Surface Integral

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Surface Integral
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Surface Integral
The definition of surface integral relies on
splitting the surface into small surface
elements.
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Surface Integral

• It is now time to think about integrating
  functions over some surface, S, in
  three-dimensional space.  Lets start off with a
  sketch of the surface S since the notation can
  get a little confusing once we get into it.  Here
  is a sketch of some surface S.

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Surface Integral
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Surface Integral
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Surface Integral
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Surface Integral
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Surface Integral Problem 1

• Water Flow

z
0,0,2
0,3,2
y
0,0,0
0,3,0
x
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Solution 1

• ? ?surface B dS
• ? ?surface Bx dy dz
• 3 ?z0z2 ?y0 y3 yzdydz
• 3y2/2 30 ? z2z0zdz
• 3x9/2 ? z2z0zdz
• 3x9/2 z2/2 20
• 27 liters min-1

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Conversion of Coordinates

• Rectangular to

Spherical
Cylindrical
r X sin? cos?
r X cos?
r y sin? sin?
r y sin?
r z cos?
r z 0
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Conversion of Coordinates

• Cylindrical to Rectangular

Ax Ar cos? - A? sin?
Ay Ar sin? - A? cos?
Az Az
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Conversion of Coordinates

• Spherical to Rectangular

Ax Ar sin? cos? A? cos? cos? A? sin?
Ay Ar sin? sin? A? cos? sin? A? cos?
Az Ar cos? - A? sin?
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Conversion of Coordinates

• Rectangular Circular Spherical

x y z
r ? z
r ? ?
cos? -sin? 0
sin?cos? cos?cos? -sin?
x y z
1 0 0
0 1 0
sin? cos? 0
sin?sin? cos?sin? cos?
0 0 1
0 0 1
Cos -sin? 0
1 0 0
r ? z
cos? sin? 0
-sin? cos? 0
0 1 0
-sin? cos? 0
0 0 1
cos? -sin? 0
0 0 1
0 0 1
r ? ?
sin?cos? sin?sin? cos?
sin? 0 cos?
1 0 0
cos?cos? cos?sin? -sin?
cos? 0 -sin?
0 1 0
-sin? cos? 0
0 1 0
0 0 1
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Cross-Product of Vector

• The Vector product of two vectors is defined as a
  third vector magnitudes multiplied by the sine of
  the angle between them.
• The direction of resultant vector is
  perpendicular to the plane containing the first
  two.
• C AB sin? AB sin?
• A x B C h AB sin?
• Where h is unit vector normal to the plane

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Cross-Product of Vector

• X-product of A B

z
h
C AB sin?
A
y
?
B
While magnitude of C is normal to both A B
x
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Cross Product

• x x y z
• y x z x
• z x x y
• x x x y x y z x z 0

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Cross Product

• Example 1
• A x8 y3 z10 B - x 15 y6 z17
• Find A x B
• Example 2
• A y20 z5 B -x6 y14
• Find A x B

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Tutorial

• Example 1  Evaluate

where S is the portion of the zy- plane
Solution
Okay, since we are looking for the portion of the
plane that lies in front of the yz-plane we are
going to need to write the equation of the
surface in the form
     
   
This is easy enough to do.
Next we need to determine just what D is.  Here
is a sketch of the surface S.
Here is a sketch of the region D
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Tutorial

• Notice that the axes are labeled differently than
  we are used to seeing in the sketch of D.  This
  was to keep the sketch consistent with the sketch
  of the surface.  We arrived at the equation of
  the hypotenuse by setting x equal to zero in the
  equation of the plane and solving for z.  Here
  are the ranges for y and z.

Now, because the surface is not in the form
  we cant use the formula above. 
However, as noted above we can modify this
formula to get one that will work for us.  Here
it is
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Tutorial

• The changes made to the formula should be the
  somewhat obvious changes.  So, lets do the
  integral

Notice that we plugged in the equation of the
plane for the x in the integrand.  At this point
weve got a fairly simple double integral to do. 
Here is that work
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Tutorial Example 2

• Evaluate

where S is the upper half of a sphere of radius 2
Solution
We gave the parameterization of a sphere in the
previous section.  Here is the parameterization
for this sphere
Since we are working on the upper half of the
sphere here are the limits on the parameters
Next, we need to determine . 
Here are the two individual vectors
Now lets take the cross product.
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Tutorial Example 2.
Finally, we need the magnitude of this