Let f be a function of two variables x and y defined on an open region U -(x, y) and dx af (x, y) exist on U. If they are totally differen- ду such that both tiable at a point (a, b) e U, then show that - (а, b) дхду a²f (a,b). дудх This is due to Young (1909) and called the fundamental theorem of differen- tials. Note that no continuity for the partial derivatives are assumed. It is easily verified that xy(x² – y²) x² + y? if (x, y) # (0,0), f(x, y) = if (x, у) %3D (0,0), satisfies a²f -(0,0) = 1. дудх -(0,0) = -1 and дхду Check that this function does not satisfy the conditions stated in the problem.
Let f be a function of two variables x and y defined on an open region U -(x, y) and dx af (x, y) exist on U. If they are totally differen- ду such that both tiable at a point (a, b) e U, then show that - (а, b) дхду a²f (a,b). дудх This is due to Young (1909) and called the fundamental theorem of differen- tials. Note that no continuity for the partial derivatives are assumed. It is easily verified that xy(x² – y²) x² + y? if (x, y) # (0,0), f(x, y) = if (x, у) %3D (0,0), satisfies a²f -(0,0) = 1. дудх -(0,0) = -1 and дхду Check that this function does not satisfy the conditions stated in the problem.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 78E
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Transcribed Image Text:Let f be a function of two variables x and y defined on an open region U
ôf
OI (x,y) and
dx
tiable at a point (a,b) e U, then show that
such that both
-(x, y) exist on U. If they are totally differen-
ду
-(a, b) =
-(а,).
дхду
дудх
This is due to Young (1909) and called the fundamental theorem of differen-
tials. Note that no continuity for the partial derivatives are assumed.
It is easily verified that
xy(x² – y²)
x² + y2
if (x, y) # (0,0),
f(x, y) =
if (x, y) = (0,0),
satisfies
a²f
²f
-(0,0) = 1.
дудх
-(0,0) = –1 and
дхду
Check that this function does not satisfy the conditions stated in the problem.
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