How Do Polar Coordinates Relate to Integral Equations in Homework Problems?

[Denver Dang]

Homework Statement

Let the curve C be paramatized into polar coordinates given by:

$$\[r\left( t \right)=\left( r\left( t \right)\cos \theta \left( t \right),\,\,\,\,\,r\left( t \right)\sin \theta \left( t \right) \right),\,\,\,\,\,a\le t\le b\]$$
where r and theta is continuous derivatives and r(t) > 0.

Show that:

$$\[F\left( r\left( t \right) \right)\cdot r'\left( t \right)=\theta '\left( t \right)\,\,\,\,\,and\,\,\,\,\,\int_{C}{F\cdot dr=\theta \left( b \right)}-\theta \left( a \right)\]$$

Homework Equations

F is given by:

$$\[F\left( x,y \right)=\frac{-y\,i+x\,j}{{{x}^{2}}+{{y}^{2}}}\]$$

The Attempt at a Solution

No idea... I get stupid when it changes into polar coords.
So anyone with a little hint maybe ?


Regards

 

[clamtrox]

Start by calculating what F is in polar coordinates: x = r cos theta, y = r sin theta.

 

[Denver Dang]

Then I get:

$$F\left( r,\theta \right)=\frac{\cos \left( \theta \right)j-\sin \left( \theta \right)i}{r}$$

 

[Dick]

Then I get:

$$F\left( r,\theta \right)=\frac{\cos \left( \theta \right)j-\sin \left( \theta \right)i}{r}$$

So far, so good. Now what's R'(t) where R(t)=r(t)cos(theta(t))i+r(t)sin(theta(t))j? Try not to confuse the scalar r(t) with the vector R(t). It's R'(t) you want to dot with F.

 

[Denver Dang]

Well, that's what confuses me :)
How do I differentiate a function with functions in it ? And with respect to what ? r or theta ? If I do it with respect to both, then I get to different R'(t), don't I ?

 

[Dick]

Differentiate with respect to t. You'll need to use the product rule.

 

[Denver Dang]

Hmmm...
But there is no t in R(t) ? :S

 

[Dick]

Hmmm...
But there is no t in R(t) ? :S

There are two functions r(t) and theta(t)!

 

[Denver Dang]

Ehhh, I feel pretty stupid right now :S
Do I know the functions r(t) and theta(t) ? :S

 

[Dick]

Ehhh, I feel pretty stupid right now :S
Do I know the functions r(t) and theta(t) ? :S

No, but you know dr(t)/dt=r'(t) and dtheta(t)/dt=theta'(t). Just write out R(t)' in terms of them.