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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![**Problem 3: Work Done by a Vector Field**
Find the work done by the vector field **F** over the curve in the direction of increasing \( t \).
\[ \mathbf{F} = (z, x, y) \]
\[ \mathbf{r}(t) = (\sin t, \cos t, t), \quad 0 \leq t \leq 2\pi \]
*Explanation:*
1. **Vector Field \(\mathbf{F}\)**: A three-dimensional vector field where each point \((x, y, z)\) in space is assigned a vector \((z, x, y)\).
2. **Parameterization \(\mathbf{r}(t)\)**: A parameterized curve where the x, y, and z components are given as:
- \( x(t) = \sin t \)
- \( y(t) = \cos t \)
- \( z(t) = t \)
The parameter \( t \) ranges from \( 0 \) to \( 2\pi \).
This setup involves calculating the line integral of the vector field \(\mathbf{F}\) along the curve \(\mathbf{r}(t)\) in the specified direction of \( t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F306db17a-c2a3-4020-aada-7a92dff91485%2F946c1c37-329c-4da4-8010-1c7c3882f565%2Fx6fit79_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3: Work Done by a Vector Field**
Find the work done by the vector field **F** over the curve in the direction of increasing \( t \).
\[ \mathbf{F} = (z, x, y) \]
\[ \mathbf{r}(t) = (\sin t, \cos t, t), \quad 0 \leq t \leq 2\pi \]
*Explanation:*
1. **Vector Field \(\mathbf{F}\)**: A three-dimensional vector field where each point \((x, y, z)\) in space is assigned a vector \((z, x, y)\).
2. **Parameterization \(\mathbf{r}(t)\)**: A parameterized curve where the x, y, and z components are given as:
- \( x(t) = \sin t \)
- \( y(t) = \cos t \)
- \( z(t) = t \)
The parameter \( t \) ranges from \( 0 \) to \( 2\pi \).
This setup involves calculating the line integral of the vector field \(\mathbf{F}\) along the curve \(\mathbf{r}(t)\) in the specified direction of \( t \).
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