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Section 18.1 Contour Integrals
Section _______________
Parametric Curve
x = f t , y = g t
Orientation of the curve
Let’s go back to:
C : parametric curve in the complex plane defined by x = x t , y = y t , a ≤ t ≤ b t real
Now let z t = x t + iy t , a ≤ t ≤ b define C
If C is piecewise smooth, we will now call it a ________.
An integral of f z on C is called a ______________.
If is simple and
closed we use
C To reinforce the positive
orientation we use
The general
symbol is
To evaluate the countour integral
In the formula for
replace by
then
f z
z
dz =
for continuous and smooth
C
f z dz
f C
Math 241 – Rimmer
18.1 Contour Integrals
2
Evaluate
where is defined by 3 2 , 2 2
C
z dz C z t = t + it − ≤ t ≤
18.1 Contour Integrals
2
f z = z so f z t =
=
=
=
z t = 3 t + 2 it ⇒
dz = z ′ t dt ⇒
z ′ t =
dz =
2
2
C 2
z dz f z t z t dt
−
Math 241 – Rimmer
18.1 Contour Integrals
2 2
Evaluate
where is left half of the ellipse 1
36 4
from 2 to 2
C
x y
dz C
z i z i
= = −
2 2
We need to parametrize : 1
36 4
x y
C + =
x
y
≤ t ≤
z t =
z ′ t =
dz = z ′ t dt =
C
dz =
=
Orientation:
Continuing to view the complex function as a flow,
we can now calculate __________ and ________.
C : positively oriented simple closed curve
The _________ around C measures the tendency of the flow to rotate C
The ________ across is the difference between the rate at which
fluid enters and the rate at which fluid leaves the region bounded by.
C
C
18.1 Contour Integrals
________ on a ______ curve
and
for all on
f C
z C
( )
⇒ L =length of C
Bounding Theorem
Math 241 – Rimmer
18.1 Contour Integrals
