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Magazine Chapter 4: Problem 164
Find a unit normal vector to the surface at the indicated point. $$\ln \left(\frac{x}{y-z}\right)=0 \text { when } x=y=1$$
Short Answer
Expert verified
The unit normal vector is \( \left( \frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \).
Step by step solution
01
Understand the Surface Equation
Given the equation \( \ln \left(\frac{x}{y-z}\right) = 0 \), we can simplify this to \( \frac{x}{y-z} = 1 \). Consequently, the equation of the surface becomes \( x = y - z \).
02
Rewrite as a Level Surface
Rearranging the surface equation, we have \( f(x, y, z) = x - y + z = 0 \). Here, \( f(x, y, z) \) represents a level surface in \( \mathbb{R}^3 \).
03
Calculate the Gradient
To find a normal vector to the surface at a point, we compute the gradient of \( f(x, y, z) \). The gradient is given by \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) = (1, -1, 1) \).
04
Evaluate the Gradient at the Indicated Point
Substitute \( x = 1, y = 1, z = 0 \) into the gradient to find its value at the given point. The gradient remains \( (1, -1, 1) \) since it is a constant vector field on this surface.
05
Normalize the Gradient Vector
To find the unit normal vector, we need to normalize the gradient vector. The magnitude of \( abla f = (1, -1, 1) \) is \( \|abla f\| = \sqrt{1^2 + (-1)^2 + 1^2} = \sqrt{3} \). Thus, the unit normal vector is \( \left( \frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Unit Vector
In mathematics and physics, a unit vector is a vector that has a magnitude of 1. Unit vectors are particularly useful in defining directions without considering magnitude. In our exercise, we aim to find a unit normal vector to a surface, which means a vector that is perpendicular to the surface at a given point and has a magnitude of 1.
- Unit vectors are often denoted with a hat above the letter, such as \( \hat{i} \), \( \hat{j} \), or \( \hat{k} \).
- To convert any vector into a unit vector, we use the process of normalization, which involves dividing the vector by its magnitude.
- A unit normal vector is essential in many fields, including physics and computer graphics, for calculations involving force directions and surface orientations.
Gradient
The gradient is a vector that contains all the partial derivatives of a multivariable function. It points in the direction of the greatest rate of increase of the function. For a function of three variables, like in our exercise, the gradient \( abla f \) is given by \( (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}) \).
- The gradient vector is extremely crucial in determining the nature of a surface at a specific point.
- In our exercise, the gradient \( (1, -1, 1) \) tells us how the surface is oriented in space.
- It also serves as a normal vector since it's perpendicular to level surfaces, thus playing a dual role in both analytical and geometrical contexts.
Level Surface
A level surface, in mathematical terms, is the set of points \((x, y, z)\) in three-dimensional space where a given scalar function \(f(x, y, z)\) is constant. For example, in the exercise, the level surface is \(x - y + z = 0\).
- Level surfaces are vital for visualizing and understanding complex mathematical surfaces and contours in 3D space.
- They can represent things like potential fields in physics or topographic maps in geography.
- A normal vector to the level surface is derived from its gradient, providing further analytical insight into the surface's properties.
Normalization
Normalization is the process used to convert a vector into a unit vector. This involves dividing each component of the vector by its magnitude. It ensures that the resulting vector has a length of 1, making it a unit vector.
- The formula for normalizing a vector \(\vec{v} = (a, b, c)\) is \( \frac{\vec{v}}{\|\vec{v}\|} \), where \(\|\vec{v}\|\) is the magnitude and is calculated as \( \sqrt{a^2 + b^2 + c^2} \).
- Normalization keeps the direction of the vector unchanged, which is particularly useful when direction is vital in applications.
- In our context, normalization was needed to ensure the gradient became a unit normal vector, crucial when applying constraints like equilibrium or field consistency.
