Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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### Problem Statement (1.3) Suppose that \( D \) is the ellipse \[ D = \left\{ (x, y) \in \mathbb{R}^2 : \frac{x^2}{4} + y^2 < 10 \right\} \] and that \( f \) is a differentiable function defined on all of \( \mathbb{R}^2 \). Suppose that \( (x_0, y_0) \) is in \( \partial D \), the boundary of \( D \). Denote by \( \frac{\partial f}{\partial n}(x_0, y_0) \) the derivative of \( f \) in the direction of the outward pointing unit normal at the point \( (x_0, y_0) \). Given that \[ \frac{\partial f}{\partial x}(4, 3) = 2 \quad \text{and} \quad \frac{\partial f}{\partial y}(4, 3) = 2, \] calculate \( \frac{\partial f}{\partial n}(4, 3) \).
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Transcribed Image Text:### Problem Statement (1.3) Suppose that \( D \) is the ellipse \[ D = \left\{ (x, y) \in \mathbb{R}^2 : \frac{x^2}{4} + y^2 < 10 \right\} \] and that \( f \) is a differentiable function defined on all of \( \mathbb{R}^2 \). Suppose that \( (x_0, y_0) \) is in \( \partial D \), the boundary of \( D \). Denote by \( \frac{\partial f}{\partial n}(x_0, y_0) \) the derivative of \( f \) in the direction of the outward pointing unit normal at the point \( (x_0, y_0) \). Given that \[ \frac{\partial f}{\partial x}(4, 3) = 2 \quad \text{and} \quad \frac{\partial f}{\partial y}(4, 3) = 2, \] calculate \( \frac{\partial f}{\partial n}(4, 3) \).
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