Q. 58 Let r1(t)聽and聽r2(t)聽be contin... [FREE SOLUTION]

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Chapter 11: Q. 58 (page 862)

Let $r_{1} (t) a n d r_{2} (t)$ be continuous vector functions with three components, and let $t_{0}$ be a point in the domains of both $r_{1} a n d r_{2}$ . Prove that
localid="1649617482201" $lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 脳 r_{2} (t)) = r_{1} (t_{0}) 脳 r_{2} (t_{0})$

Short Answer

Expert verified

Ans: It is proved that $lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 脳 r_{2} (t)) = r_{1} (t_{0}) 脳 r_{2} (t_{0})$

Step by step solution

Step 1. Given information.

given,

$lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 脳 r_{2} (t)) = r_{1} (t_{0}) 脳 r_{2} (t_{0})$

Step 2. Solution

Consider two continuous vector functions with three components. Let

$r_{1} (t) = <x_{1} (t), y_{1} (t), z_{1} (t)> and r_{2} (t) = <x_{2} (t), y_{2} (t), z_{2} (t)>$

The objective is to prove that

$lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 脳 r_{2} (t)) = r_{1} (t_{0}) 脳 r_{2} (t_{0})$ where $t_{0}$ belongs to the domains of $r_{1} a n d r_{2}$ .

Since role="math" localid="1649617811675" $r_{1} (t) a n d r_{2} (t)$ are continuous, their components $x_{1} (t), y_{1} (t), z_{1} (t), x_{2} (t), y_{2} (t), z_{2} (t)$ are continuous. Thus product and sum of the continuous functions are also continuous.

Therefore, $r_{1} (t) 脳 r_{2} (t)$ is continuous.

Step 3. Consider

$\begin{aligned} lim_{t 鈫� t_{0}} 鈥� <r_{1} (t) 脳 r_{2} (t)> \\ = lim_{t 鈫� t_{0}} 鈥� <y_{1} (t) z_{2} (t) 鈭� y_{2} (t) z_{1} (t), x_{2} (t) z_{1} (t) 鈭� x_{1} (t) z_{2} (t), x_{1} (t) y_{2} (t) 鈭� x_{2} (t) y_{1} (t)> \\ = lim_{t 鈫� t_{0}} 鈥� (y_{1} (t) z_{2} (t) 鈭� y_{2} (t) z_{1} (t) i) + lim_{t 鈫� t_{0}} 鈥� (x_{2} (t) z_{1} (t) 鈭� x_{1} (t) z_{2} (t)) j \\ + lim_{t 鈫� r_{0}} 鈥� (x_{1} (t) y_{2} (t) 鈭� x_{2} (t) y_{1} (t)) k \end{aligned}$

From the definition of a continuous function.

$\begin{aligned} = (y_{1} (t_{0}) z_{2} (t_{0}) 鈭� y_{2} (t_{0}) z_{1} (t_{0})) i + (x_{2} (t_{0}) z_{1} (t_{0}) z_{1} (t_{0}) 鈭� x_{1} (t_{0}) z_{2} (t_{0})) j \\ + lim_{t 鈫� t_{0}} 鈥� (x_{1} (t_{0}) y_{2} (t_{0}) 鈭� x_{2} (t_{0}) y_{1} (t_{0})) k \\ = <\begin{matrix} y_{1} (t_{0}) z_{2} (t_{0}) 鈭� y_{2} (t_{0}) z_{1} (t_{0}), x_{2} (t_{0}) z_{1} (t_{0}) 鈭� x_{1} (t_{0}) z_{2} (t_{0}) \\ x_{1} (t_{0}) y_{2} (t_{0}) 鈭� x_{2} (t_{0}) y_{1} (t_{0}) \end{matrix}> \\ = r_{1} (t_{0}) 脳 r_{2} (t_{0}) \end{aligned}$

Thus, $lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 脳 r_{2} (t)) = r_{1} (t_{0}) 脳 r_{2} (t_{0})$

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91影视

Let $r_{1} (t) a n d r_{2} (t)$ be continuous vector functions with three components, and let $t_{0}$ be a point in the domains of both $r_{1} a n d r_{2}$ . Prove that
localid="1649617482201" $lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 脳 r_{2} (t)) = r_{1} (t_{0}) 脳 r_{2} (t_{0})$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solution

Step 3. Consider

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Let r1(t)andr2(t)be continuous vector functions with three components, and let t0be a point in the domains of both r1andr2. Prove that localid="1649617482201" limt鈫�t0鈥�r1(t)脳r2(t)=r1t0脳r2t0

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solution

Step 3. Consider

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Most popular questions from this chapter

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Logic and Functions

Mechanics Maths

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Calculus

Study anywhere. Anytime. Across all devices.

Let $r_{1} (t) a n d r_{2} (t)$ be continuous vector functions with three components, and let $t_{0}$ be a point in the domains of both $r_{1} a n d r_{2}$ . Prove that
localid="1649617482201" $lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 脳 r_{2} (t)) = r_{1} (t_{0}) 脳 r_{2} (t_{0})$