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Chapter 13: Problem 6

Find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla \boldsymbol{F}(\boldsymbol{x}, \boldsymbol{y}, z) .]$$x^{2}+y^{2}+z^{2}=11\)

Short Answer

Expert verified
The unit normal vector to the surface \(x^{2}+y^{2}+z^{2}-11=0\) at any point (x, y, z) is \(\frac{1}{2}\langle x, y, z \rangle\).

Step by step solution

01

Calculate the gradient of F

The gradient of the scalar field \(F(x, y, z)\) is defined as \(\nabla F = \langle \frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z} \rangle \). Applying this to calculate the gradient of \(F = x^{2}+y^{2}+z^{2}-11\), we get: \(\nabla F = \langle 2x, 2y, 2z \rangle \).
02

Normalize the gradient vector

The normalization of a vector involves dividing each of its components by its magnitude. The magnitude of a vector \(v = \langle a, b, c \rangle \) is calculated as \(\|v\| = \sqrt{a^{2} + b^{2} + c^{2}} \). The magnitude of \(\nabla F = \langle 2x, 2y, 2z \rangle\) is \(\|\nabla F\| = \sqrt{(2x)^{2} + (2y)^{2} + (2z)^{2}} = 2\sqrt{x^{2} + y^{2} + z^{2}}\). Thus, the normalized gradient vector is \(\frac{\nabla F}{\|\nabla F\|} = \frac{1}{2}\langle x, y, z \rangle \).
03

Validate by checking magnitude

To ensure that the obtained vector is indeed a unit vector, its magnitude should be checked to be equal to 1. The magnitude of the unit normal vector \(\frac{1}{2}\langle x, y, z \rangle\) is calculated as \(\left\|\frac{1}{2}\langle x, y, z \rangle\right\| = \frac{1}{2}\sqrt{x^{2} + y^{2} + z^{2}} = 1\). It is therefore confirmed that \(\frac{1}{2}\langle x, y, z \rangle\) is a unit normal vector to the surface at any point (x, y, z)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gradient Vector
In vector calculus, the gradient vector is a powerful concept used to denote the direction of the steepest ascent of a scalar field. Essentially, it points to where the function increases most rapidly. The gradient of a scalar field \( F(x, y, z) \) is represented by the symbol \( abla F \), and it combines all the partial derivatives of \( F \) with respect to each variable:

\[abla F = \left\langle \frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z} \right\rangle.\]
  • Directional Insight: The gradient provides insight into how the scalar field changes in all directions, and it is perpendicular to level surfaces of the field.
  • Example: For the function \( F = x^2 + y^2 + z^2 - 11 \), its gradient is \( abla F = \langle 2x, 2y, 2z \rangle \).
To find the gradient, always compute the partial derivatives with respect to each variable, giving you a vector that summarizes how each component affects the function's value.
Normal Vector
A normal vector is one that is perpendicular, or normal, to a surface at a given point. This concept is particularly useful in geometry and physics, for calculating tangents, slopes, or boundaries.
  • The gradient vector \( abla F \) of a scalar field serves as the normal vector to its level surface at any point.
  • For the equation \( x^2+y^2+z^2=11 \), the normal to the surface at any point (x, y, z) is \( \langle 2x, 2y, 2z \rangle \).
The normal vector is crucial in defining the orientation of the surface. With this orientation, properties like surface integrals and flux calculations can be understood and calculated.
Unit Vector
A unit vector is a vector with a magnitude of 1. Normalizing a vector involves changing its magnitude to 1 while preserving its direction. This is done by dividing the vector by its magnitude.

For a vector \( v = \langle a, b, c \rangle \), the magnitude is given by:
\[\|v\| = \sqrt{a^2 + b^2 + c^2}.\]
  • Process: Divide each component of the vector by \( \|v\| \) to normalize it.
  • Example: The normalized vector of \( abla F = \langle 2x, 2y, 2z \rangle \) is \( \frac{1}{2}\langle x, y, z \rangle \), ensuring its magnitude is 1.
Unit vectors are indispensable because they allow you to express directions independently of magnitude, making them ideal for calculations involving directions and orientations.
Scalar Field
A scalar field assigns a scalar value to every point in space, representing phenomena like temperature distribution or gravitational potential. In mathematical terms, a scalar field is a function \( F(x, y, z) \) that maps each point to a single scalar value.
  • Visualization: Imagine a scalar field as a landscape with hills and valleys, where the height at each point is the value of the field.
  • Application: In the exercise, \( F = x^2 + y^2 + z^2 - 11 \) defines a spherical surface, where each point on the surface satisfies the equation \( x^2 + y^2 + z^2 = 11 \).
Scalar fields are fundamental in physics and engineering, aiding in understanding spatial variations and designing systems that interact with these variations efficiently.

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Most popular questions from this chapter

Consider the objective function \(g(\alpha, \beta, \gamma)=\) \(\cos \alpha \cos \beta \cos \gamma\) subject to the constraint that \(\alpha, \beta\), and \(\gamma\) are the angles of a triangle. (a) Use Lagrange multipliers to maximize \(g\). (b) Use the constraint to reduce the function \(g\) to a function of two independent variables. Use a computer algebra system to graph the surface represented by \(g .\) Identify the maximum values on the graph.

A measure of what hot weather feels like to two average persons is the Apparent Temperature Index. A model for this index is \(A=0.885 t-22.4 h+1.20 t h-0.544\) where \(A\) is the apparent temperature in degrees Celsius, \(t\) is the air temperature, and \(h\) is the relative humidity in decimal form. (Source: The UMAP Journal, Fall 1984) (a) Find \(\partial A / \partial t\) and \(\partial A / \partial h\) when \(t=30^{\circ}\) and \(h=0.80\). (b) Which has a greater effect on \(A\), air temperature or humidity? Explain.

Prove that the angle of inclination \(\theta\) of the tangent plane to the surface \(z=f(x, y)\) at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) is given by \(\cos \theta=\frac{1}{\sqrt{\left[f_{x}\left(x_{0}, y_{0}\right)\right]^{2}+\left[f_{y}\left(x_{0}, y_{0}\right)\right]^{2}+1}}\)

Use Lagrange multipliers to find the indicated extrema of \(f\) subject to two constraints. In each case, assume that \(x, y\), and \(z\) are nonnegative. Maximize \(f(x, y, z)=x y z\) Constraints: \(x^{2}+z^{2}=5, \quad x-2 y=0\)

Ideal Gas Law According to the Ideal Gas Law, \(P V=k T\), where \(P\) is pressure, \(V\) is volume, \(T\) is temperature (in Kelvins), and \(k\) is a constant of proportionality. A tank contains 2600 cubic inches of nitrogen at a pressure of 20 pounds per square inch and a temperature of \(300 \mathrm{~K}\). (a) Determine \(k\). (b) Write \(P\) as a function of \(V\) and \(T\) and describe the level curves.
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