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  • 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

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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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