Question

If `u=x^2+y^2+z^2` where `x=e^t, y=e^tsint,z=e^tcost`

Prove that `(du)/(dt)=4e^(2t)`

Answer 1

`(du)/(dt)=(delu)/(delx).(dx)/(dt)+(delu)/(dely).(dy)/(dt)+(delu)/(delz).(dz)/(dt)`

`(delu)/(delx)=2x,(delu)/(dely)=2y,(delu)/(delz)=2z`

`(dx)/(dt)=e^t, (dy)/(dt)=e^t(sint+cost) ,(dz)/(dt)=e^t(-sint+cost)`

`(du)/(dt)=(delu)/(delx).(dx)/(dt)+(delu)/(dely).(dy)/(dt)+(delu)/(delz).(dz)/(dt)`

`=2x(e^t)+2y(e^t(sint+cost))+2z(e^t(-sint+cost))`

= 2x(x)+2y(y+z)+2z(z-y)

`=2x^2+2y^2+2z^2+2xy-2xy`

`=2x^2+2y^2+2z^2`

`=2(x^2+y^2+z^2)`

=2u ………………………………………(1)

`u=x^2+y^2+z^2=(e^t)^2+(e^tsint)^2+(e^tcost)^2`

`=e^(2t)+e^(2t)(sin^2t+cos^2t)`

`=e^(2t)+e^(2t)=2e^(2t)`

Substituting value of u in equation (1)

`(du)/(dt)=2u=2(2e^(2t))=4e^(2t)`

Hence proved

`(Du)/(dt)=4e.`