Chapter 11: Q. 15 (page 879)

Under what conditions does a twice-differentiable vector valued function $r (t)$ not have a binormal vector at a point in the domain of $r (t)$ ?

Short Answer

Expert verified

The function $r (t)$ doesn't exists when $r' (t) = 0$ .

Step by step solution

Step 1. Given information.

Consider the given question,

A twice-differentiable vector-valued function is $r (t)$ .

Step 2. Definition of binomial vector.

Assume $t_{0}$ to be any point in the domain of $r (t)$ .

Assume $r (t)$ to be a differential vector function on the interval $I \subseteq R$ such that $T' (t_{0}) \neq 0$ , where, $t_{0} \in I$ .

The binomial vector B at $r (t_{0})$ is defined as,

$B (t_{0}) = T (t_{0}) \times N (t_{0})$

But the unit vector $(T (t_{0}))$ exist only when $r' (t_{0}) \neq 0$ and the principal unit vector normal , $N (t_{0}) = \frac{T' (t_{0})}{||T' (t_{0})||}$ exists only when $T' (t_{0}) \neq 0$ .

So the basic condition for the existence of $B (t_{0})$ is $r' (t_{0}) = 0$ .

Thus, the binomial vector doesn't exists at any point $t_{0}$ at which $r' (t_{0}) = 0$ .

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.

Under what conditions does a twice-differentiable vector valued function $r (t)$ not have a binormal vector at a point in the domain of $r (t)$ ?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Definition of binomial vector.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Pure Maths

Applied Mathematics

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Under what conditions does a twice-differentiable vector valued function r(t) not have a binormal vector at a point in the domain of r(t)?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Definition of binomial vector.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Pure Maths

Applied Mathematics

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Under what conditions does a twice-differentiable vector valued function $r (t)$ not have a binormal vector at a point in the domain of $r (t)$ ?