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Sahithyan's S2
Sahithyan's S2 — Methods of Mathematics

Theorems

Mean Value Theorem

Let

  • δ>0\delta > 0
  • D={(x,y)R2:(xa)2+(yb)2<δ}D = \big\{ (x,y) \in \mathbb{R}^2 : (x-a)^2 + (y-b)^2 \lt \delta \big\}
  • f:DRf : D \rightarrow \mathbb{R}
  • (a,b)D(a,b) \in D
  • fxf_x and fyf_y exists

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

Δf=f(Px,Py)f(a,b)=Δxfx(a+θΔx,b)+Δyfy(a+Δx,b+αΔy)\Delta f = f(P_x, P_y) - f(a,b) = \Delta{x} f_x (a+\theta \Delta{x}, b) + \Delta{y} f_y (a+\Delta{x}, b+\alpha \Delta{y})

Schwarz’s theorem

Aka. Clairaut’s theorem on equality of mixed partials.

Let DR2D \subset \mathbb{R}^2 be open. Let f:DRf: D \rightarrow \mathbb{R}. If fxyf_{xy} and fyxf_{yx} are continous on DD then fxy=fyxf_{xy}=f_{yx}.