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Let P denote the path that goes in straight lines from (1, 1, 1) to (-1,-1,1) to (-1,-1,-1) to (1,1,-1) and then back to (1,1,1). 2xyz+z eầu +2 x²z+sin(2) Compute fp F. dr, where F(x, y, z): = ().

Let P denote the path that goes in straight lines from (1, 1, 1) to (-1,-1,1) to (-1,-1,-1) to (1,1,-1) and then back to (1,1,1). 2xyz+z eầu +2 x²z+sin(2) Compute fp F. dr, where F(x, y, z): = ().

Advanced Engineering Mathematics
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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### Problem Statement 2. Let \( P \) denote the path that goes in straight lines from \( (1, 1, 1) \) to \( (-1, -1, 1) \) to \( (-1, -1, -1) \) to \( (1, 1, -1) \) and then back to \( (1, 1, 1) \). Compute \(\int_{P} \vec{F} \cdot d\vec{r}\), where \(\vec{F}(x, y, z) = \left( \begin{array}{c} \frac{2xy + z}{e^{xy} + 2} \\ x^2z + \sin(z) \end{array} \right)\). ### Explanation - **Path Description:** The problem describes a closed path \( P \) in three-dimensional space. The path consists of straight-line segments connecting the points sequentially: - \( (1, 1, 1) \to (-1, -1, 1) \) - \( (-1, -1, 1) \to (-1, -1, -1) \) - \( (-1, -1, -1) \to (1, 1, -1) \) - \( (1, 1, -1) \to (1, 1, 1) \). - **Vector Field:** The vector field \(\vec{F}(x, y, z)\) is given with components: - First component: \(\frac{2xy + z}{e^{xy} + 2}\) - Second component: \(x^2z + \sin(z)\) - **Line Integral Task:** The task is to evaluate the line integral of the vector field \(\vec{F}\) along the path \( P \). The line integral \(\int_{P} \vec{F} \cdot d\vec{r}\) involves integrating the dot product of the vector field \(\vec{F}\) and the differential path vector \(d\vec{r}\) along the closed path \( P \).
Transcribed Image Text:### Problem Statement 2. Let \( P \) denote the path that goes in straight lines from \( (1, 1, 1) \) to \( (-1, -1, 1) \) to \( (-1, -1, -1) \) to \( (1, 1, -1) \) and then back to \( (1, 1, 1) \). Compute \(\int_{P} \vec{F} \cdot d\vec{r}\), where \(\vec{F}(x, y, z) = \left( \begin{array}{c} \frac{2xy + z}{e^{xy} + 2} \\ x^2z + \sin(z) \end{array} \right)\). ### Explanation - **Path Description:** The problem describes a closed path \( P \) in three-dimensional space. The path consists of straight-line segments connecting the points sequentially: - \( (1, 1, 1) \to (-1, -1, 1) \) - \( (-1, -1, 1) \to (-1, -1, -1) \) - \( (-1, -1, -1) \to (1, 1, -1) \) - \( (1, 1, -1) \to (1, 1, 1) \). - **Vector Field:** The vector field \(\vec{F}(x, y, z)\) is given with components: - First component: \(\frac{2xy + z}{e^{xy} + 2}\) - Second component: \(x^2z + \sin(z)\) - **Line Integral Task:** The task is to evaluate the line integral of the vector field \(\vec{F}\) along the path \( P \). The line integral \(\int_{P} \vec{F} \cdot d\vec{r}\) involves integrating the dot product of the vector field \(\vec{F}\) and the differential path vector \(d\vec{r}\) along the closed path \( P \).
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