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Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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The image contains handwritten notes on a mathematics problem involving vector fields and line integrals. Herein is a transcription and explanation:

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**Problem Statement:**

Let \( F(x, y) = e^{2y} \mathbf{i} + (1 + 2x) e^{2y} \mathbf{j} \).

1. Show that \( F \) is conservative.
2. Compute the line integral \( \int_C F \cdot d\mathbf{s} \) along any curve joining the points \( (0, 1) \) to \( (1, 2) \).
3. Find a scalar potential \( \phi : \mathbb{R}^3 \to \mathbb{R} \) such that \( F = \nabla \phi \).

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**Explanation:**

- **Conservative Vector Field:** A vector field \( F \) is said to be conservative if it can be expressed as the gradient of a scalar potential function \( \phi \), i.e., \( F = \nabla \phi \). In a conservative field, the line integral between two points is path-independent.

- **Line Integral:** The computation of a line integral \( \int_C F \cdot d\mathbf{s} \) involves integrating the field along a specified curve \( C \). For conservative fields, this integral depends only on the endpoints of the curve.

- **Scalar Potential \( \phi \):** Finding \( \phi \) involves identifying a function such that the gradient of \( \phi \) recovers the vector field \( F \).

This problem involves identifying and working with properties of vector fields and calculus techniques related to their integrals and potentials.
Transcribed Image Text:The image contains handwritten notes on a mathematics problem involving vector fields and line integrals. Herein is a transcription and explanation: --- **Problem Statement:** Let \( F(x, y) = e^{2y} \mathbf{i} + (1 + 2x) e^{2y} \mathbf{j} \). 1. Show that \( F \) is conservative. 2. Compute the line integral \( \int_C F \cdot d\mathbf{s} \) along any curve joining the points \( (0, 1) \) to \( (1, 2) \). 3. Find a scalar potential \( \phi : \mathbb{R}^3 \to \mathbb{R} \) such that \( F = \nabla \phi \). --- **Explanation:** - **Conservative Vector Field:** A vector field \( F \) is said to be conservative if it can be expressed as the gradient of a scalar potential function \( \phi \), i.e., \( F = \nabla \phi \). In a conservative field, the line integral between two points is path-independent. - **Line Integral:** The computation of a line integral \( \int_C F \cdot d\mathbf{s} \) involves integrating the field along a specified curve \( C \). For conservative fields, this integral depends only on the endpoints of the curve. - **Scalar Potential \( \phi \):** Finding \( \phi \) involves identifying a function such that the gradient of \( \phi \) recovers the vector field \( F \). This problem involves identifying and working with properties of vector fields and calculus techniques related to their integrals and potentials.
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