Chapter 14: Q. 3 (page 1140)

Write two different normal vectors for a smooth surface S given by (x, y, g(x, y)) at the point $(x_{0}, y_{0}, g (x_{0}, y_{0})) .$

Short Answer

Expert verified

$The required two different normal vectors for a smooth surface S described by, (x, y, g (x, y)) at the point (x_{0}, y_{0}, g (x_{0}, y_{0})) are <- \frac{\partial g}{\partial x}, - \frac{\partial g}{\partial y}, 1> and <\frac{\partial g}{\partial x}, \frac{\partial g}{\partial y}, - 1>$

Step by step solution

Step 1. Given

smooth surface S given by (x, y, g(x, y)) at the point $(x_{0}, y_{0}, g (x_{0}, y_{0})) .$

Step 2. Objective

$The objective is to calculate the two different normal vectors for a smooth surface S described by (x, y, g (x, y)) at the point (x_{0}, y_{0}, g (x_{0}, y_{0})) . A parametrized surface of (x, y, g (x, y)) in terms of x and y is, r (x, y) = <x, y, g (x, y))> . Then, the desired normal vector will be, n = \pm (r_{x} + r_{y})$

Step 3. Calculation

$Now find r_{x} and r_{y} By above step, the surface S is parametrized by as follows, r (x, y) = <x, y, g (x, y))> . then, r_{x} = \frac{\partial}{\partial x} r (x, y) = \frac{\partial}{\partial x} <x, y, g (x, y))> = <1, 0, \frac{\partial g}{\partial x}> . and, r_{y} = \frac{\partial}{\partial y} r (x, y) = \frac{\partial}{\partial y} <x, y, g (x, y))> = <1, 0, \frac{\partial g}{\partial y}> . Next find r_{x} \times r_{y} . r_{x} \times r_{y} = <1, 0, \frac{\partial g}{\partial x}> \times <1, 0, \frac{\partial g}{\partial y}> . = |\begin{matrix} i & j & k \\ 1 & 0 & \frac{\partial g}{\partial x} \\ 0 & 1 & \frac{\partial g}{\partial y} \end{matrix}| = [0 . \frac{\partial g}{\partial y} - \frac{\partial g}{\partial x} . 1] i - [1 . \frac{\partial g}{\partial y} - \frac{\partial g}{\partial x} . 0] j + [1.1 - 0.0] k = - \frac{\partial g}{\partial x} i - \frac{\partial g}{\partial y} j + k = <- \frac{\partial g}{\partial x}, - \frac{\partial g}{\partial y}, 1> .$

Step 4. Result

$Then the normal vector will be, n = \pm (r_{x} + r_{y}), or n = \pm <- \frac{\partial g}{\partial x}, - \frac{\partial g}{\partial y}, 1> implies that, n = <- \frac{\partial g}{\partial x}, - \frac{\partial g}{\partial y}, 1> and n = - <- \frac{\partial g}{\partial x}, - \frac{\partial g}{\partial y}, 1> implies that, n = <- \frac{\partial g}{\partial x}, - \frac{\partial g}{\partial y}, 1> and n = <\frac{\partial g}{\partial x}, \frac{\partial g}{\partial y}, - 1> Therefore the required two different normal vectors for a smooth surface S described by, (x, y, g (x, y)) at the point (x_{0}, y_{0}, g (x_{0}, y_{0})) are <- \frac{\partial g}{\partial x}, - \frac{\partial g}{\partial y}, 1> and <\frac{\partial g}{\partial x}, \frac{\partial g}{\partial y}, - 1>$

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Write two different normal vectors for a smooth surface S given by (x, y, g(x, y)) at the point $(x_{0}, y_{0}, g (x_{0}, y_{0})) .$

Short Answer

Step by step solution

Step 1. Given

Step 2. Objective

Step 3. Calculation

Step 4. Result

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Write two different normal vectors for a smooth surface S given by (x, y, g(x, y)) at the point(x0,y0,g(x0,y0)).

Short Answer

Step by step solution

Step 1. Given

Step 2. Objective

Step 3. Calculation

Step 4. Result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Statistics

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Write two different normal vectors for a smooth surface S given by (x, y, g(x, y)) at the point $(x_{0}, y_{0}, g (x_{0}, y_{0})) .$