Mean Value Theorem
Let
- δ>0
- D={(x,y)∈R2:(x−a)2+(y−b)2<δ}
- f:D→R
- (a,b)∈D
- fx and fy exists
Then ∃θ,α∈(0,1) such that:
Δf=f(Px,Py)−f(a,b)=Δxfx(a+θΔx,b)+Δyfy(a+Δx,b+αΔy)
Schwarz’s theorem
Aka. Clairaut’s theorem on equality of mixed partials.
Let D⊂R2 be open. Let f:D→R. If
fxy and fyx are continous on D then fxy=fyx.