# 4.4 Tangent planes and linear approximations  (Page 5/11)

 Page 5 / 11

## Continuity of first partials implies differentiability

Let $z=f\left(x,y\right)$ be a function of two variables with $\left({x}_{0},{y}_{0}\right)$ in the domain of $f.$ If $f\left(x,y\right),$ ${f}_{x}\left(x,y\right),$ and ${f}_{y}\left(x,y\right)$ all exist in a neighborhood of $\left({x}_{0},{y}_{0}\right)$ and are continuous at $\left({x}_{0},{y}_{0}\right),$ then $f\left(x,y\right)$ is differentiable there.

Recall that earlier we showed that the function

$f\left(x,y\right)=\left\{\begin{array}{cc}\frac{xy}{\sqrt{{x}^{2}+{y}^{2}}}\hfill & \left(x,y\right)\ne \left(0,0\right)\hfill \\ 0\hfill & \left(x,y\right)=\left(0,0\right)\hfill \end{array}$

was not differentiable at the origin. Let’s calculate the partial derivatives ${f}_{x}$ and ${f}_{y}\text{:}$

$\frac{\partial f}{\partial x}=\frac{{y}^{3}}{{\left({x}^{2}+{y}^{2}\right)}^{3\text{/}2}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{\partial f}{\partial y}=\frac{{x}^{3}}{{\left({x}^{2}+{y}^{2}\right)}^{3\text{/}2}}.$

The contrapositive of the preceding theorem states that if a function is not differentiable, then at least one of the hypotheses must be false. Let’s explore the condition that ${f}_{x}\left(0,0\right)$ must be continuous. For this to be true, it must be true that $\underset{\left(x,y\right)\to \left(0,0\right)}{\text{lim}}{f}_{x}\left(0,0\right)={f}_{x}\left(0,0\right)\text{:}$

$\underset{\left(x,y\right)\to \left(0,0\right)}{\text{lim}}{f}_{x}\left(x,y\right)=\underset{\left(x,y\right)\to \left(0,0\right)}{\text{lim}}\frac{{y}^{3}}{{\left({x}^{2}+{y}^{2}\right)}^{3\text{/}2}}.$

Let $x=ky.$ Then

$\begin{array}{cc}\hfill \underset{\left(x,y\right)\to \left(0,0\right)}{\text{lim}}\frac{{y}^{3}}{{\left({x}^{2}+{y}^{2}\right)}^{3\text{/}2}}& =\underset{y\to 0}{\text{lim}}\frac{{y}^{3}}{{\left({\left(ky\right)}^{2}+{y}^{2}\right)}^{3\text{/}2}}\hfill \\ & =\underset{y\to 0}{\text{lim}}\frac{{y}^{3}}{{\left({k}^{2}{y}^{2}+{y}^{2}\right)}^{3\text{/}2}}\hfill \\ & =\underset{y\to 0}{\text{lim}}\frac{{y}^{3}}{{|y|}^{3}{\left({k}^{2}+1\right)}^{3\text{/}2}}\hfill \\ & =\frac{1}{{\left({k}^{2}+1\right)}^{3\text{/}2}}\underset{y\to 0}{\text{lim}}\frac{|y|}{y}.\hfill \end{array}$

If $y>0,$ then this expression equals $1\text{/}{\left({k}^{2}+1\right)}^{3\text{/}2};$ if $y<0,$ then it equals $\text{−}\left(1\text{/}{\left({k}^{2}+1\right)}^{3\text{/}2}\right).$ In either case, the value depends on $k,$ so the limit fails to exist.

## Differentials

In Linear Approximations and Differentials we first studied the concept of differentials. The differential of $y,$ written $dy,$ is defined as ${f}^{\prime }\left(x\right)dx.$ The differential is used to approximate $\text{Δ}y=f\left(x+\text{Δ}x\right)-f\left(x\right),$ where $\text{Δ}x=dx.$ Extending this idea to the linear approximation of a function of two variables at the point $\left({x}_{0},{y}_{0}\right)$ yields the formula for the total differential for a function of two variables.

## Definition

Let $z=f\left(x,y\right)$ be a function of two variables with $\left({x}_{0},{y}_{0}\right)$ in the domain of $f,$ and let $\text{Δ}x$ and $\text{Δ}y$ be chosen so that $\left({x}_{0}+\text{Δ}x,{y}_{0}+\text{Δ}y\right)$ is also in the domain of $f.$ If $f$ is differentiable at the point $\left({x}_{0},{y}_{0}\right),$ then the differentials $dx$ and $dy$ are defined as

$dx=dx\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}dy=\text{Δ}y.$

The differential $dz,$ also called the total differential    of $z=f\left(x,y\right)$ at $\left({x}_{0},{y}_{0}\right),$ is defined as

$dz={f}_{x}\left({x}_{0},{y}_{0}\right)dx+{f}_{y}\left({x}_{0},{y}_{0}\right)dy.$

Notice that the symbol $\partial$ is not used to denote the total differential; rather, $d$ appears in front of $z.$ Now, let’s define $\text{Δ}z=f\left(x+\text{Δ}x,y+\text{Δ}y\right)-f\left(x,y\right).$ We use $dz$ to approximate $\text{Δ}z,$ so

$\text{Δ}z\approx dz={f}_{x}\left({x}_{0},{y}_{0}\right)dx+{f}_{y}\left({x}_{0},{y}_{0}\right)dy.$

Therefore, the differential is used to approximate the change in the function $z=f\left({x}_{0},{y}_{0}\right)$ at the point $\left({x}_{0},{y}_{0}\right)$ for given values of $\text{Δ}x$ and $\text{Δ}y.$ Since $\text{Δ}z=f\left(x+\text{Δ}x,y+\text{Δ}y\right)-f\left(x,y\right),$ this can be used further to approximate $f\left(x+\text{Δ}x,y+\text{Δ}y\right)\text{:}$

$\begin{array}{cc}\hfill f\left(x+\text{Δ}x,y+\text{Δ}y\right)& =f\left(x,y\right)+\text{Δ}z\hfill \\ & \approx f\left(x,y\right)+{f}_{x}\left({x}_{0},{y}_{0}\right)\text{Δ}x+{f}_{y}\left({x}_{0},{y}_{0}\right)\text{Δ}y.\hfill \end{array}$

See the following figure.

One such application of this idea is to determine error propagation. For example, if we are manufacturing a gadget and are off by a certain amount in measuring a given quantity, the differential can be used to estimate the error in the total volume of the gadget.

## Approximation by differentials

Find the differential $dz$ of the function $f\left(x,y\right)=3{x}^{2}-2xy+{y}^{2}$ and use it to approximate $\text{Δ}z$ at point $\left(2,-3\right).$ Use $\text{Δ}x=0.1$ and $\text{Δ}y=-0.05.$ What is the exact value of $\text{Δ}z?$

First, we must calculate $f\left({x}_{0},{y}_{0}\right),{f}_{x}\left({x}_{0},{y}_{0}\right),\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{f}_{y}\left({x}_{0},{y}_{0}\right)$ using ${x}_{0}=2$ and ${y}_{0}=-3\text{:}$

$\begin{array}{ccc}\hfill f\left({x}_{0},{y}_{0}\right)& =\hfill & f\left(2,-3\right)=3{\left(2\right)}^{2}-2\left(2\right)\left(-3\right)+{\left(-3\right)}^{2}=12+12+9=33\hfill \\ \hfill {f}_{x}\left(x,y\right)& =\hfill & 6x-2y\hfill \\ \hfill {f}_{y}\left(x,y\right)& =\hfill & -2x+2y\hfill \\ \hfill {f}_{x}\left({x}_{0},{y}_{0}\right)& =\hfill & {f}_{x}\left(2,-3\right)=6\left(2\right)-2\left(-3\right)=12+6=18\hfill \\ \hfill {f}_{y}\left({x}_{0},{y}_{0}\right)& =\hfill & {f}_{y}\left(2,-3\right)=-2\left(2\right)+2\left(-3\right)=-4-6=-10.\hfill \end{array}$

Then, we substitute these quantities into [link] :

$\begin{array}{c}dz={f}_{x}\left({x}_{0},{y}_{0}\right)dx+{f}_{y}\left({x}_{0},{y}_{0}\right)dy\hfill \\ dz=18\left(0.1\right)-10\left(-0.05\right)=1.8+0.5=2.3.\hfill \end{array}$

This is the approximation to $\text{Δ}z=f\left({x}_{0}+\text{Δ}x,{y}_{0}+\text{Δ}y\right)-f\left({x}_{0},{y}_{0}\right).$ The exact value of $\text{Δ}z$ is given by

$\begin{array}{cc}\text{Δ}z\hfill & =f\left({x}_{0}+\text{Δ}x,{y}_{0}+\text{Δ}y\right)-f\left({x}_{0},{y}_{0}\right)\hfill \\ & =f\left(2+0.1,-3-0.05\right)-f\left(2,-3\right)\hfill \\ & =f\left(2.1,-3.05\right)-f\left(2,-3\right)\hfill \\ & =2.3425.\hfill \end{array}$

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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
Privacy Information Security Software Version 1.1a
Good
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?