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Suppose S is a subset of R 2 , and that f is a continuous real-valued function on S . If both partial derivatives of f exist at each point of the interior S 0 of S , and both are continuous on S 0 , then f is said to belong to C 1 ( S ) . If all k th order mixed partial derivatives exist at each point of S 0 , and all of them are continuous on S 0 , then f is said to belong to C k ( S ) . Finally, if all mixed partial derivatives, of arbitrary orders, exist and are continuous on S 0 , then f is said to belong to C ( S ) .

  1. Suppose f is a real-valued function of two real variables and that it is differentiable, as a function of two real variables, at the point ( a , b ) . Show that the numbers L 1 and L 2 in the definition are exactly the partial derivatives of f at ( a , b ) . That is,
    L 1 = t i a l f t i a l x ( a , b ) = lim h 0 f ( a + h , b ) - f ( a , b ) h
    and
    L 2 = t i a l f t i a l y ( a , b ) = lim h 0 f ( a , b + h ) - f ( a , b ) h .
  2. Define f on R 2 as follows: f ( 0 , 0 ) = 0 , and if ( x , y ) ( 0 , 0 ) , then f ( x , y ) = x y / ( x 2 + y 2 ) . Show that both partial derivatives of f at ( 0 , 0 ) exist and are 0. Show also that f is not , as a function of two real variables, differentiable at ( 0 , 0 ) . HINT: Let h and k run through the numbers 1 / n .
  3. What do parts (a) and (b) tell about the relationship between a function of two real variables being differentiable at a point ( a , b ) and its having both partial derivatives exist at ( a , b ) ?
  4. Suppose f = u + i v is a complex-valued function of a complex variable, and assume that f is differentiable, as a function of a complex variable, at a point c = a + b i ( a , b ) . Prove that the real and imaginary parts u and v of f are differentiable, as functions of two real variables. Relate the five quantities
    t i a l u t i a l x ( a , b ) , t i a l u t i a l y ( a , b ) , t i a l v t i a l x ( a , b ) , t i a l v t i a l y ( a , b ) , and f ' ( c ) .

Perhaps the most interesting theorem about partial derivatives is the “mixed partials are equal” theorem. That is, f x y = f y x . The point is that this is not always the case. An extra hypothesis is necessary.

Theorem on mixed partials

Let f : S R be such that both second order partials derivatives f x y and f y x exist at a point ( a , b ) of the interior of S , and assume in addition that one of these second order partials exists at every point in a disk B r ( a , b ) around ( a , b ) and that it is continuous at the point ( a , b ) . Then f x y ( a , b ) = f y x ( a , b ) .

Suppose that it is f y x that is continuous at ( a , b ) . Let ϵ > 0 be given, and let δ 1 > 0 be such that if | ( c , d ) - ( a , b ) | < δ 1 then | f y x ( c , d ) - f y x ( a , b ) | < ϵ . Next, choose a δ 2 such that if 0 < | k | < δ 2 , then

| f x y ( a , b ) - f x ( a , b + k ) - f x ( a , b ) k | < ϵ ,

and fix such a k . We may also assume that | k | < δ 1 / 2 . Finally, choose a δ 3 > 0 such that if 0 < | h | < δ 3 , then

| f x ( a , b + k ) - f ( a + h , b + k ) - f ( a , b + k ) h | < | k | ϵ ,

and

| f x ( a , b ) - f ( a + h , b ) - f ( a , b ) h | < | k | ϵ ,

and fix such an h . Again, we may also assume that | h | < δ 1 / 2 .

In the following calculation we will use the Mean Value Theorem twice.

0 | f x y ( a , b ) - f y x ( a , b ) | | f x y ( a , b ) - f x ( a , b + k ) - f x ( a , b ) k | + | f x ( a , b + k ) - f x ( a , b ) k - f y x ( a , b ) | ϵ + | f x ( a , b + k ) - f ( a + h , b + k ) - f ( a , b + k ) h k | + | f ( a + h , b ) - f ( a , b ) h - f x ( a , b ) k | + | f ( a + h , b + k ) - f ( a , b + k ) + ( f ( a + h , b ) - f ( a , b ) ) h k - f y x ( a , b ) | < 3 ϵ + | f ( a + h , b + k ) - f ( a , b + k ) + ( f ( a + h , b ) - f ( a , b ) ) h k - f y x ( a , b ) | = 3 ϵ + | f y ( a + h , b ' ) - f y ( a , b ' ) h - f y x ( a , b ) | = 3 ϵ + | f y x ( a ' , b ' ) - f y x ( a , b ) | < 4 ϵ ,

because b ' is between b and b + k , and a ' is between a and a + h , so that | ( a ' , b ' ) - ( a , b ) | < δ 1 / 2 < δ 1 . Hence, | f x y ( a , b ) - f y x ( a , b ) < 4 ϵ , for an arbitrary ϵ , and so the theorem is proved.

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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