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Math 241 Chapter 15

Dr
...
Wyss-Gallifent

§15
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Define a vector field: Assigns a vector to each point in the plane or in 3-space
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Can represent a force field or fluid flow - both are useful
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Two important definitions
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∂
∂x

ˆ+ ∂y  + ∂z k so that gradient,
ı ∂ ˆ ∂ ˆ

¯
(a) The divergence ∇ · F = Mx + Ny + Pz gives the net fluid flow in/out of a point (very small
ball)
...

3
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In fact it’s a special kind of VF
...

There are two facts to note:
¯
¯ 0
¯ ¯
¯
(a) If F is conservative then ∇× F = ¯ and consequently if ∇× F = 0 then F is not conservative
...

¯
(b) If we have F we can tell if it’s conservative by the above method and we can find the potential
function too using the iterative method
...

§15
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If C is a curve and f gives the density at any point then we can define the line integral of f over/on
¯
C, denoted C f ds, as the total mass of C
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The result is independent of the parametrization
and the orientation
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¯
¯
2
...
The most basic way to
¯
over/on C, denoted C F r
b ¯
¯ r
¯
evaluate it is by parametrizing C as r(t) on [a, b] and then C F ·d¯ = a F (x(t), y(t), z(t))· r′ (t) dt
...
If −C is the same curve in the opposite direction
¯ r
¯ r
then −C F · d¯ = − C F · d¯
...

(b) The parametrization in that direction doesn’t matter
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We can write C M dx + N dy + P dz which
ˆ
ı
ˆ
r
means the same as C (M ˆ + N  + P k) · d¯
...

¯
Sample units: C in cm, F in g · cm/s (dynes) and the result in g · cm2 /s2 (ergs)
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3 The Fundamental Theorem of Line Integrals
¯
1
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2
...

¯ r
¯
(b) If F is conservative we say that the integral C F · d¯ is independent of path because only the
start and endpoints matter, not the path taken
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4 Green’s Theorem
1
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C
2
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(b) This is the same as

C

(M ˆ + N ) · d¯
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C
(d) If R contains holes then C is all the edges (made up of pieces) and the inner holes must have
clockwise orientation
...

§15
...
If Σ is a surface and f gives the density at any point then we can define the surface integral of f
¯
over/on Σ, denoted Σ f dS, as the total mass of Σ
...

Sample units: Σ in cm2 , f in g/cm2 and the result in g
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In this section I’ll do parametrizations where one variable depends on the other
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§15
...
Comment on oriented versus nonoriented surfaces and on fluid flow
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¯
2
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The most basic
way to evaluate it is by parametrizing Σ as r(u, v) on the region R in the uv-plane and then
¯
¯
¯ ¯
r
¯
¯
¯
F · n dS = ± R F (x(u, v), y(u, v), z(u, v)) · (¯u × rv ) dA where we use + if ru × rv points in
Σ
the same direction as the preferred orientation and − otherwise
...

3
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However If
¯
the surface is very very simple (like a horizontal plane) then we can find n directly and just do
¯
¯ ¯
F · n first and then it becomes an integral from §15
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§15
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Discuss induced orientations
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Thm: If Σ is a surface with oriented edge C then C F ·d¯ = Σ (∇×F )·¯ dS where the orientation
on Σ is induced from C
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3
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¯
(b) It’s interesting (not heavily used by us) that this can be used when integrating ∇ × F over
some Σ1 because we can replace Σ1 by another surface Σ2 provided they have the same
¯
¯ ¯
¯
boundary curve C via Σ1 (∇ × F ) · n dS = C F · n dS = Σ2 (∇ × F ) · n dS provided we’re
careful about orientations
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8 The Divergence Theorem (Gauss’ Theorem)
1
...

then Σ F · n dS =
D
2
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(b) If Σ is oriented inwards we just reverse, meaning put on a negative sign
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