Chapter 12: Q 58 (page 917)

In Exercises $53 - 58$ , determine the level surfaces role="math" $c = - 3, - 2, - 1, 0, 1, 2, 3$ if they exist for the specified function.
$f (x, y, z) = \sqrt{1 - (x^{2} + y^{2} + z^{2})}$ .

Short Answer

Expert verified

The level surfaces are determined by the equation $x^{2} + y^{2} + z^{2} = 1 - c^{2}$ .

For $c = - 1, 1$ , $x^{2} + y^{2} + z^{2} = 0$ represents the origin of 3-D plane.

Also the level surface is sphere with equation :-

$x^{2} + y^{2} + z^{2} = 1 - c^{2}$ , $c = 0$ .

For $c = - 3, - 2, 2, 3$ level surfaces are undefined.

Step by step solution

Step 1. Given Information

We have given the following function :-

$f (x, y, z) = \sqrt{1 - (x^{2} + y^{2} + z^{2})}$ .

We have to determine level surfaces $c = - 3, - 2, - 1, 0, 1, 2, 3$ for the given function.

Step 2. Determine level surfaces

The given function is :-

$f (x, y, z) = \sqrt{1 - (x^{2} + y^{2} + z^{2})}$ .

We know that the level surfaces are determined as :-

$f (x, y, z) = c$ .

That is we have :-

role="math" localid="1653401635917" $\sqrt{1 - (x^{2} + y^{2} + z^{2})} = c \Rightarrow 1 - (x^{2} + y^{2} + z^{2}) = c^{2} \Rightarrow x^{2} + y^{2} + z^{2} = 1 - c^{2}$

This is the equation of sphere.

So the level surfaces are undefined for role="math" localid="1653401835726" $c = - 3, - 2, 2, 3$ as radius of sphere cannot be negative.

Also for role="math" localid="1653401848401" $c = - 1, 1$ represents the origin of 3-D plane.

So that we have :-

The level surfaces are sphere with the equation $x^{2} + y^{2} + z^{2} = 1 - c^{2}$ forrole="math" localid="1653401820381" $c = 0$ .

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In Exercises $53 - 58$ , determine the level surfaces role="math" $c = - 3, - 2, - 1, 0, 1, 2, 3$ if they exist for the specified function.
$f (x, y, z) = \sqrt{1 - (x^{2} + y^{2} + z^{2})}$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Determine level surfaces

One App. One Place for Learning.

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In Exercises 53-58, determine the level surfaces role="math" c=-3,-2,-1,0,1,2,3if they exist for the specified function.fx,y,z=1-x2+y2+z2.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Determine level surfaces

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Statistics

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Exercises $53 - 58$ , determine the level surfaces role="math" $c = - 3, - 2, - 1, 0, 1, 2, 3$ if they exist for the specified function.
$f (x, y, z) = \sqrt{1 - (x^{2} + y^{2} + z^{2})}$ .