EXERCISE 3.2 The implicit function of an ellipsoid surface is x²/a² + y²/b² + ²/² −1 = 0. Derive the general expression of the surface normal. Compute the surface normal at point (0, 0, -c).
EXERCISE 3.2 The implicit function of an ellipsoid surface is x²/a² + y²/b² + ²/² −1 = 0. Derive the general expression of the surface normal. Compute the surface normal at point (0, 0, -c).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.6: Quadratic Functions
Problem 35E
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![## Exercise 3.2
The implicit function of an ellipsoid surface is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} - 1 = 0. \]
1. **Derive the General Expression of the Surface Normal:**
To find the surface normal of an ellipsoid, we need to compute the gradient of the implicit function.
2. **Compute the Surface Normal at the Point (0, 0, -c):**
Substitute the point \((0, 0, -c)\) into the derived expression for the normal vector to find the surface normal at this specific point.
Make sure to include the detailed steps for computing the gradient (∇F = (∂F/∂x, ∂F/∂y, ∂F/∂z)) and plugging in the point coordinates to clearly show the process.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F780f9839-f737-4aba-91a2-6210989911b1%2F3c725b23-9bb9-4f6c-b719-a3233893b892%2F8v04f4d_processed.png&w=3840&q=75)
Transcribed Image Text:## Exercise 3.2
The implicit function of an ellipsoid surface is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} - 1 = 0. \]
1. **Derive the General Expression of the Surface Normal:**
To find the surface normal of an ellipsoid, we need to compute the gradient of the implicit function.
2. **Compute the Surface Normal at the Point (0, 0, -c):**
Substitute the point \((0, 0, -c)\) into the derived expression for the normal vector to find the surface normal at this specific point.
Make sure to include the detailed steps for computing the gradient (∇F = (∂F/∂x, ∂F/∂y, ∂F/∂z)) and plugging in the point coordinates to clearly show the process.
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