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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![### Calculating the k-component of Curl for a Given Vector Field
In this exercise, you are asked to find the \( k \)-component of the curl, denoted as \( \text{curl} \, \mathbf{F} \), for the following vector field \( \mathbf{F} \) in the plane:
\[ \mathbf{F} = (x e^y) \mathbf{i} + (8y e^x) \mathbf{j} \]
This involves determining the partial derivatives of the vector field's components and then finding their difference as per the definition of curl in a two-dimensional context.
To find the \( k \)-component of \( \text{curl} \, \mathbf{F} \):
\[ (\text{curl} \, \mathbf{F}) \cdot \mathbf{k} = \left( \frac{\partial (8y e^x)}{\partial x} - \frac{\partial (x e^y)}{\partial y} \right) \mathbf{k} \]
Perform the partial derivatives:
1. Calculate \( \frac{\partial (8y e^x)}{\partial x} \):
\[ \frac{\partial (8y e^x)}{\partial x} = 8y e^x \]
2. Calculate \( \frac{\partial (x e^y)}{\partial y} \):
\[ \frac{\partial (x e^y)}{\partial y} = x e^y \]
Now, subtract these results:
\[ (\text{curl} \, \mathbf{F}) \cdot \mathbf{k} = \left( 8y e^x - x e^y \right) \mathbf{k} \]
Hence, the \( k \)-component of the curl for the given vector field \( \mathbf{F} \) is:
\[ (\text{curl} \, \mathbf{F}) \cdot \mathbf{k} = \left( 8y e^x - x e^y \right) \mathbf{k} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c6d6b24-4b7a-4949-954a-bc8418c314a2%2F52f407c9-f82a-4621-8962-e62de659596d%2F6kke9avr_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the k-component of Curl for a Given Vector Field
In this exercise, you are asked to find the \( k \)-component of the curl, denoted as \( \text{curl} \, \mathbf{F} \), for the following vector field \( \mathbf{F} \) in the plane:
\[ \mathbf{F} = (x e^y) \mathbf{i} + (8y e^x) \mathbf{j} \]
This involves determining the partial derivatives of the vector field's components and then finding their difference as per the definition of curl in a two-dimensional context.
To find the \( k \)-component of \( \text{curl} \, \mathbf{F} \):
\[ (\text{curl} \, \mathbf{F}) \cdot \mathbf{k} = \left( \frac{\partial (8y e^x)}{\partial x} - \frac{\partial (x e^y)}{\partial y} \right) \mathbf{k} \]
Perform the partial derivatives:
1. Calculate \( \frac{\partial (8y e^x)}{\partial x} \):
\[ \frac{\partial (8y e^x)}{\partial x} = 8y e^x \]
2. Calculate \( \frac{\partial (x e^y)}{\partial y} \):
\[ \frac{\partial (x e^y)}{\partial y} = x e^y \]
Now, subtract these results:
\[ (\text{curl} \, \mathbf{F}) \cdot \mathbf{k} = \left( 8y e^x - x e^y \right) \mathbf{k} \]
Hence, the \( k \)-component of the curl for the given vector field \( \mathbf{F} \) is:
\[ (\text{curl} \, \mathbf{F}) \cdot \mathbf{k} = \left( 8y e^x - x e^y \right) \mathbf{k} \]
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