\iint_S \text{curl}(\mathbf{F}) \cdot d\mathbf{S}.
\]
**Given Vector Field:**
\[
\mathbf{F}(x, y, z) = e^{xy} \cos(z) \, \mathbf{i} + x^2 z \, \mathbf{j} + xy \, \mathbf{k}
\]
where:
- \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) are the unit vectors in the x, y, and z directions respectively.
**Surface Definition:**
The surface \( S \) is defined as the hemisphere:
\[
x = \sqrt{49 - y^2 - z^2}
\]
This hemisphere is oriented in the direction of the positive x-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c3b109f-12a8-4027-af77-f6318b8ef96d%2F4552bd29-d8cd-4ffb-8f15-4d569f3cf761%2Fe7h0ac_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use Stokes' theorem to evaluate the surface integral:
\[
\iint_S \text{curl}(\mathbf{F}) \cdot d\mathbf{S}.
\]
**Given Vector Field:**
\[
\mathbf{F}(x, y, z) = e^{xy} \cos(z) \, \mathbf{i} + x^2 z \, \mathbf{j} + xy \, \mathbf{k}
\]
where:
- \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) are the unit vectors in the x, y, and z directions respectively.
**Surface Definition:**
The surface \( S \) is defined as the hemisphere:
\[
x = \sqrt{49 - y^2 - z^2}
\]
This hemisphere is oriented in the direction of the positive x-axis.
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