Chapter 11: Q 15. (page 871)

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

Short Answer

Expert verified

$\frac{d (r (f (τ)))}{d τ} = r' (f (τ)) . f' (τ)$

Step by step solution

Step 1. Given information is:

$r (t) = <x (t), y (t), z (t)> be a differentiable vector function with three components . t = f (τ) be a differentiable real - valued function of τ, and f (τ) is in the domain of r for every value of τ on some interval I .$

Step 2. Calculating drfτdτ

$\frac{d (r (f (τ)))}{d τ} = \frac{d (r (t))}{d τ} (since f (τ) = t) \frac{d (r (f (τ)))}{d τ} = \frac{d <x (t), y (t), z (t)>}{d τ} (substitute for r (t)) \frac{d (r (f (τ)))}{d τ} = <\frac{d (x (t))}{d τ}, \frac{d (y (t))}{d τ}, \frac{d (z (t))}{d τ}> \frac{d (r (f (τ)))}{d τ} = <\frac{d x}{d t} . \frac{d t}{d τ}, \frac{d y}{d t} . \frac{d t}{d τ}, \frac{d z}{d t} . \frac{d t}{d τ}> \frac{d (r (f (τ)))}{d τ} = <\frac{d x}{d t}, \frac{d y}{d t}, \frac{d z}{d t}> \frac{d t}{d τ} \frac{d (r (f (τ)))}{d τ} = \frac{d r}{d t} . \frac{d t}{d τ} \frac{d (r (f (τ)))}{d τ} = r' (t) . f' (τ) \frac{d (r (f (τ)))}{d τ} = r' (f (τ)) . f' (τ)$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Calculating drfτdτ

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Probability and Statistics

Pure Maths

Logic and Functions

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Lett=f(τ)beadifferentiablereal-valuedfunctionofτ,andletr(t)beadifferentiablevectorfunctionwitheithertwoorthreecomponentssuchthatf(τ)isinthedomainofrforeveryvalueofτonsomeintervalI.Finddrfτdτ.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Calculating drfτdτ

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Probability and Statistics

Pure Maths

Logic and Functions

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

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