Theorems

Mean Value Theorem

Let

• $\delta > 0$
• $D = \big\{ (x,y) \in \mathbb{R}^2 : (x-a)^2 + (y-b)^2 \lt \delta \big\}$
• $f : D \rightarrow \mathbb{R}$
• $(a,b) \in D$
• $f_x$ and $f_y$ exists

Then $\exists \theta, \alpha \in (0,1)$ such that:

Schwarz’s Theorem

Also known as Clairaut’s theorem on equality of mixed partials. Let $D \subset \mathbb{R}^2$ be open. Let $f: D \rightarrow \mathbb{R}$. If $f_{xy}$ and $f_{yx}$ are continous on $D$ then $f_{xy}=f_{yx}$.