Page 74 - Introduction to Continuum Mechanics

PartD Polar Coordinates 59

But from Calculus,

Since Eqs. (2D1.4) and (2D1.5) must yield the same result for all increments dr, d6, we have

Thus,

(ii) Components of Vv
Let

By definition of Vv, we have

Let TsVv, then

Now,

Therefore,

But from Eq. (2D1.7)

and from calculus, we have,

From the above three equations and Eqs. (2D1.2), we have

In order that Eqs. (2D1.8) and (2D1.9) agree for all increments dr,dO, we have