Math2111: Chapter 4: Surface integrals. Section 2: Surface area and sur...
1 of 6
http://quasirandomideas.wordpress.com/2010/05/13/math2111-chapter-...
Front Page
About
Papers
Writing LaTeX in WordPress
Book
Teaching
← Math2111: Chapter 4: Surface integrals. Section 1: Parameterisations of surfaces
May 13, 2010 · Leave a Comment · Edit This
In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of
contents for this course. This blog entry printed to pdf is available here.
In the following we assume that the surfaces are smooth, that is, they are assumed to be images of
for which:
parameterised surfaces
is a Jordan-measurable subset of ;
the mapping is one-to-one;
the normal vector
except possibly at a finite number of points;
Surface area
In a previous post we discussed parameterised surfaces. Now we calculate the area of parameterised
surfaces.
Recall that the area of a parallelogram in
spanned by two vectors and is given by the Euclidean
norm
of the vector obtained by taking the cross product of these two vectors, that is, by
From the parameterisation
we obtain two tangent vectors and
We can
(
) by a parallelogram spanned by
approximate a piece of the surface at some point
the vectors
and
whose area can be approximated by
By summing over all pieces which approximate the whole surface, i.e. forming the sum
14/05/2010 1:04 AM
Math2111: Chapter 4: Surface integrals. Section 2: Surface area and sur...
2 of 6
http://quasirandomideas.wordpress.com/2010/05/13/math2111-chapter-...
and considering the limit when the size of the pieces goes to zero we obtain the integral
(Here,
is the normal vector defined here.) We call
Definition
Let
defined by
the surface area of the surface .
be a parameterisation of a surface
Then the surface area
of
is
The last formula can also be written as
If the surface is a graph of a function
then
and hence the surface of
the graph is given by
Example
To calculate its surface area, notice that, because of symmetry, we can
Consider a sphere of radius
calculate the surface area of the upper hemisphere and multiply the result by to obtain the surface area
where
We
of the whole sphere. The upper hemisphere is given by the equation
can set
and use the parameterisation of surfaces for functions as shown in
Section 1. The parameter domain is in this case
and the normal vector
is
The length of this vector is
Hence the surface area of the hemisphere (which we shall denote by
) is given by
The last integral can be evaluated using polar coordinates by which we obtain
14/05/2010 1:04 AM
Math2111: Chapter 4: Surface integrals. Section 2: Surface area and sur...
3 of 6
http://quasirandomideas.wordpress.com/2010/05/13/math2111-chapter-...
Hence the area of the sphere is given by
Exercise
Calculate the surface area of a cone parameterised by
and
and
where
Scalar surface integrals
We now integrate scalar fields over surfaces. This is in analogy to scalar line integrals considered in
Chapter 3, Section 1.
Definition
Let
be a parameterisation of the surface
continuous. Then the integral of over is given by
and let
be
The last formula can also be written as
If the surface is the graph of a function
Example
Let a surface
be given by
then we also have
with
and
and let
Then set
Then
Hence
Exercise
The surface in the previous example is the graph of a function. Use this to parameterise the surface and
calculate the scalar surface integral using this approach.
Applications
Scalar surface integrals can be used to calculate mass, center of mass and moments of inertia of thin
shells. Let be the density function of a very thin shell.
Mass
Center of mass
and
14/05/2010 1:04 AM
Math2111: Chapter 4: Surface integrals. Section 2: Surface area and sur...
4 of 6
Moment of inertia
http://quasirandomideas.wordpress.com/2010/05/13/math2111-chapter-...
and
Categories: Math2111 Several Variable Calculus · Teaching
Tagged: scalar surface integral, Several Variable Calculus, surface area
0 responses so far ↓
There are no comments yet...Kick things off by filling out the form below.
Leave a Comment
Logged in as quasirandomideas. Logout »
You are the author of this post.
You are subscribed to this blog (manage).
ICIAM 2011 Vancouver
Math2111 Several Variable Calculus
Research
Teaching
About
Book
Papers
Teaching
Math2111: Several Variable Calculus: Table of Contents
Writing LaTeX in WordPress
14/05/2010 1:04 AM