Chapter 11: Q. 2 (page 900)

$Give precise mathematical definitions or descriptions of each of the concepts that follow . Then illustrate the definition with a graph or an algebraic example . a vector function or vector - valued function in ℝ^{2} and ℝ^{3}$

Short Answer

Expert verified

$For ℝ^{2} : r (t) = <x (t), y (t)> = x (t) i + y (t) j . For ℝ^{3} : r (t) = <x (t), y (t), z (t)> = x (t) i + y (t) j + z (t) k .$

Step by step solution

Step 1. Given information is:

$a vector function or vector - valued function in ℝ^{2} and ℝ^{3}$

Step 2. Definition for vector-valued function in ℝ2

$Let x = x (t), y = y (t) be t w o real - valued functions, each of which is defined on some interval I \subseteq R . A vector function, or vector - valued function, r (t), with t w o components is a function of the form r (t) = <x (t), y (t)> = x (t) i + y (t) j . The variable t in the vector function is called the parameter . The functions x = x (t), y = y (t), are called the components of r (t) .$

Step 3. Definition for vector-valued function in ℝ3

$Let x = x (t), y = y (t), and z = z (t) be three real - valued functions, each of which is defined on some interval I \subseteq R . A vector function, or vector - valued function, r (t), with three components is a function of the form r (t) = <x (t), y (t), z (t)> = x (t) i + y (t) j + z (t) k . The variable t in the vector function is called the parameter . The functions x = x (t), y = y (t), and z = z (t) are called the components of r (t) .$

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$Give precise mathematical definitions or descriptions of each of the concepts that follow . Then illustrate the definition with a graph or an algebraic example . a vector function or vector - valued function in ℝ^{2} and ℝ^{3}$

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Definition for vector-valued function in ℝ2

Step 3. Definition for vector-valued function in ℝ3

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Giveprecisemathematicaldefinitionsordescriptionsofeachoftheconceptsthatfollow.Thenillustratethedefinitionwithagraphoranalgebraicexample.avectorfunctionorvector-valuedfunctioninℝ2andℝ3

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Definition for vector-valued function in ℝ2

Step 3. Definition for vector-valued function in ℝ3

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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$Give precise mathematical definitions or descriptions of each of the concepts that follow . Then illustrate the definition with a graph or an algebraic example . a vector function or vector - valued function in ℝ^{2} and ℝ^{3}$