Advanced Engineering Mathematics
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ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Let F= x^3 i + j + k. Find the general equation of a flow line. Find the flow line through the origin.
![### Vector Field and Flow Lines
**Problem Statement:**
1. Let \( \mathbf{F} = x^3 \mathbf{i} + \mathbf{j} + \mathbf{k} \). Find the general equation of a flow line. Find the flow line through the origin.
---
**Explanation:**
Given a vector field \( \mathbf{F} \) represented as \( x^3 \mathbf{i} + \mathbf{j} + \mathbf{k} \), your goal is to determine the general equation for the flow line, and specifically, the flow line that passes through the origin.
To solve this, you can follow these steps:
1. **General Equation of a Flow Line:**
- Flow lines or streamlines in a vector field are curves whose tangent at each point is in the direction of the vector field at that point.
- The equations for the flow lines can be obtained by solving the differential equations given by the components of \( \mathbf{F} \):
\[
\frac{dx}{x^3} = dy = dz
\]
2. **Finding the Specific Flow Line Through the Origin:**
- To determine the flow line that passes through the origin (0, 0, 0), substitute the initial conditions into your general solution.
By working through these steps, you can understand how to model and interpret flow lines in this vector field. This exercise is essential in fields such as fluid dynamics, electromagnetic theory, and other areas involving vector calculus.](https://content.bartleby.com/qna-images/question/ebd9624c-6bf1-4607-b017-42b6f178ff82/0a8e9f41-2267-4a5a-a026-405f07c54b2c/h7oz90n.png)
Transcribed Image Text:### Vector Field and Flow Lines
**Problem Statement:**
1. Let \( \mathbf{F} = x^3 \mathbf{i} + \mathbf{j} + \mathbf{k} \). Find the general equation of a flow line. Find the flow line through the origin.
---
**Explanation:**
Given a vector field \( \mathbf{F} \) represented as \( x^3 \mathbf{i} + \mathbf{j} + \mathbf{k} \), your goal is to determine the general equation for the flow line, and specifically, the flow line that passes through the origin.
To solve this, you can follow these steps:
1. **General Equation of a Flow Line:**
- Flow lines or streamlines in a vector field are curves whose tangent at each point is in the direction of the vector field at that point.
- The equations for the flow lines can be obtained by solving the differential equations given by the components of \( \mathbf{F} \):
\[
\frac{dx}{x^3} = dy = dz
\]
2. **Finding the Specific Flow Line Through the Origin:**
- To determine the flow line that passes through the origin (0, 0, 0), substitute the initial conditions into your general solution.
By working through these steps, you can understand how to model and interpret flow lines in this vector field. This exercise is essential in fields such as fluid dynamics, electromagnetic theory, and other areas involving vector calculus.
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