Q29E Give an example of a function wh... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q29E (page 120)

Give an example of a function whose image is the unit sphere
$x^{2} + y^{2} + z^{2} = 1$
inR³.

Short Answer

Expert verified

f $[\begin{matrix} 蠒 \\ 胃 \end{matrix}] = [\begin{matrix} \sin 蠒 \cos 胃 \\ \sin 蠒 \sin 胃 \\ \cos 胃 \end{matrix}]$ is a function whose image is the unit sphere $x^{2} + y^{2} + z^{2} = 1$ in ${鈩�}^{3} {鈩�}^{3}$

Step by step solution

Consider the parameters.

$i m a g e (f) = \{f (x) : x i n X\} = \{b i n Y : b = f (x), f o r s o m e x i n X\} x = \sin 蠒 \cos 胃 L e t, Y = \sin 蠒 \sin 胃 z = \cos 蠒$ The image of a function consists of all the values the function takes in its target space. If f is a function from X to Y, then

$x^{2} + y^{2} + z^{2} = 1 鈬� {(\sin 蠒 \cos 胃)}^{2} + {(\sin 蠒 \sin 胃)}^{2} + {(\cos 蠒)}^{2} = 1 鈬� \sin^{2} 蠒 (\cos^{2} 胃 + \sin^{2} 胃) + \cos^{2} 蠒 = 1$

Let, $y = \sin 蠒 \sin 胃 z = \cos 蠒$

Substitute these in the equation of sphere, thus,

$x^{2} + y^{2} + z^{2} = 1 鈬� {(\sin 蠒 \cos 胃)}^{2} + {(\sin 蠒 \sin 胃)}^{2} + {(\cos 蠒)}^{2} 鈬� \sin^{2} 蠒 (\cos^{2} 胃 + \sin^{2} 胃) + \cos^{2} 蠒 = 1 鈬� \sin^{2} 蠒 + \cos^{2} 蠒 = 1 鈭� 1 = 1$

Final answer.

$f [\begin{matrix} 蠒 \\ 胃 \end{matrix}] = [\begin{matrix} \sin 蠒 \cos 胃 \\ \sin 蠒 \sin 胃 \\ \cos 蠒 \end{matrix}]$ is a parametrization of the ellipse $x^{2} + y^{2} + z^{2} = 1$ in ${鈩�}^{3} {鈩�}^{3}$ .

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91影视

Give an example of a function whose image is the unit sphere
$x^{2} + y^{2} + z^{2} = 1$
inR³.

Short Answer

Step by step solution

Consider the parameters.

Final answer.

One App. One Place for Learning.

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91影视

Give an example of a function whose image is the unit sphere x2+y2+z2=1inR3.

Short Answer

Step by step solution

Consider the parameters.

Final answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Discrete Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Probability and Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

Give an example of a function whose image is the unit sphere
$x^{2} + y^{2} + z^{2} = 1$
inR³.