User:Egm6321.f11.team2.Xia/RP2.9

=Problem 2.9- commutative partial derivations=

Given
Review calculus, and differentiation $$\displaystyle \frac{\partial^{2}\phi(x,y)}{\partial x\partial y}=\frac{\partial^{2}\phi(x,y)}{\partial y\partial x}$$

Find
Find the minimum degree of differentiability of the function $$\displaystyle \phi(x,y) $$ such that the above equation is satisfied, State the full theorem and provide a proof.

Clairaut's theorem
According to the Schwarz's theorem,, if


 * $$\displaystyle f \colon \mathbb{R}^n \to \mathbb{R}$$

has continuous second partial derivatives at any given point in $$ \displaystyle \mathbb{R}^n $$, then for $$ \displaystyle 1 \leq i,j \leq n,$$


 * $$ \displaystyle \frac{\partial^2 f}{\partial x_i\, \partial x_j}(a_1, \dots, a_n) = \frac{\partial^2 f}{\partial x_j\, \partial x_i}(a_1, \dots, a_n).\,\!$$

And we then know that the minimum degree of differentiability of the function $$\displaystyle \phi(x,y) $$ must be three when it satisfies eqn (2.09.1)

Proof of Clairaut's theorem
The result was first discovered by L.Euler around 1734 in connection with problems in hydrodynamics. In order to prove this theorem, we shall first introduce the Mean Value theorem as follows: Let $$ \displaystyle f:[a,b]\rightarrow\mathbb{R} $$ be a continuous function on the closed interval $$ \displaystyle [a,b]$$, and derivative on the open interval $$ \displaystyle (a,b)$$, where $$ \displaystyle a(2.15.1)

Suppose Q is a closed rectangle with sides parallel to the coordinate axes, put
 * $$\triangle(f,Q)=f(a+h,b+k)-f(a+h,b)-f(a,b+k)+f(a,b)$$

(2.15.2)


 * $$\ u(t)=f(t,b+k)-f(t,b)$$


 * $$\triangle(f,Q)=u(a+h)-u(a)$$


 * $$\ =hu'(x)$$


 * $$ =h(\frac{\partial f}{\partial x}(x,b+k)-\frac{\partial f}{\partial x}(x,b)$$


 * $$=hk\frac{\partial^{2}}{\partial y\partial x}f(x,y) $$

(2.15.3) So, we put


 * $$A=\frac{\partial^{2}}{\partial y\partial x}f(a,b)$$,

Chose $$ \displaystyle \epsilon>0 $$. if $$ \displaystyle h $$ and $$ \displaystyle k $$ are sufficiently small, we have


 * $$|\frac{\triangle(f,Q)}{hk}-A|<\epsilon$$

(2.15.4) Fix $$ \displaystyle h $$, and let $$ \displaystyle k\rightarrow0 $$, then


 * $$|\frac{(\frac{\partial f}{\partial y}(a+h,b)-\frac{\partial f}{\partial y}(a,b)}{h}-A|\leq\epsilon$$

(2.15.5) Since $$ \displaystyle \epsilon$$ was is arbitrary, and (2.15.5) holds for all sufficiently small $$ \displaystyle h\neq0$$


 * $$\frac{\partial^{2}}{\partial y\partial x}f(a,b)=\frac{\partial^{2}}{\partial x\partial y}f(a,b)$$

(2.15.6) Since $$ \displaystyle (a,b)$$ is the arbitrary point in $$ \displaystyle \mathbb{R}^2$$, so
 * $$\frac{\partial^{2}}{\partial y\partial x}f(x,y)=\frac{\partial^{2}}{\partial x\partial y}f(x,y)$$