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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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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