Originally Posted by

**bigdoggy** I have a problem relating to the extremal function for a surface of revolution.

The Euler-Lagrange equation is $\displaystyle {d \over dx}({\partial F \over \partial y'})- {\partial F \over \partial y}=0.$

Surface area of revolution =$\displaystyle \int {2\pi y \sqrt{1+(y')^2}}dx$

So $\displaystyle F(x,y,y') = {2\pi y \sqrt{1+(y')^2}}dx$ [eqn. (1)] is a function of y and y' only and so the Euler-Lagrange equation is $\displaystyle F-y'{\partial f \over \partial y'}=0$ because x is absent from F. Mr F says: This is wrong. It should be $\displaystyle {\color{red} F-y'{\partial {\color{blue}F} \over \partial y'}= {\color{blue}C}}$ where C is a constant. This is a first integral of the Euler-Lagrange equation.

But in a similar problem $\displaystyle F(x,y,y')= (y')^2+yy'+y^2$ [eqn. (2)] and the E-L equation is stated as $\displaystyle {d \over dx}({\partial F \over \partial y'})- {\partial F \over \partial y}=0$ because x is not absent from F....

why is eqn. (1) independent of x but eqn. (2) isn't !?