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Calculus of Variation 
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#1
Sep1513, 11:51 AM

P: 28

In calculus of variation, we use Euler's equation to minimize the integral.
e.g. ∫f{y,y' ;x}dx why we treat y and y' independent ? 


#2
Sep1913, 03:54 AM

Sci Advisor
Thanks
PF Gold
P: 1,911

Because there is no algebraic relation between a function and its derivative.
This is why you need boundary conditions to solve differential equations. 


#3
Sep2113, 11:28 PM

Thanks
P: 5,869

The real reason is that we use the partial derivatives to obtain an expression for the difference ## F(z + \Delta z, y + \Delta y, x)  F(z, y, x) ##, which is approximately ## F_z \Delta z + F_y \Delta y ## when ##\Delta z## and ##\Delta y## are sufficiently small. This expression is true generally, and is true when ## z ## represents the derivative of ## y ##  all it takes is that the variations of both must be small enough. If ## y = f(x) ##, its variation is ## \delta y = \epsilon g(x) ##, and ## \delta y' = \epsilon g'(x)##. If ## \epsilon ## is small enough, then using the result above, ## F((y + \delta y)', (y + \delta y), x)  F(y', y, x)) \approx \epsilon F_{y'}g'(x) + \epsilon F_y g(x) ##, where ##F_{y'}## is just a fancy symbol equivalent to ##F_z##, meaning partial differentiation with respect to the first argument. Then we use integration by parts and convert that to ## \epsilon ((F_{y'})' + F_y) g(x)##. Observe that we do use the relationship between ## y ## and ## y' ## in the final step. 


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