Find the line integral of F = (9x² - 7x) i+7z j+ k from (0,0,0) to (1,1,1) over each of the following paths in the accompanying figure. a. C₁: r(t)=ti+tj+tk, Osts1 b. C₂: r(t)=ti+t²j+t k, Osts1 c. C3UC4: the path consisting of the line segment from (0,0,0) to (1,1,0) followed by the segment from (1,1,0) to (1,1,1) (0,0,0) C₁ 5 (1,1,1)
Find the line integral of F = (9x² - 7x) i+7z j+ k from (0,0,0) to (1,1,1) over each of the following paths in the accompanying figure. a. C₁: r(t)=ti+tj+tk, Osts1 b. C₂: r(t)=ti+t²j+t k, Osts1 c. C3UC4: the path consisting of the line segment from (0,0,0) to (1,1,0) followed by the segment from (1,1,0) to (1,1,1) (0,0,0) C₁ 5 (1,1,1)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Line Integrals of Vector Field
#### Problem Statement:
Find the line integral of \( \mathbf{F} = \left( 9x^2 - 7x \right) \mathbf{i} + 7z \mathbf{j} + \mathbf{k} \) from \( (0,0,0) \) to \( (1,1,1) \) over each of the following paths in the accompanying figure:
**a. Path \( C_1 \):**
\[ \mathbf{r}(t) = t \mathbf{i} + t \mathbf{j} + t \mathbf{k}, \; 0 \leq t \leq 1 \]
**b. Path \( C_2 \):**
\[ \mathbf{r}(t) = t \mathbf{i} + \frac{t^2}{4} \mathbf{j} + t \mathbf{k}, \; 0 \leq t \leq 1 \]
**c. Path \( C_3 \cup C_4 \):**
The path consisting of the line segment from \( (0,0,0) \) to \( (1,1,0) \) followed by the segment from \( (1,1,0) \) to \( (1,1,1) \).
#### Explanation of Figure:
The accompanying figure illustrates three paths in a 3-dimensional Cartesian coordinate system:
- **Path \( C_1 \)** is represented by a yellow curve. It is a straight line from \( (0,0,0) \) to \( (1,1,1) \).
- **Path \( C_2 \)** is illustrated by the blue parabola-like curve, starting at \( (0,0,0) \) and curving slightly up to \( (1,1,1) \). This represents a non-linear route.
- **Path \( C_3 \cup C_4 \)** is depicted with red and brown line segments. Path \( C_3 \) (red) extends from \( (0,0,0) \) to \( (1,1,0) \), and Path \( C_4 \) (brown) continues from \( (1,1,0) \) to \( (1,1,1) \).
Each path provides a different route from the origin to the point \( (](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd49838c-9fe9-4c2b-b877-1f4ac641c60c%2F738a4858-a85e-4e8e-8836-ea5b483b7897%2F948umuo_processed.png&w=3840&q=75)
Transcribed Image Text:### Line Integrals of Vector Field
#### Problem Statement:
Find the line integral of \( \mathbf{F} = \left( 9x^2 - 7x \right) \mathbf{i} + 7z \mathbf{j} + \mathbf{k} \) from \( (0,0,0) \) to \( (1,1,1) \) over each of the following paths in the accompanying figure:
**a. Path \( C_1 \):**
\[ \mathbf{r}(t) = t \mathbf{i} + t \mathbf{j} + t \mathbf{k}, \; 0 \leq t \leq 1 \]
**b. Path \( C_2 \):**
\[ \mathbf{r}(t) = t \mathbf{i} + \frac{t^2}{4} \mathbf{j} + t \mathbf{k}, \; 0 \leq t \leq 1 \]
**c. Path \( C_3 \cup C_4 \):**
The path consisting of the line segment from \( (0,0,0) \) to \( (1,1,0) \) followed by the segment from \( (1,1,0) \) to \( (1,1,1) \).
#### Explanation of Figure:
The accompanying figure illustrates three paths in a 3-dimensional Cartesian coordinate system:
- **Path \( C_1 \)** is represented by a yellow curve. It is a straight line from \( (0,0,0) \) to \( (1,1,1) \).
- **Path \( C_2 \)** is illustrated by the blue parabola-like curve, starting at \( (0,0,0) \) and curving slightly up to \( (1,1,1) \). This represents a non-linear route.
- **Path \( C_3 \cup C_4 \)** is depicted with red and brown line segments. Path \( C_3 \) (red) extends from \( (0,0,0) \) to \( (1,1,0) \), and Path \( C_4 \) (brown) continues from \( (1,1,0) \) to \( (1,1,1) \).
Each path provides a different route from the origin to the point \( (
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