# 4.10 More on partial derivatives  (Page 2/3)

 Page 2 / 3

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 ${C}^{1}\left(S\right).$ 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 ${C}^{k}\left(S\right).$ Finally, if all mixed partial derivatives, of arbitrary orders, exist and are continuous on ${S}^{0},$ then $f$ is said to belong to ${C}^{\infty }\left(S\right).$

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 $\left(a,b\right).$ Show that the numbers ${L}_{1}$ and ${L}_{2}$ in the definition are exactly the partial derivatives of $f$ at $\left(a,b\right).$ That is,
${L}_{1}=\frac{tialf}{tialx}\left(a,b\right)=\underset{h\to 0}{lim}\frac{f\left(a+h,b\right)-f\left(a,b\right)}{h}$
and
${L}_{2}=\frac{tialf}{tialy}\left(a,b\right)=\underset{h\to 0}{lim}\frac{f\left(a,b+h\right)-f\left(a,b\right)}{h}.$
2. Define $f$ on ${R}^{2}$ as follows: $f\left(0,0\right)=0,$ and if $\left(x,y\right)\ne \left(0,0\right),$ then $f\left(x,y\right)=xy/\left({x}^{2}+{y}^{2}\right).$ Show that both partial derivatives of $f$ at $\left(0,0\right)$ exist and are 0. Show also that $f$ is not , as a function of two real variables, differentiable at $\left(0,0\right).$ 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 $\left(a,b\right)$ and its having both partial derivatives exist at $\left(a,b\right)?$
4. Suppose $f=u+iv$ 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 $c=a+bi\equiv \left(a,b\right).$ Prove that the real and imaginary parts $u$ and $v$ of $f$ are differentiable, as functions of two real variables. Relate the five quantities
$\frac{tialu}{tialx}\left(a,b\right),\frac{tialu}{tialy}\left(a,b\right),\frac{tialv}{tialx}\left(a,b\right),\frac{tialv}{tialy}\left(a,b\right),\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{f}^{\text{'}}\left(c\right).$

Perhaps the most interesting theorem about partial derivatives is the “mixed partials are equal” theorem. That is, ${f}_{xy}={f}_{yx}.$ The point is that this is not always the case. An extra hypothesis is necessary.

## Theorem on mixed partials

Let $f:S\to R$ be such that both second order partials derivatives ${f}_{xy}$ and ${f}_{yx}$ exist at a point $\left(a,b\right)$ of the interior of $S,$ and assume in addition that one of these second order partials exists at every point in a disk ${B}_{r}\left(a,b\right)$ around $\left(a,b\right)$ and that it is continuous at the point $\left(a,b\right).$ Then ${f}_{xy}\left(a,b\right)={f}_{yx}\left(a,b\right).$

Suppose that it is ${f}_{yx}$ that is continuous at $\left(a,b\right).$ Let $ϵ>0$ be given, and let ${\delta }_{1}>0$ be such that if $|\left(c,d\right)-\left(a,b\right)|<{\delta }_{1}$ then $|{f}_{yx}\left(c,d\right)-{f}_{yx}\left(a,b\right)|<ϵ.$ Next, choose a ${\delta }_{2}$ such that if $0<|k|<{\delta }_{2},$ then

$|{f}_{xy}\left(a,b\right)-\frac{{f}_{x}\left(a,b+k\right)-{f}_{x}\left(a,b\right)}{k}|<ϵ,$

and fix such a $k.$ We may also assume that $|k|<{\delta }_{1}/2.$ Finally, choose a ${\delta }_{3}>0$ such that if $0<|h|<{\delta }_{3},$ then

$|{f}_{x}\left(a,b+k\right)-\frac{f\left(a+h,b+k\right)-f\left(a,b+k\right)}{h}|<|k|ϵ,$

and

$|{f}_{x}\left(a,b\right)-\frac{f\left(a+h,b\right)-f\left(a,b\right)}{h}|<|k|ϵ,$

and fix such an $h.$ Again, we may also assume that $|h|<{\delta }_{1}/2.$

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

$\begin{array}{ccc}\hfill 0& \le & |{f}_{xy}\left(a,b\right)-{f}_{yx}\left(a,b\right)|\hfill \\ & \le & |{f}_{xy}\left(a,b\right)-\frac{{f}_{x}\left(a,b+k\right)-{f}_{x}\left(a,b\right)}{k}|\hfill \\ & & \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}+|\frac{{f}_{x}\left(a,b+k\right)-{f}_{x}\left(a,b\right)}{k}-{f}_{yx}\left(a,b\right)|\hfill \\ & \le & ϵ+|\frac{{f}_{x}\left(a,b+k\right)-\frac{f\left(a+h,b+k\right)-f\left(a,b+k\right)}{h}}{k}|\hfill \\ & & \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}+|\frac{\frac{f\left(a+h,b\right)-f\left(a,b\right)}{h}-{f}_{x}\left(a,b\right)}{k}|\hfill \\ & & \phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}+|\frac{f\left(a+h,b+k\right)-f\left(a,b+k\right)+\left(f\left(a+h,b\right)-f\left(a,b\right)\right)}{hk}-{f}_{yx}\left(a,b\right)|\hfill \\ & <& 3ϵ+|\frac{f\left(a+h,b+k\right)-f\left(a,b+k\right)+\left(f\left(a+h,b\right)-f\left(a,b\right)\right)}{hk}-{f}_{yx}\left(a,b\right)|\hfill \\ & =& 3ϵ+|\frac{{f}_{y}\left(a+h,{b}^{\text{'}}\right)-{f}_{y}\left(a,{b}^{\text{'}}\right)}{h}-{f}_{yx}\left(a,b\right)|\hfill \\ & =& 3ϵ+|{f}_{yx}\left({a}^{\text{'}},{b}^{\text{'}}\right)-{f}_{yx}\left(a,b\right)|\hfill \\ & <& 4ϵ,\hfill \end{array}$

because ${b}^{\text{'}}$ is between $b$ and $b+k,$ and ${a}^{\text{'}}$ is between $a$ and $a+h,$ so that $|\left({a}^{\text{'}},{b}^{\text{'}}\right)-\left(a,b\right)|<{\delta }_{1}/\sqrt{2}<{\delta }_{1}.$ Hence, $|{f}_{xy}\left(a,b\right)-{f}_{yx}\left(a,b\right)<4ϵ,$ for an arbitrary $ϵ,$ and so the theorem is proved.

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
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what is the k.e before it land
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what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
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not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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