Find a potential function f for the field F. F = (Inx+ sec c²(10x+10y))i + sec (10x+10y)+ 7y 2 y²+z² |j+ 7z 2 V +Z A general expression for the infinitely many potential functions is f(x,y,z) = (Type an exact answer.)
Find a potential function f for the field F. F = (Inx+ sec c²(10x+10y))i + sec (10x+10y)+ 7y 2 y²+z² |j+ 7z 2 V +Z A general expression for the infinitely many potential functions is f(x,y,z) = (Type an exact answer.)
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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![### Finding a Potential Function for the Field F
Given the vector field \( F \), we aim to find a potential function \( f \) such that \( \nabla f = F \).
The given field \( F \) is represented as:
\[ F = \left( \ln x + \sec^2 (10x + 10y) \right) \mathbf{i} + \left( \sec^2 (10x + 10y) + \frac{7y}{y^2 + z^2} \right) \mathbf{j} + \left( \frac{7z}{y^2 + z^2} \right) \mathbf{k} \]
Where:
- \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) are the unit vectors in the x, y, and z directions, respectively.
A general expression for the infinitely many potential functions is \( f(x,y,z) = \) [Input box for students to type an exact answer].
The vector components on the right-hand side of the equation incorporate composite functions and logarithms. Here are the separated components for better understanding:
- The \( i \)-component is \( \ln x + \sec^2 (10x + 10y) \)
- The \( j \)-component is \( \sec^2 (10x + 10y) + \frac{7y}{y^2 + z^2} \)
- The \( k \)-component is \( \frac{7z}{y^2 + z^2} \)
### Explanation:
To derive the potential function \( f \), students should integrate each component of \( F \) with respect to its relevant variable while treating other variables as constants. The final potential function will include an arbitrary function of the remaining variables after each integration step.
### Student Interactive Input
There is a text box provided where students are prompted to type their exact answer for the general expression representing the function \( f(x,y,z) \).
### Educational Goal
The goal is to reinforce concepts of vector calculus, particularly in finding scalar potential functions from vector fields, involving partial derivatives and integrals.
For explanations of any graphs or diagrams:
No additional diagrams or graphs are included within this image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1840ebe-b0df-4fe8-9210-d8e3dcfa32cc%2F84fb5eb4-884c-4407-b08a-fcf894546458%2Ftdxjecm_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding a Potential Function for the Field F
Given the vector field \( F \), we aim to find a potential function \( f \) such that \( \nabla f = F \).
The given field \( F \) is represented as:
\[ F = \left( \ln x + \sec^2 (10x + 10y) \right) \mathbf{i} + \left( \sec^2 (10x + 10y) + \frac{7y}{y^2 + z^2} \right) \mathbf{j} + \left( \frac{7z}{y^2 + z^2} \right) \mathbf{k} \]
Where:
- \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) are the unit vectors in the x, y, and z directions, respectively.
A general expression for the infinitely many potential functions is \( f(x,y,z) = \) [Input box for students to type an exact answer].
The vector components on the right-hand side of the equation incorporate composite functions and logarithms. Here are the separated components for better understanding:
- The \( i \)-component is \( \ln x + \sec^2 (10x + 10y) \)
- The \( j \)-component is \( \sec^2 (10x + 10y) + \frac{7y}{y^2 + z^2} \)
- The \( k \)-component is \( \frac{7z}{y^2 + z^2} \)
### Explanation:
To derive the potential function \( f \), students should integrate each component of \( F \) with respect to its relevant variable while treating other variables as constants. The final potential function will include an arbitrary function of the remaining variables after each integration step.
### Student Interactive Input
There is a text box provided where students are prompted to type their exact answer for the general expression representing the function \( f(x,y,z) \).
### Educational Goal
The goal is to reinforce concepts of vector calculus, particularly in finding scalar potential functions from vector fields, involving partial derivatives and integrals.
For explanations of any graphs or diagrams:
No additional diagrams or graphs are included within this image.
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