Solving Euler Theorem Doubts with Partial Derivatives

In summary: I get\frac{\partial^2 z}{\partial x^2}= xf"(y/x)+ 2f'(y/x)(-y/x)+ g"(y/x) (1) Similarly,\frac{\partial z}{\partial y}= x f'(y/x)(1/x)+ g'(y/x)(1/x) Differentiating that,\frac{\partial^2 z}{\partial y\partial x}= f'(y/x)(1/x)+ xf"(y/x)(-y/x^2)(1/x)+ xf'(y/x)(-1/x^2)+ g'(y/x)(-y/x^2)(1/x)= xf"(y/x)(1/x)+
  • #1
Nina2905
1
0
firstly, all d's i use will mean partial derivative 'do' because i don't have the font installed. sorry :(

please help me with these.. u got to use euler theorem
1. if z= xf(y/x) + g(y/x), show that x2(d2z/dx2) + 2xy(d2z/dxdy) + y2(d2z/dy2) =0
2. if z= (xy)/(x-y), PT (d2z/dx2) + 2(d2z/dxdy) + (d2z/dy2) = 2/(x-y)

thanks...
 
Physics news on Phys.org
  • #2
First, it's not a matrer of having "fonts" installed, just use LaTex with [ tex ] and [ /tex ] (without the spaces) beginning and ending. To see LaTex commands, click on any formula on this board.

I'm not sure which "Euler Theorem" you mean (there are many). It looks to me like like you only need to differentiate.

If z= xf(y/x)+ g(y/x), then
[tex]\frac{\partial z}{\partial x}= f(y/x)+ x f'(y/x)(-y/x^2)+ g'(y/x)(-y/x^2)[/tex]
by the chain rule. Doing that again,
[tex]\frac{\partial^2 z}{\partial x^2}= [f'(y/x)(-y/x^2)]+ [f'(y/x)(-y/x^2)+ xf"(y/x)(-y/x^2)^2+ xf'(y/x)(2y/x^3)]+ g"(y/x)(-y/x^2)^2+ g'(y/x)(-2y/x^3)][/tex]
Of course, that can be simplified a lot.
 

FAQ: Solving Euler Theorem Doubts with Partial Derivatives

What is Euler's Theorem and how is it used in solving problems?

Euler's Theorem is a mathematical theorem that relates the partial derivatives of a multivariable function to its total derivative. It is used to simplify the process of solving problems involving functions with multiple variables by breaking it down into smaller steps.

What are partial derivatives and how do they relate to Euler's Theorem?

Partial derivatives are the derivatives of a function with respect to one variable while holding all other variables constant. They are used in Euler's Theorem to calculate the total derivative of a multivariable function by taking the sum of the partial derivatives multiplied by the corresponding variables.

Can Euler's Theorem be applied to any type of function?

Yes, Euler's Theorem can be applied to any function that has multiple variables. It is particularly useful in solving problems involving functions with two or more variables, such as optimization problems or finding the critical points of a function.

How do partial derivatives help in solving problems?

Partial derivatives help in solving problems by breaking down a complex function into smaller parts and allowing us to focus on one variable at a time. This makes it easier to calculate the total derivative and find the solution to the problem.

Are there any limitations to using Euler's Theorem for solving problems?

One limitation of Euler's Theorem is that it only applies to smooth functions, meaning that the function must have continuous partial derivatives. It also may not work for functions with highly nonlinear relationships between variables. In these cases, other methods may be needed to solve the problem.

Similar threads

Calculate this Integral around the Circular Path using Green's Theorem
Replies
3
Views
1K
Partial differentiaton question, how to distinguish between u and f (u )
Replies
3
Views
2K
Find f(x,y) given partial derivative and initial condition
Replies
6
Views
1K
Question about second-order partial derivatives
Replies
3
Views
4K
Doubt about the derivative of a Taylor series
2
Replies
42
Views
3K
Solving Geodesic Equations with Euler-Lagrange and Noether's Theorem
Replies
6
Views
1K
Multivariable calculus problem involving partial derivatives along a surface
Replies
9
Views
1K
Lagrange multipliers understanding
Replies
8
Views
1K
Calculating the partial derivative in polar coordinates
Replies
3
Views
2K
Derivation of Euler-Lagrange equation?
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top