Euler's Theorem
An Euler theorem is a theorem that relates the derivatives of functions of several variables. It states that if a function is differentiable and the partial derivatives of the function with respect to each variable are continuous, then the function's derivative is equal to the sum of the partial derivatives of the function with respect to each variable.
Formally:
If a function f(x, y, z) has continuous partial derivatives, then:
∂x∂[f(x,y,z)]=∂y∂[f(x,y,z)]=∂z∂[f(x,y,z)]
Examples:
- The function f(x, y, z) = x^2 + y^2 - z^3 is differentiable. The partial derivatives of the function are:
∂x∂[f(x,y,z)]=2x
∂y∂[f(x,y,z)]=2y
∂z∂[f(x,y,z)]=−3z2
From these partial derivatives, we can see that the function's derivative is equal to the sum of the partial derivatives:
∂x∂[f(x,y,z)]=∂y∂[f(x,y,z)]=∂z∂[f(x,y,z)]
Therefore, the Euler theorem holds for this function.
- The function f(x, y, z) = x^2 + y^3 - z^4 is also differentiable. Its partial derivatives are:
∂x∂[f(x,y,z)]=2x
∂y∂[f(x,y,z)]=3y2
∂z∂[f(x,y,z)]=−4z3
Applying the Euler theorem, we get:
∂x∂[f(x,y,z)]=∂y∂[f(x,y,z)]=∂z∂[f(x,y,z)]
which also holds for this function