Menu Close

if-w-f-x-y-and-x-r-cos-y-rsin-then-prove-that-w-rr-w-0-

Tinku Tara 0 Comments
Pin




Question Number 164121 by mkam last updated on 14/Jan/22
if w = f(x,y) and x = r cosθ , y = rsinθ then prove that w_(rr) + w_(θθ) = 0?
$${if}\:{w}\:=\:{f}\left({x},{y}\right)\:{and}\:{x}\:=\:{r}\:{cos}\theta\:,\:{y}\:=\:{rsin}\theta \\ $$$$ \\ $$$${then}\:{prove}\:{that}\:{w}_{{rr}} \:+\:{w}_{\theta\theta} \:=\:\mathrm{0}? \\ $$
Commented by mkam last updated on 14/Jan/22
????
$$???? \\ $$
Commented by mkam last updated on 15/Jan/22
???
$$??? \\ $$
Commented by mr W last updated on 15/Jan/22
you cant prove! it′s false!
$${you}\:{cant}\:{prove}! \\ $$$${it}'{s}\:{false}! \\ $$
Answered by mr W last updated on 15/Jan/22
w_r =f_x cos θ+f_y sin θ w_(rr) =f_(xx) cos^2 θ+f_(xy) cos θ sin θ+f_(yx) sin θ cos θ+f_(yy) sin^2 θ w_θ =−f_x r sin θ+f_y r cos θ w_(θθ) =−f_x r cos θ−r sin θ (−f_(xx) r sin θ+f_(xy) r cos θ)−f_y r sin θ+r cos θ (−f_(xy) r sin θ+f_(yy) r cos θ) w_(θθ) =−f_x r cos θ+f_(xx) r^2 sin^2 θ−f_(xy) r^2 sin θ cos θ−f_y r sin θ−f_(xy) r^2 sin θ cos θ+f_(yy) r^2 cos^2 θ w_(θθ) =−f_x r cos θ−f_y r sin θ+f_(xx) r^2 sin^2 θ+f_(yy) r^2 cos^2 θ−2f_(xy) r^2 sin θ cos θ w_(rr) +w_(θθ) ≠0
$${w}_{{r}} ={f}_{{x}} \mathrm{cos}\:\theta+{f}_{{y}} \mathrm{sin}\:\theta \\ $$$${w}_{{rr}} ={f}_{{xx}} \mathrm{cos}^{\mathrm{2}} \:\theta+{f}_{{xy}} \mathrm{cos}\:\theta\:\mathrm{sin}\:\theta+{f}_{{yx}} \mathrm{sin}\:\theta\:\mathrm{cos}\:\theta+{f}_{{yy}} \mathrm{sin}^{\mathrm{2}} \:\theta \\ $$$${w}_{\theta} =−{f}_{{x}} {r}\:\mathrm{sin}\:\theta+{f}_{{y}} {r}\:\mathrm{cos}\:\theta \\ $$$${w}_{\theta\theta} =−{f}_{{x}} {r}\:\mathrm{cos}\:\theta−{r}\:\mathrm{sin}\:\theta\:\left(−{f}_{{xx}} {r}\:\mathrm{sin}\:\theta+{f}_{{xy}} {r}\:\mathrm{cos}\:\theta\right)−{f}_{{y}} {r}\:\mathrm{sin}\:\theta+{r}\:\mathrm{cos}\:\theta\:\left(−{f}_{{xy}} {r}\:\mathrm{sin}\:\theta+{f}_{{yy}} {r}\:\mathrm{cos}\:\theta\right) \\ $$$${w}_{\theta\theta} =−{f}_{{x}} {r}\:\mathrm{cos}\:\theta+{f}_{{xx}} {r}^{\mathrm{2}} \:\mathrm{sin}^{\mathrm{2}} \:\theta−{f}_{{xy}} {r}^{\mathrm{2}} \:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta−{f}_{{y}} {r}\:\mathrm{sin}\:\theta−{f}_{{xy}} {r}^{\mathrm{2}} \:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta+{f}_{{yy}} {r}^{\mathrm{2}} \:\mathrm{cos}^{\mathrm{2}} \:\theta \\ $$$${w}_{\theta\theta} =−{f}_{{x}} {r}\:\mathrm{cos}\:\theta−{f}_{{y}} {r}\:\mathrm{sin}\:\theta+{f}_{{xx}} {r}^{\mathrm{2}} \:\mathrm{sin}^{\mathrm{2}} \:\theta+{f}_{{yy}} {r}^{\mathrm{2}} \:\mathrm{cos}^{\mathrm{2}} \:\theta−\mathrm{2}{f}_{{xy}} {r}^{\mathrm{2}} \:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta \\ $$$${w}_{{rr}} +{w}_{\theta\theta} \neq\mathrm{0} \\ $$

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *