- #1

Cauchy1789

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## Homework Statement

Here is difficult one guys,

Lets imagine that an object movement along a curve is described by the parameterized function called

[tex]\omega: I \rightarrow \mathbb{R}^3[/tex] which moves on the interval [tex][a,b]\subset I[/tex]. and this depended on motor which supplies the constant effect of |v| = 1.

With this in mind show that

[tex](\omega(b) - \omega(a)) \cdot v = \int_{a}^{b} \omega(t)' \cdot v dt \leq \int_{a}^{b} |\omega(t)'| dt[/tex]

## Homework Equations

## The Attempt at a Solution

From what I learned in Calculus is relatively easy to show that according to the fundametal theorem of Calculus which states that [tex]\int_{a}^{b} f(x) dx = F(b) - F(a)[/tex] where F is the anti-derivate of f.

such that [tex](\omega(b) - \omega(a)) = \int_{a}^{b} \omega'(t) dt[/tex] where [tex]\omega(t)[/tex] is the anti-derivative, and since the movement depends of the constant, then both on side are the same aren't they?

If I expand the inequaliy then [tex]v \cdot(\omega(b) - \omega(a)) \leq |\omega(b) - \omega(a)| [/tex] which is only true if [tex]|v| \leq 1[/tex]

Haven't I covered what needs to be covered in this?

Sincerely

Cauchy

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