Answer to Question #105383 in Calculus for Ajay

Question

(1) Calculate the Jacobian of ∂(x,y,z)/∂(u,v,w) for x=√w ucosv,y=√w usinv and z=w–1 at the point (5,π/2 ,3)
(2) Find the minimum value of the function f(x,y)=x^2+2y^2 on circle x2+y^2=1
(3) If u=Sin^(–1) (x^2+u^2)^(1/5) then show that x∂u/∂x +y∂u/∂y=(2/5)tanu
(4) Let the function f be defined by
f(x,y)=3x^2y^4/x^4+y^8 ,(x,y)≠(0,0)
=0 (x,y)=(0,0)
Show that f has directional derivatives in
all direction at (0,0)
(5) Verify that f(x,y) =exp(–k^2 t)Sin(Kx)
satisfies the heat equation ∂f/∂t=∂^2f/∂x^2 where k is a constant

Answer 1

1.$x=\sqrt{w}ucos v ; y= \sqrt{w}usin v ; z=w-1$

$\frac{∂(x,y,z)}{∂(u,v,w)}=\begin{bmatrix} \sqrt{w}cos v & -u\sqrt{w}cosv & \frac{ucosv}{2\sqrt{w}}\\ \sqrt{w} sin v &u\sqrt{w}sinv & \frac{usinv}{2\sqrt{w}}\\ 0&0 & 1 \end{bmatrix}$

$\frac{∂(x,y,z)}{∂(u,v,w)}=\sqrt{w}cosv(u\sqrt{w}cosv)+u\sqrt{w}sinv(\sqrt{w}sinv)$

$\frac{∂(x,y,z)}{∂(u,v,w)}=uwcos^2v+uwsin^2v=uw=15$

2.$f(x,y)=x^2+2y^2$

We know on the circle $x^2+y^2=1$

So, $f(x,y)=1+y^2$

Minimum value of f(x,y) is equal to 1 when coordinates are (1,0).

3.$u=sin^{-1}(x^2+y^2)^{1/5}$

$∂u/∂x = (2/5)(x^2+y^2)^{-4/5}x^2 / \sqrt {1-(x^2+y^2)^{2/5}}$

$∂u/∂y = (2/5)(x^2+y^2)^{-4/5}y^2 / \sqrt {1-(x^2+y^2)^{2/5}}$

$x∂u/∂x +y∂u/∂y=(2/5)(x^2+y^2)^{1/5}/\sqrt{1-(x^2+y^2)^{2/5}}$

$x∂u/∂x +y∂u/∂y=(2/5)tan u$

4.Fractional Derivative is defined as $D_uf(x,y)=lim_{t\to0} \frac{f(x+tu_1,y+tu_2)-f(x,y)}{t}$

$D_uf(0,0)=lim_{t\to0} \frac{f(tu_1,tu_2)-f(0,0)}{t}$

$D_uf(0,0)=lim_{t\to0} \frac{f(tu_1,tu_2)-0}{t}$

$D_uf(0,0)=lim_{t\to0} \frac{3(tu_1)^2(tu_2)^4/(tu_1)^4+(tu_2)^8)-0}{t}$

$D_uf(0,0)=\frac{0-0}{u_1^4}=0$ ,Thus, Fractional Derivative exists in all direction at (0,0).

5.$f(x,t) =e^{–k^2 t}Sin(kx)$

$∂f/∂t=e^{–k^2 t}Sin(kx)(-k^2)$

$∂^2f/∂x^2=e^{–k^2 t}Sin(kx)(-k^2)$

$∂f/∂t=∂^2f/∂x^2$ (Proved)