Chapter 15 Curves, Surfaces and Volumes Alberto J. Benavides, Adewale Awoniyi, Xiaohong Cui

1
Chapter 15 Curves, Surfaces and VolumesAlberto
J. Benavides, Adewale Awoniyi, Xiaohong Cui
Department of Chemical Engineering Texas AM
University, College Station, TX
2
Agenda
Surfaces
15.4
Surface Integrals
15.5
Volume Integrals
15.6
3
15.4. Surfaces Alberto J. Benavides, Adewale
Awoniyi, Xiaohong Cui
Department of Chemical Engineering Texas AM
University, College Station, TX
4
15.4.1 Parametric representation
A parametric surface is a surface in the
Euclidean space R3 which is defined by a
parametric equation with two parameters, u and v.
That is x x(u,v), y y(u,v), z z(u,v) or
r x(u,v)i y(u,v)j z(u,v)k Example 1
Sphere
x sin(v)cos(u) y sin(v)sin(u) z cos(v)
(0,p)
Source http//www.math.uri.edu/bkaskosz/flashmo/
parsur/
(2p,0)
(0,0)
5
15.4.1 Parametric representation
A parametric surface is a surface in the
Euclidean space R3 which is defined by a
parametric equation with two parameters, u and v.
That is x x(u,v), y y(u,v), z z(u,v)
or R x(u,v)i y(u,v)j z(u,v)k Example 2
Cone
x vcos(u) y vsin(u) z v
(0,1)
Source http//www.math.uri.edu/bkaskosz/flashmo/
parsur/
(2p,-1)
(0,-1)
6
15.4.1 Parametric representation
A parametric surface is a surface in the
Euclidean space R3 which is defined by a
parametric equation with two parameters, u and v.
That is x x(u,v), y y(u,v), z z(u,v)
or R x(u,v)i y(u,v)j z(u,v)k Example 3
Mobius Band
x 2cos(u)vcos(u/2) y 2sin(u)vcos(u/2) z
vsin(u/2)
(0,0.5)
Source http//www.math.uri.edu/bkaskosz/flashmo/
parsur/
(2p,-0.5)
(0,-0.5)
7
15.4.1 Parametric representation
A parametric surface is a surface in the
Euclidean space R3 which is defined by a
parametric equation with two parameters, u and v.
That is x x(u,v), y y(u,v), z z(u,v)
or R x(u,v)i y(u,v)j z(u,v)k Example 4
Snake
x (1-u)(3cosv)cos(2pu) y (1-u)(3cosv)sin(2pu
) z 6u(1-u)sin(v)
(0,1)
Source http//www.math.uri.edu/bkaskosz/flashmo/
parsur/
(2p,0)
(0,0)
8
15.4.1 Parametric representation
A parametric surface is a surface in the
Euclidean space R3 which is defined by a
parametric equation with two parameters, u and v.
That is x x(u,v), y y(u,v), z z(u,v)
or R x(u,v)i y(u,v)j z(u,v)k Example 5
Shell
x (4/3)usin(v)sin(v)cos(u) y
(4/3)usin(v)sin(v)sin(u) z
(4/3)usin(v)cos(v)
(0,p)
Source http//www.math.uri.edu/bkaskosz/flashmo/
parsur/
(1.1p,0)
(-6,0)
9
15.4.2 Tangent plane and normal
Consider the parametric surface S specified by r.
If we keep u constant by setting u u0, then
r(u0,v) moves along a curve C that lies on S as v
varies. The curve C is referred to as a grid
curve of constant u. Similarly, by keeping v
constant, we obtain grid curves of constant v
that lie on S as u varies.
Source Advanced Engineering Mathematics (2nd
Edition), Michael Greenberg, Prentice Hall
10
15.4.2 Tangent plane and normal
Let p (x(u0,v0) y(u0,v0) z(u0,v0)) be a point
on S. Let C1 and C2 be the grid curves
parametrized by r(u0,v) and r(u,v0) respectively.
Then the derivative of r(u0,v) at the point P
gives rise to a tangent vector of C1 at P, i.e.,
the vector
Likewise,
is a tangent vector to C2 at p. In particular,
the vectors ru(u0,v0) and rv(u0,v0) are contained
in the tangent plane to the surface S at the
point p, i.e., the plane that contains all the
tangent vectors to curves lying in S and passing
through p. Consequently, if ru(u0,v0) and
rv(u0,v0) are not parallel, then ru(u0,v0)
rv(u0,v0) is a normal vector to the tangent plane
at p, from which it follows that an equation for
the tangent plane is
Adapted from http//www.math.osu.edu/khkwa/254au
10/16.6withScannedExamples.pdf
11
15.4.2 Tangent plane and normal
 
Or,
Where d axp byp czp.
 
Adapted from http//www.math.osu.edu/khkwa/254au
10/16.6withScannedExamples.pdf
12
15.4.2 Tangent plane and normal
Example 6. Find an equation of the tangent plane
to the given parametric surface at the specified
point.
Solution Compute the tangent vectors.
The cross product of these two will give us the
normal.
At the point ( 2, 3, 0 ), the parametric
equations are
Solving this system of equations gives us u 1,
v 1. The normal vector at point ( 2, 3, 0 ) is
lt-6,2,-6gt. Therefore, the tangent plane at point
( 2, 3, 0 ) is
Source http//www.math.ucla.edu/ronmiech/Calculu
s_Problems/32B/chap14/section6/937d15/937_15.html
13
15.4.2 Tangent plane and normal
 
14
15.4.2 Tangent plane and normal
 
Adapted from planetmath.org/encyclopedia/TangentP
lane.html
15
15.4.2 Tangent plane and normal
 
Adapted from planetmath.org/encyclopedia/TangentP
lane.html
16
15.5. Surface Integrals Alberto J. Benavides,
Adewale Awoniyi, Xiaohong Cui
Department of Chemical Engineering Texas AM
University, College Station, TX
17
15.5 Surface Integrals Motivation Outline
- Why Surface Integrals Are Important
Applications of surface integrals include (1)
Calculating surface area (2) Calculating flux
(rate of fluid flow across a surface area, etc)
- Outline of This Section (1) Area element dA
(2) Surface integrals
18
15.5.1 Area Element dA
Consider a surface S given parametrically by
R(u,v) x(u,v)i y(u,v)j z(u,v)k where
R(u,v) is C1 and Ru Rv ? 0 on S. Such a
surface is said to be smooth.
The two vectors along u constant and v
constant curves through P, dR Rvdv and dR
Rudu respectively, define a parallelogram (called
area element on S) lying in the tangent plane to
S at P. According to the geometrical significance
of the cross product of two vectors, the area of
the parallelogram is dA Rudu Rvdv Ru
Rvdudv
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
19
15.5.1 Area Element dA
dA Rudu Rvdv Ru Rvdudv
To obtain a computational version of the above
expression, cross Ru xui yuj zuk with Rv
xvi yvj zvk. The norm of the resulting vector
is the square root of the sum of the square of
its components. Thus we obtain
 
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
20
15.5.1 Area Element dA
 
 
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
21
15.5.1 Area Element dA
 
 
Figure 2. dA in polar coordinates (Source
Advanced Engineering Mathematics (2nd Edition),
Michael Greenberg, Prentice Hall)
 
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
22
15.5.1 Area Element dA
 
 
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
23
15.5.1 Area Element dA
Calculating Surface Area According to we define
the area of the curved surface S by the double
integral
dA Rudu Rvdv Ru Rvdudv
 
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
24
15.5.1 Area Element dA
 
 
 
 
 
Source http//tutorial.math.lamar.edu/Classes/Ca
lcIII/SurfaceArea.aspx
25
15.5.2 Surface Integrals
 
 
 
 
 
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
26
15.5.2 Surface Integrals
Calculation (1) Surface is parametrized in the
form R(u,v)
 
 
27
15.5.2 Surface Integrals
 
 
 
Source http//tutorial.math.lamar.edu/Classes/Ca
lcIII/SurfaceIntegrals.aspx
28
15.5.2 Surface Integrals
 
 
 
Source http//tutorial.math.lamar.edu/Classes/Ca
lcIII/SurfaceIntegrals.aspx
29
15.5.2 Surface Integrals
 
 
 
Source http//www.math.oregonstate.edu/home/prog
rams/undergrad/CalculusQuestStudyGuides/vcalc/flux
/flux.html
30
15.5.2 Surface Integrals
 
 
Source http//www.math.oregonstate.edu/home/prog
rams/undergrad/CalculusQuestStudyGuides/vcalc/flux
/flux.html
31
15.6. Volume Integrals Alberto J. Benavides,
Adewale Awoniyi, Xiaohong Cui
Department of Chemical Engineering Texas AM
University, College Station, TX
32
15.6 Volume Integrals Motivation Outline
- Why Volume Integrals Are Important
Applications of volume integrals include (1)
Calculating volume of a given region (2)
Physical applications calculating the moment of
inertia, gravitational force, etc (3) Solving
partial differential equations
- Outline of This Section (1) Volume element
dV (2) Volume integrals
33
15.6.1 Volume Element dV
Consider the position vector R given
parametrically by R(u,v,w) x(u,v,w)i
y(u,v,w)j z(u,v,w)k where R(u,v,w) is C1 and
Ru, Rv and Rw are linearly independent. For each
fixed w, the parametrization defines a surface.
As we vary w we produce a family of such surfaces
which will generate a volume. The three vectors
dR Rudu, dR Rvdv and dR Rwdw determine a
parallelepiped of nonzero volume (called volume
element).
Figure 1. Volume element dV (Source Advanced
Engineering Mathematics (2nd Edition), Michael
Greenberg, Prentice Hall)
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
34
15.6.1 Volume Element dV
 
Figure 1. Volume element dV (Source Advanced
Engineering Mathematics (2nd Edition), Michael
Greenberg, Prentice Hall)
 
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
35
15.6.1 Volume Element dV
 
Figure 2. Volume element in cylindrical
coordinates (Sourcehttp//keep2.sjfc.edu/faculty/
kgreen/vector/Block3/jacob/node9.html)
 
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
36
15.6.1 Volume Element dV
Two special cases of calculating dV(1)
Cyclindrical Coordinates Example 1. (continue)
Summary of information obtained from the position
vector in cylindrical coordinates is as follows.
Source Advanced Engineering Mathematics (2nd
Edition), Michael Greenberg, Prentice Hall
37
15.6.1 Volume Element dV
 
Figure 3. Volume element in spherical
coordinates (Sourcehttp//keep2.sjfc.edu/faculty/
kgreen/vector/Block3/jacob/node9.html)
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
38
15.6.1 Volume Element dV
 
Figure 3. Volume element in spherical
coordinates (Source Salas, Hille, Etgen
Calculus One and Several Variables, John Wiley
Sons, Inc)
 
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
39
15.6.1 Volume Element dV
Two special cases of calculating dV(2) Spherical
Coordinates Example 2. (continue) Summary of
information obtained from the position vector in
spherical coordinates is as follows.
Source Advanced Engineering Mathematics (2nd
Edition), Michael Greenberg, Prentice Hall
40
15.6.1 Volume Element dV
Calculating Volume of a Given Region According
to we define the volume of a region V by the
triple integral
 
 
Figure 1. Volume element dV (Source Advanced
Engineering Mathematics (2nd Edition), Michael
Greenberg, Prentice Hall)
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
41
15.6.1 Volume Element dV
 
Figure 4. The region described in Example
3 (Source http//tutorial.math.lamar.edu/Classes/
CalcIII/TripleIntegrals.aspx/)
Source http//tutorial.math.lamar.edu/Classes/Ca
lcIII/TripleIntegrals.aspx
42
15.6.1 Volume Element dV
 
Solution
Source http//tutorial.math.lamar.edu/Classes/Ca
lcIII/TripleIntegrals.aspx
43
15.6.2 Volume Integrals
Definition We define volume integral of a given
function f over a given region V in 3-space for
if x, y, z are parametrized as u, v, w, then
 
Where R is the region in u, v, w space
corresponding to the region V in the x, y, z
space.
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
44
15.6.2 Volume Integrals
 
 
 
Source http//tutorial.math.lamar.edu/Classes/Ca
lcIII/TripleIntegrals.aspx
45
15.6.2 Volume Integrals
 
 
 
Source http//tutorial.math.lamar.edu/Classes/Ca
lcIII/TripleIntegrals.aspx
46
15.6.2 Volume Integrals
 
 
 
Source http//tutorial.math.lamar.edu/Classes/Ca
lcIII/TripleIntegrals.aspx
47
15.6.2 Volume Integrals
 
 
Figure 6. The hollow sphere described in Example
5 (Source http//spiff.rit.edu/classes/phy
s350/phys_iii.html/)
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
48
15.6.2 Volume Integrals
 
Figure 6. The hollow sphere described in Example
5 (Source http//spiff.rit.edu/classes/phy
s350/phys_iii.html/)
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall
49
15.6.2 Volume Integrals
 
Figure 6. The hollow sphere described in Example
5 (Source http//spiff.rit.edu/classes/phy
s350/phys_iii.html/)
Adapted from Advanced Engineering Mathematics
(2nd Edition), Michael Greenberg, Prentice Hall