Section 16.7 Surface Integrals

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Section 16.7 Surface Integrals
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Surface Integrals
We now consider integrating functions over a
surface S that lies above some plane region D.
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Surface Integrals
Lets suppose the surface S is described by
zg(x,y) and we are considering a function
f(x,y,z) defined on S
The surface integral is given by
Where dS is the change in the surface AREA!
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More useful
i.e. we convert a surface integral into a
standard double integral that we can compute!
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Example

• Evaluate

Where
And S is the portion of the plane 2xy2z6 in
the first octant.
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Surface Integrals

• We still need to discuss surface integrals of
  vector fieldsbut we need a few new notions about
  surfaces first.
• Recall the vector form of a line integral (which
  used the tangent vector to the curve)
• For surface integrals we will make use of the
  normal vector to the surface!

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Normal vector to a surface

• If a surface S is given by zg(x,y), what is the
  normal vector to the surface at a point
  (x,y,g(x,y)) on the surface?

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Definition Oriented Surface

• Suppose our surface has a tangent plane defined
  at every point (x,y,z) on the surface
• Then at each tangent plane there are TWO unit
  normal vectors with n1 -n2
• If it is possible to choose a unit normal vector
  n at every point (x,y,z) so that n varies
  continuously over S, we say S is an oriented
  surface

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Example
Remark An oriented surface has two distinct
sides
Positive orientation
Negative orientation
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Surface Integrals of Vector Fields

• If F is a continuous vector field defined on an
  oriented surface S with unit normal vector n,
  then the surface integral of F over S is

This is often called the flux of F across S
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Using our knowledge of the normal vector and the
surface area, this can be simplified
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i.e. a more simplified look at this
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An application

• If is the density of a fluid
  that is moving through a surface S with velocity
  given by a vector field, F(x,y,z), then

Represents the mass of the fluid flowing across
the surface S per unit of time.
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Example

• Let S be the portion of the paraboloid
• Lying above the xy-plane oriented by an upward
  normal vector. A fluid with a constant density is
  flowing through the surface S according to the
  velocity field F(x,y,z) ltx,y,zgt. Find the rate
  of mass flow through S.