Chapter 11: Q. 67 (page 873)

Let $t = f (τ)$ be a differentiable real-valued function of $τ$ , and let $r (t)$ be a differentiable vector function with three components such that $f (τ)$ is in the domain of $r$ for every value of $τ$ on some interval I. Prove that $\frac{d r}{d τ} = \frac{d r}{d t} \frac{d t}{d τ}$ . (This is Theorem 11.8.)

Short Answer

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Ans:

$\frac{d r}{d τ} = \frac{d}{d τ} (r (t)) = \frac{d}{d τ} ⟨ x (t), y (t), z (t) ⟩ = \frac{d r}{d t} \cdot \frac{d t}{d τ}$

Step by step solution

Step 1. Given information:

Part (a). $t = f (τ)$ be a differentiable real-valued function of $τ$ .

Part (b). $r (t)$ is differentiable vector function with three components such that $f (τ)$ is in the domain of $r$ for every value of $τ$ on some interval I.

Step 2. Proving:

Let $r (t) = ⟨ x (t), y (t), z (t) ⟩$

The objective is to prove that $\frac{d r}{d τ} = \frac{d r}{d t}, \frac{d t}{d τ}$

$\frac{d r}{d τ} = \frac{d}{d τ} (r (t)) = \frac{d}{d τ} ⟨ x (t), y (t), z (t) ⟩ = <\frac{d}{d τ} (x (t)), \frac{d}{d τ} (y (t)), \frac{d}{d τ} (z (t))> Derivative of a vector function.$

$= <\frac{d x}{d t}, \frac{d t}{d τ}, \frac{d y}{d t}, \frac{d t}{d τ}, \frac{d z}{d t}, \frac{d t}{d τ}> chain rule for scalar functions.$

$= <\frac{d x}{d t}, \frac{d y}{d t}, \frac{d z}{d t}> \frac{d t}{d τ}$ Multiplication of a vector

function with a scalar function.

$= \frac{d r}{d t} \cdot \frac{d t}{d τ}$

Derivative of a vector function.

Thus $\frac{d r}{d τ} = \frac{d r}{d t} \cdot \frac{d t}{d τ}$

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Let t=f(τ)be a differentiable real-valued function of τ, and let r(t)be a differentiable vector function with three components such that f(τ)is in the domain of rfor every value of τon some interval I. Prove that drdτ=drdtdtdτ. (This is Theorem 11.8.)

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Proving:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Logic and Functions

Probability and Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.