# 4.4 Tangent planes and linear approximations

 Page 1 / 11
• Determine the equation of a plane tangent to a given surface at a point.
• Use the tangent plane to approximate a function of two variables at a point.
• Explain when a function of two variables is differentiable.
• Use the total differential to approximate the change in a function of two variables.

In this section, we consider the problem of finding the tangent plane to a surface, which is analogous to finding the equation of a tangent line to a curve when the curve is defined by the graph of a function of one variable, $y=f\left(x\right).$ The slope of the tangent line at the point $x=a$ is given by $m=f\prime \left(a\right);$ what is the slope of a tangent plane? We learned about the equation of a plane in Equations of Lines and Planes in Space ; in this section, we see how it can be applied to the problem at hand.

## Tangent planes

Intuitively, it seems clear that, in a plane, only one line can be tangent to a curve at a point. However, in three-dimensional space, many lines can be tangent to a given point. If these lines lie in the same plane, they determine the tangent plane at that point. A more intuitive way to think of a tangent plane is to assume the surface is smooth at that point (no corners). Then, a tangent line to the surface at that point in any direction does not have any abrupt changes in slope because the direction changes smoothly. Therefore, in a small-enough neighborhood around the point, a tangent plane touches the surface at that point only.

## Definition

Let ${P}_{0}=\left({x}_{0},{y}_{0},{z}_{0}\right)$ be a point on a surface $S,$ and let $C$ be any curve passing through ${P}_{0}$ and lying entirely in $S.$ If the tangent lines to all such curves $C$ at ${P}_{0}$ lie in the same plane, then this plane is called the tangent plane    to $S$ at ${P}_{0}$ ( [link] ).

For a tangent plane to a surface to exist at a point on that surface, it is sufficient for the function that defines the surface to be differentiable at that point. We define the term tangent plane here and then explore the idea intuitively.

## Definition

Let $S$ be a surface defined by a differentiable function $z=f\left(x,y\right),$ and let ${P}_{0}=\left({x}_{0},{y}_{0}\right)$ be a point in the domain of $f.$ Then, the equation of the tangent plane to $S$ at ${P}_{0}$ is given by

$z=f\left({x}_{0},{y}_{0}\right)+{f}_{x}\left({x}_{0},{y}_{0}\right)\left(x-{x}_{0}\right)+{f}_{y}\left({x}_{0},{y}_{0}\right)\left(y-{y}_{0}\right).$

To see why this formula is correct, let’s first find two tangent lines to the surface $S.$ The equation of the tangent line to the curve that is represented by the intersection of $S$ with the vertical trace given by $x={x}_{0}$ is $z=f\left({x}_{0},{y}_{0}\right)+{f}_{y}\left({x}_{0},{y}_{0}\right)\left(y-{y}_{0}\right).$ Similarly, the equation of the tangent line to the curve that is represented by the intersection of $S$ with the vertical trace given by $y={y}_{0}$ is $z=f\left({x}_{0},{y}_{0}\right)+{f}_{x}\left({x}_{0},{y}_{0}\right)\left(x-{x}_{0}\right).$ A parallel vector to the first tangent line is $\text{a}=\text{j}+{f}_{y}\left({x}_{0},{y}_{0}\right)\text{k;}$ a parallel vector to the second tangent line is $\text{b}=\text{i}+{f}_{x}\left({x}_{0},{y}_{0}\right)\text{k}.$ We can take the cross product of these two vectors:

$\begin{array}{cc}\hfill a\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}b& =\left(j+{f}_{y}\left({x}_{0},{y}_{0}\right)k\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(i+{f}_{x}\left({x}_{0},{y}_{0}\right)k\right)\hfill \\ & =|\begin{array}{ccc}i\hfill & j\hfill & k\hfill \\ 0\hfill & 1\hfill & {f}_{y}\left({x}_{0},{y}_{0}\right)\hfill \\ 1\hfill & 0\hfill & {f}_{x}\left({x}_{0},{y}_{0}\right)\hfill \end{array}|\hfill \\ & ={f}_{x}\left({x}_{0},{y}_{0}\right)i+{f}_{y}\left({x}_{0},{y}_{0}\right)j-k.\hfill \end{array}$

This vector is perpendicular to both lines and is therefore perpendicular to the tangent plane. We can use this vector as a normal vector to the tangent plane, along with the point ${P}_{0}=\left({x}_{0},{y}_{0},f\left({x}_{0},{y}_{0}\right)\right)$ in the equation for a plane:

Introduction about quantum dots in nanotechnology
what does nano mean?
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?
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
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
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
what is system testing?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!