<< Chapter < Page Chapter >> Page >
  • 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 ( x ) . The slope of the tangent line at the point x = a is given by m = f ( a ) ; 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.


Let P 0 = ( x 0 , y 0 , z 0 ) 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] ).

A surface S is shown with a point P0 = (x0, y0, z0). There are two intersecting curves shown on S that pass through P0. There are tangents drawn for each of these curves at P0, and these tangent lines create a plane, namely, the tangent plane at P0.
The tangent plane to a surface S at a point P 0 contains all the tangent lines to curves in S that pass through P 0 .

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.


Let S be a surface defined by a differentiable function z = f ( x , y ) , and let P 0 = ( x 0 , y 0 ) 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 ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x x 0 ) + f y ( x 0 , y 0 ) ( y y 0 ) .

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 ( x 0 , y 0 ) + f y ( x 0 , y 0 ) ( y y 0 ) . 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 ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x x 0 ) . A parallel vector to the first tangent line is a = j + f y ( x 0 , y 0 ) k; a parallel vector to the second tangent line is b = i + f x ( x 0 , y 0 ) k . We can take the cross product of these two vectors:

a × b = ( j + f y ( x 0 , y 0 ) k ) × ( i + f x ( x 0 , y 0 ) k ) = | i j k 0 1 f y ( x 0 , y 0 ) 1 0 f x ( x 0 , y 0 ) | = f x ( x 0 , y 0 ) i + f y ( x 0 , y 0 ) j k .

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 = ( x 0 , y 0 , f ( x 0 , y 0 ) ) in the equation for a plane:

Questions & Answers

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
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
characteristics of micro business
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?