Intereting Posts

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Why do we think of a vector as being the same as a differential operator?
Quadratic variation of Brownian motion and almost-sure convergence
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Definition of $ 1 + \frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{\ddots}}}}$
Geometrical Proof of a Rotation
Transforming from one spherical coordinate system to another
Extending partial sums of the Taylor series of $e^x$ to a smooth function on $\mathbb{R}^2$?
What is the rank of COCHIN
Spectrum of Laplace operator with potential acting on $L^2(\mathbb R)$ is discrete
Showing that left invariant vector fields commute with right invariant vector fields
Does solvability of Lie algebra have useful application in study of PDEs?
How to prove that the rank of a matrix is a lower semi-continuous function?
Difference between proof of negation and proof by contradiction
Is there a series where the terms tend to zero faster than the harmonic series but it still diverges?

Suppose $f(\textbf x)=f(x_1,x_2) $ has mixed partial derivatives $f”_{12}=f”_{21}=0$, so can I say: there exist $f_1(x_1)$ and $f_2(x_2)$ such that $\min_{\textbf x} f(\textbf x)\equiv \min_{x_1}f_1(x_1)+ \min_{x_2}f_2(x_2)$? Or even further, as follows:

$$f(\textbf x)\equiv f_1(x_1)+ f_2(x_2)$$

A positive simple case is $f(x_1,x_2)=x_1^2+x_2^3$. I can not think of any opposite cases, but I am not so sure about it and may need a proof.

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For a mixed derivative $f_{xy} = 0$, integrating with respect to $y$ gives:

$$

f_x(x,y) = \int f_{xy} \,dy + h(x).

$$

Integrating with respect to $x$:

$$

f(x,y) = \iint f_{xy} \,dydx + \int h(x)dx + g(y).

$$

Similar result yields if we start from $f_{yx}$, now this implies

$$

f(x,y) = f_1(x) + f_2(y),

$$

and there goes your conclusion in the question.

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