MHB Solving Polar Coordinates in System of Equations

[onie mti]

given that

x^'=f(x,y)
y'=g(x,y)
iff the vector function (r, θ) is a sloution of the system
r^'=f(rcosθ,rsinθ)cosθ +g(rcosθ,rsinθ)sinθ

am trying to show that this is true but i just don't get where the sinθ and cosθ come from, how do i get to that

 

[Prove It]

Are x and y functions of some other variable, say t?

 

[chisigma]

given that

x^'=f(x,y)
y'=g(x,y)
iff the vector function (r, θ) is a sloution of the system
r^'=f(rcosθ,rsinθ)cosθ +g(rcosθ,rsinθ)sinθ

am trying to show that this is true but i just don't get where the sinθ and cosθ come from, how do i get to that

Operating in polar coordinates You have $\displaystyle x= r\ \cos \theta$ and $\displaystyle y= r\ \sin \theta$, so that is...

$\displaystyle r^{\ '} = \frac{d}{d t} \sqrt{x^{2} + y^{2}} = \frac{1}{2}\ \frac{2\ x\ x^{\ '} + 2\ y\ y^{\ '}}{\sqrt{x^{2} + y^{2}}} = f(r\ \cos \theta, r\ \sin \theta)\ \cos \theta + g(r\ \cos \theta, r\ \sin \theta)\ \sin \theta $

Kind regards

$\chi$ $\sigma$