1. Chapter 15 Curves, Surfaces and Volumes Alberto J
Chapter 15 Curves, Surfaces and Volumes Alberto J. Benavides, Adewale Awoniyi, Xiaohong Cui
Department of Chemical Engineering
Texas A&M 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 A&M 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,π)
Source: t=v, s=u
Source:
(0,0)
(2π,0)
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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)
t=v, s=u
Source:
(0,-1)
(2π,-1)
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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)
t=v, s=u
Source:
(0,-0.5)
(2π,-0.5)
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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)(3+cosv)cos(2πu)
y = (1-u)(3+cosv)sin(2πu)
z = 6u+(1-u)sin(v)
(0,1)
t=v, s=u
Source:
(0,0)
(2π,0)
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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)^u*sin(v)*sin(v)*cos(u)
y = (4/3)^u*sin(v)*sin(v)*sin(u)
z = (4/3)^u*sin(v)*cos(v)
(0,π)
t=v, s=u
Source:
(-6,0)
(1.1π,0)
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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
4/26/2017

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:
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11. 15.4.2 Tangent plane and normal
Or,
Where d = axp + byp + czp.
Adapted from:
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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 <-6,2,-6>. Therefore, the tangent plane at point ( 2, 3, 0 ) is:
Source:
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13. 15.4.2 Tangent plane and normal
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14. 15.4.2 Tangent plane and normal
Adapted from: planetmath.org/encyclopedia/TangentPlane.html
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15. 15.4.2 Tangent plane and normal
Adapted from: planetmath.org/encyclopedia/TangentPlane.html
4/26/2017

16. 15. 5. Surface Integrals Alberto J
15.5. Surface Integrals Alberto J. Benavides, Adewale Awoniyi, Xiaohong Cui
Department of Chemical Engineering
Texas A&M 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
4/26/2017

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 × Rv║dudv
Figure 1. Area element dA
(Adapted from: File:Surface_integral1.svg) dA P
dR = Rudu
v = constant
dR = Rvdv
u = constant
Adapted from:
Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall
4/26/2017

19. dA = ║Rudu × Rvdv║ = ║Ru × Rv║dudv
Area Element dA
dA = ║Rudu × Rvdv║ = ║Ru × Rv║dudv
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
Figure 1. Area element dA
(Adapted from: File:Surface_integral1.svg) dA P
dR = Rudu
v = constant
dR = Rvdv
u = constant
Adapted from:
Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall
4/26/2017

20. 15.5.1 Area Element dA 4/26/2017 Adapted from:
Adapted from:
Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall
4/26/2017

21. Figure 2. dA in polar coordinates
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
4/26/2017

22. 15.5.1 Area Element dA 4/26/2017 Adapted from:
Adapted from:
Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall
4/26/2017

23. dA = ║Rudu × Rvdv║ = ║Ru × Rv║dudv
Area Element dA
Calculating Surface Area
According to
we define the area of the curved surface S by the double integral
Figure 1. Area element dA
(Adapted from: File:Surface_integral1.svg) dA P
dR = Rudu
v = constant
dR = Rvdv
u = constant
dA = ║Rudu × Rvdv║ = ║Ru × Rv║dudv
Adapted from:
Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall
4/26/2017

24. Area Element dA
Source:
4/26/2017

25. Surface Integrals
Adapted from:
Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall
4/26/2017

26. 15.5.2 Surface Integrals Calculation
(1) Surface is parametrized in the form R(u,v)
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27. Surface Integrals
Source:
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28. Surface Integrals
Source:
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29. Surface Integrals
Source:
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30. Surface Integrals
Source:
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31. 15. 6. Volume Integrals Alberto J
15.6. Volume Integrals Alberto J. Benavides, Adewale Awoniyi, Xiaohong Cui
Department of Chemical Engineering
Texas A&M 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
4/26/2017

33. 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
4/26/2017

34. Figure 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
4/26/2017

35. Volume Element dV
Figure 2. Volume element in cylindrical coordinates
(Source:
Adapted from:
Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall
4/26/2017

36. 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
4/26/2017

37. Volume Element dV
Figure 3. Volume element in spherical coordinates
(Source:
Adapted from:
Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall
4/26/2017

38. 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
4/26/2017

39. 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
4/26/2017

40. Figure 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
4/26/2017

41. Figure 4. The region described in Example 3
Volume Element dV
Figure 4. The region described in Example 3
(Source:
Source:
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42. Figure 4. The region described in Example 3
Volume Element dV
Figure 4. The region described in Example 3
(Source: /CalcIII/TripleIntegrals.aspx/)
Solution:
Source:
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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
4/26/2017

44. Volume Integrals
Source:
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45. Volume Integrals
Source:
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46. Volume Integrals
Source:
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47. Volume Integrals
Figure 6. The hollow sphere described in Example 5
(Source: s350/phys_iii.html/)
Adapted from:
Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall
4/26/2017

48. Volume Integrals
Figure 6. The hollow sphere described in Example 5
(Source: s350/phys_iii.html/)
Adapted from:
Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall
4/26/2017

49. Volume Integrals
Figure 6. The hollow sphere described in Example 5
(Source: s350/phys_iii.html/)
Adapted from:
Advanced Engineering Mathematics (2nd Edition), Michael Greenberg, Prentice Hall
4/26/2017