Free Essay

No Way

In: People

Submitted By AMIDO
Words 4089
Pages 17
3

Contour integrals and Cauchy’s Theorem

3.1

Line integrals of complex functions

Our goal here will be to discuss integration of complex functions f (z) = u + iv, with particular regard to analytic functions. Of course, one way to think of integration is as antidifferentiation. But there is also the definite integral. For a function f (x) of a real variable x, we have the integral b f (x) dx. In case f (x) = u(x) + iv(x) is a complex-valued function of a a real variable x, the definite integral is the complex number obtained by b integrating the real and imaginary parts of f (x) separately, i.e. b b

u(x) dx + i a f (x) dx = a v(x) dx. For vector fields F = (P, Q) in the plane we have a the line integral

P dx+Q dy, where C is an oriented curve. In case P and
C

Q are complex-valued, in which case we call P dx + Q dy a complex 1-form, we again define the line integral by integrating the real and imaginary parts separately. Next we recall the basics of line integrals in the plane:
1. The vector field F = (P, Q) is a gradient vector field g, which we can write in terms of 1-forms as P dx + Q dy = dg, if and only if
C P dx+Q dy only depends on the endpoints of C, equivalently if and only if C P dx+Q dy = 0 for every closed curve C. If P dx+Q dy = dg, and C has endpoints z0 and z1 , then we have the formula dg = g(z1 ) − g(z0 ).

P dx + Q dy =
C

C

2. If D is a plane region with oriented boundary ∂D = C, then
P dx + Q dy =
C

D

∂Q ∂P

∂x
∂y

dxdy.

3. If D is a simply connected plane region, then F = (P, Q) is a gradient vector field g if and only if F satisfies the mixed partials condition
∂Q
∂P
=
.
∂x
∂y
(Recall that a region D is simply connected if every simple closed curve in
D is the boundary of a region contained in D. Thus a disk {z ∈ C : |z| < 1}
1

is simply connected, whereas a “ring” such as {z ∈ C : 1 < |z| < 2} is not.)
In case P dx + Q dy is a complex 1-form, all of the above still makes sense, and in particular Green’s theorem is still true.
We will be interested in the following integrals. Let dz = dx + idy, a complex 1-form (with P = 1 and Q = i), and let f (z) = u + iv. The expression f (z) dz = (u + iv)(dx + idy) = (u + iv) dx + (iu − v) dy
= (udx − vdy) + i(vdx + udy) is also a complex 1-form, of a very special type.

Then we can define

f (z) dz for any reasonable closed oriented curve C. If C is a parametrized
C

curve given by r(t), a ≤ t ≤ b, then we can view r (t) as a complex-valued curve, and then b f (r(t)) · r (t) dt,

f (z) dz = a C

where the indicated multiplication is multiplication of complex numbers
(and not the dot product). Another notation which is frequently used is the following. We denote a parametrized curve in the complex plane by z(t), a ≤ t ≤ b, and its derivative by z (t). Then b f (z) dz =
C

f (z(t))z (t) dt. a For example, let C be the curve parametrized by r(t) = t + 2t2 i, 0 ≤ t ≤ 1, and let f (z) = z 2 . Then
1

z 2 dz =
C

1

(t + 2t2 i)2 (1 + 4ti) dt =
0

(t2 − 4t4 + 4t3 i)(1 + 4ti) dt
0

1

[(t2 − 4t4 − 16t4 ) + i(4t3 + 4t3 − 16t5 )] dt

=
0
3

= t /3 − 4t5 + i(2t4 − 8t6 /3)]1 = −11/3 + (−2/3)i.
0
For another example, let let C be the unit circle, which can be efficiently parametrized as r(t) = eit = cos t + i sin t, 0 ≤ t ≤ 2π, and let f (z) = z .
¯
Then r (t) = − sin t + i cos t = i(cos t + i sin t) = ieit . d Note that this is what we would get by the usual calculation of eit . Then dt 2π
C



eit · ieit dt =

z dz =
¯
0

0



e−it · ieit dt =

i dt = 2πi.
0

2

One final point in this section: let f (z) = u + iv be any complex valued function. Then we can compute f , or equivalently df . This computation is important, among other reasons, because of the chain rule: if r(t) =
(x(t), y(t)) is a parametrized curve in the plane, then d f (r(t)) = dt f · r (t) =

∂f dx ∂f dy
+
.
∂x dt
∂y dt

d
(Here · means the dot product.) We can think of obtaining f (r(t)) roughly dt ∂f
∂f
by taking the formal definition df = dx + dy and dividing both sides
∂x
∂y by dt.
Of course we expect that df should have a particularly nice form if f (z) is analytic. In fact, for a general function f (z) = u + iv, we have df =

∂u
∂v
+i
∂x
∂x

dx +

∂u
∂v
+i
∂y
∂y

dy

and thus, if f (z) is analytic, df =
=

∂u
∂v
+i
∂x
∂x
∂u
∂v
+i
∂x
∂x

∂v
∂u
+i dy ∂x
∂x
∂u
∂v
+i idy =
∂x
∂x

dx + − dx +

∂u
∂v
+i
∂x
∂x

(dx + idy) = f (z) dz.

Hence: if f (z) is analytic, then df = f (z) dz and thus, if z(t) = (x(t), y(t)) is a parametrized curve, then d f (z(t)) = f (z(t))z (t) dt This is sometimes called the chain rule for analytic functions. For example, if α = a + bi is a complex number, then applying the chain rule to the analytic function f (z) = ez and z(t) = αt = at + (bt)i, we see that d αt e = αeαt . dt 3

3.2

Cauchy’s theorem

Suppose now that C is a simple closed curve which is the boundary ∂D of a f (z) dz.

region in C. We want to apply Green’s theorem to the integral
C

Working this out, since

f (z) dz = (u + iv)(dx + idy) = (u dx − v dy) + i(v dx + u dy), we see that


f (z) dz =
D

C

∂v
∂u

∂x ∂y

dA + i
D

∂u ∂v

∂x ∂y

dA.

Thus, the integrand is always zero if and only if the following equations hold:
∂v
∂u
=− ;
∂x
∂y

∂u
∂v
=
.
∂x
∂y

Of course, these are just the Cauchy-Riemann equations! This gives:
Theorem (Cauchy’s integral theorem): Let C be a simple closed curve which is the boundary ∂D of a region in C. Let f (z) be analytic in D. Then f (z) dz = 0.
C

Actually, there is a stronger result, which we shall prove in the next section: Theorem (Cauchy’s integral theorem 2): Let D be a simply connected region in C and let C be a closed curve (not necessarily simple) contained in D. Let f (z) be analytic in D. Then f (z) dz = 0.
C

Example: let D = C and let f (z) be the function z 2 + z + 1. Let C be the unit circle. Then as before we use the parametrization of the unit circle given by r(t) = eit , 0 ≤ t ≤ 2π, and r (t) = ieit . Thus

C



(e2it + eit + 1)ieit dt = i

f (z) dz =
0

(e3it + e2it + eit ) dt.
0

4

It is easy to check directly that this integral is 0, for example because terms

such as 0 cos 3t dt (or the same integral with cos 3t replaced by sin 3t or cos 2t, etc.) are all zero.
On the other hand, again with C the unit circle,

C

1 dz = z 2π



e−it ieit dt = i

0

dt = 2πi = 0.
0

The difference is that 1/z is analytic in the region C−{0} = {z ∈ C : z = 0}, but this region is not simply connected. (Why not?) f (z) dz = 0

Actually, the converse to Cauchy’s theorem is also true: if
C

for every closed curve in a region D (simply connected or not), then f (z) is analytic in D. We will see this later.

3.3

Antiderivatives

If D is a simply connected region, C is a curve contained in D, P , Q are de∂Q
∂P
fined in D and
=
, then the line integral
P dx+Q dy only depends
∂x
∂y
C
on the endpoints of C. However, if P dx + Q dy = dF , then

P dx + Q dy
C

only depends on the endpoints of C whether or not D is simply connected.
We see what this condition means in terms of complex function theory: Let f (z) = u + iv and suppose that f (z) dz = dF , where we write F in terms of its real and imaginary parts as F = U + iV . This says that
(u dx − v dy) + i(v dx + u dy) =

∂U
∂U
dx + dy + i
∂x
∂y

∂V
∂V
dx + dy .
∂x
∂y

Equating terms, this says that
∂V
∂U
=
∂x
∂y
∂U
∂V
v=−
=
.
∂y
∂x

u=

In particular, we see that F satisfies the Cauchy-Riemann equations, and its complex derivative is
F (z) =

∂U
∂V
+i
= u + iv = f (z).
∂x
∂x

5

We say that F (z) is a complex antiderivative for f (z), i.e. F (z) = f (z). In this case
∂u
∂2U
∂2V
∂v
=
=
=
;
2
∂x
∂x
∂x∂y
∂y
∂u
∂2V
∂2U
∂v
=−
=
=− .
2
∂y
∂y
∂x∂y
∂x
It follows that, if f (z) has a complex antiderivative, then f (z) satisfies the
Cauchy-Riemann equations: f (z) is necessarily analytic.
Thus we see:
Theorem: If the 1-form f (z) dz is of the form dF , or equivalently the vector field (u + iv, −v + iu) is a gradient vector field (U + iV ), then both
F (z) and f (z) are analytic, and F (z) is a complex antiderivative for f (z):
F (z) = f (z). Conversely, if F (z) is a complex antiderivative for f (z), then
F (z) and f (z) are analytic and f (z) dz = dF .
The theorem tells us a little more: Suppose that F (z) is a complex antiderivative for f (z), i.e. F (z) = f (z). If C has endpoints z0 and z1 , and is oriented so that z0 is the starting point and z1 the endpoint, then we have the formula f (z) dz = dF = F (z1 ) − F (z0 ).
C

C

For example, we have seen that, if C is the curve parametrized by r(t) = z 2 dz = −11/3+(−2/3)i. But z 3 /3

t+2t2 i, 0 ≤ t ≤ 1 and f (z) = z 2 , then
C

is clearly an antiderivative for z 2 , and C has starting point 0 and endpoint
1 + 2i. Hence z 2 dz = (1 + 2i)3 /3 − 0 = (1 + 6i − 12 − 8i)/3 = (−11 − 2i)/3,
C

which agrees with the previous calculation.
When does an analytic function have a complex antiderivative? From vector calculus, we know that f (z) dz = dF if and only if C f (z) dz only depends on the endpoints of C, if and only if C f (z) dz = 0 for every closed curve C. In particular, if C f (z) dz = 0 for every closed curve C then f (z) is analytic (converse to Cauchy’s theorem).
If f (z) is analytic in a simply connected region D, then the fact that f (z) dz = P dx + Q dy satisfies ∂Q/∂x = ∂P/∂y (here P and Q are complex valued) says that (P, Q) is a gradient vector field, or equivalently that f (z) dz = dF , in other words that f (z) has an antiderivative. Hence:
6

Theorem: Let D be a simply connected region and let f (z) be an analytic function in D. Then there exists a complex antiderivative F (z) for f (z).
Fixing a base point p0 ∈ D, a complex antiderivative F (z) for f (z) is given f (z) dz, where f (z) is any curve in D joining p0 to z.

by
C

As a consequence, we see that, if D is simply connected, f (z) is analytic f (z) dz = 0 (Cauchy’s integral

in D and C is a closed curve in D, then
C

theorem 2), since f (z) dz = dF , where F is a complex antiderivative for f (z), and hence f (z) dz =
C

dF = 0,
C

by the Fundamental Theorem for line integrals.
1
dz = 2πi = 0, where C z C is the unit circle. The antiderivative of 1/z is log z, and so the expected answer (viewing the unit circle as starting at 1 = e0 and ending at e2πi = 1 is log 1 − log 1. But log is not a single-valued function, and in fact as z = eit turns along the unit circle, the value of log changes by 2πi. So the correct answer is really log 1 − log 1, viewed as log e2πi − log e0 = 2πi − 0 = 2πi.
Of course, 1/z is analytic except at the origin, but {z ∈ C : z = 0} is not simply connected, and so 1/z need not have an antiderivative.
The real point, however, in the above example is something special about log z, or 1/z, but not the fact that 1/z fails to be defined at the origin. We could have looked at other negative powers of z, say z n where n is a negative integer less than −1, or in fact any integer = −1. In this case, z n has an antiderivative z n+1 /(n + 1), and so by the fundamental theorem for line
From this point of view, we can see why

z n dz = 0 for every closed curve C. To see this directly for the

integrals
C

case n = −2 and the unit circle C,


z −2 dz =
C



e−2it ieit dt = i

0

e−it dt = 0.

0

This calculation can be done somewhat differently as follows. Let r(t) = eαt , where α is a nonzero complex number. Then, by the chain rule for analytic functions, an antiderivative for the complex curve r(t) is checked to be s(t) =

eαt dt =

7

1 αt e . α Hence, b eαt dt = a 1 αb e − eαa . α z n dz = 0 for every integer n = −1, where

In general, we have seen that
C

C is a closed curve. To verify this for the case of the unit circle, we have



0

0

C

e(n+1)it dt =

enit ieit dt = i

z n dz =

i e(n+1)it i(n + 1)


0

i
1
= e2(n+1)πi − e0 =
(1 − 1) = 0. i(n + 1) n+1 Finally, returning to 1/z, a calculation shows that
1
dz = z x dx y dy
+
x2 + y 2 x2 + y 2

+i

−y dx x dy
+
x2 + y 2 x2 + y 2

.

The real part is the gradient of the function 1 ln(x2 + y 2 ) = d ln r. But the
2
imaginary part corresponds to the vector field
F=

−y x , x2 + y 2 x2 + y 2

,

which is a standard example of a vector field F for which Green’s theorem fails, because F is undefined at the origin. In fact, in terms of 1-forms, x dy
−y dx
+ 2
= d arg z = dθ.
2 + y2 x x + y2
In the next section, we will see how to systematically use the fact that the integral of 1/z dz around a closed curve enclosing the origin to get a formula for the value of an analytic function in terms of an integral.

3.4

Cauchy’s integral formula

Let C be a simple closed curve in C. Then C = ∂R for some region R (in other words, C is simply connected). If z0 is a point which does not lie on
C, we say that C encloses z0 if z0 ∈ R, and that C does not enclose z0 if z0 ∈ R. For example, if C is the unit circle, then C is the boundary of the
/
unit disk B = {z : |z| < 1}. Thus C encloses a point z0 if z0 lies inside the unit disk (|z0 | < 1), and C does not enclose z0 if z0 lies outside the unit disk
(|z0 | > 1). We always orient C by viewing it as ∂R and using the orientation coming from the statement of Green’s theorem.
8

Theorem (Cauchy’s integral formula): Let D be a simply connected region in C and let C be a simple closed curve contained in D. Let f (z) be analytic in D. Suppose that z0 is a point enclosed by C. Then f (z0 ) =

1
2πi

C

f (z) dz. z − z0

For example, if C is a circle of radius 5 about 0, then
2

C

ez dz = 2πie4 . z−2 2

But if C is instead the unit circle, then
C

ez dz = 0, as follows from z−2 Cauchy’s integral theorem.
Before we discuss the proof of Cauchy’s integral formula, let us look at the special case where f (z) is the constant function 1, C is the unit circle, and z0 = 0. The theorem says in this case that
1
2πi

1 = f (0) =

C

1 dz, z

as we have seen. In fact, the theorem is true for a circle of any radius: if
Cr is a circle of radius r centered at 0, then Cr can be parametrized by reit ,
0 ≤ t ≤ 2π. Then

Cr

1 dz = z 2π
0

1 ireit dt = i reit independent of r. The fact that
Cr



dt = 2πi,
0

1 dz is independent of r also follows z from Green’s theorem.
The general case is obtained from this special case as follows. Let C =
∂R, with R ⊆ D since D is simply connected. We know that C encloses z0 , which says that z0 ∈ R. Let Cr be a circle of radius r with center z0 . If r is small enough, Cr will be contained in R, as will the ball Br of radius r with center z0 . Let Rr be the region obtained by deleting Br from R.
Then ∂Rr = C − Cr , where this is to be understood as saying that the boundary of Rr has two pieces: one is C with the usual orientation coming from the fact that C is the boundary of R, and the other is Cr with the clockwise orientation, which we record by putting a minus sign in front
9

of Cr . Now z0 does not lie in Rr , so we can apply Green’s theorem to the function f (z)/(z − z0 ) which is analytic in D except at z0 and hence in Rr :

∂Rr

f (z) dz = 0. z − z0

But we have seen that ∂Rr = C − Cr , so this says that

C

f (z) dz − z − z0

Cr

f (z) dz = 0, z − z0

or in other words that

C

f (z) dz = z − z0

Cr

f (z) dz. z − z0

Now suppose that r is small, so that f (z) is approximately equal to f (z0 ) f (z) on Cr . Then the second integral dz is approximately equal to z − z0
Cr

Cr

f (z0 ) dz = f (z0 ) z − z0

Cr

1 dz, z − z0

where Cr is a circle of radius r centered at z0 . Thus we can parametrize Cr by z0 + reit , 0 ≤ t ≤ 2π, and

Cr

1 dz = z − z0


0

as before. Thus f (z0 )
Cr

1 ireit dt = i reit 2π

dt = 2πi,
0

1 dz = 2πif (z0 ), z − z0

f (z) dz is approximately equal to 2πif (z0 ). In
Cr z − z0
C
f (z) fact, this becomes an equality in the limit as r → 0. But dz is
C z − z0 independent of r, and so in fact and so

f (z) dz = z − z0

C

f (z) dz = 2πif (z0 ). z − z0

Dividing through by 2πi gives Cauchy’s formula.
The main theoretical application of Cauchy’s theorem is to think of the point z0 as a variable point inside of the region R such that C = ∂R; note
10

that the z in the formula is a dummy variable. Thus we could equally well write: 1 f (w) f (z) = dw, 2πi C w − z for all z enclosed by C. This description of the analytic function f (z) by an integral depending only on its values on the boundary curve of R turns out to have many very surprising consequences. For example, it turns out that an analytic function actually has derivatives of all orders, not just first derivatives, which is very unlike the situation for functions of a real variable.
In fact, every analytic function can be expressed as a power series. This fact can be seen by rewriting Cauchy’s formula above as f (z) =

1
2πi

C

f (w) dw, w(1 − z/w)

1 as a geometric series. The fact that every
1 − z/w analytic function is given by a convergent power series is yet another way of characterizing analytic functions. and then expanding

3.5

Homework

1. Let f (z) = x2 + iy 2 . Evaluate

f (z) dz, where (a) C is the straight
C

line joining 1 to 2 + i; (b) C is the curve (1 + t) + t2 i, 0 ≤ t ≤ 1. Are the results the same? Why or why not might you expect this?
2. Let α = c + di be a complex number. Verify directly that d αt e = αeαt . dt 3. Let D be a region in C and let u(x, y) be a real-valued function on D.
We seek another real-valued function v(x, y) such that f (z) = u + iv is analytic, i.e. satisfies the Cauchy-Riemann equations. Equivalently, we want to find a function v such that
∂u
∂v
∂u
∂v
=−
and
=
,
∂x
∂y
∂y
∂x
∂u ∂u
,
. Show that
∂y ∂x
F satisfies the mixed partials condition exactly when u is harmonic.
Conclude that, if D is simply connected, then F is a gradient vector field v and hence that u is the real part of an analytic function. which says that

v is the vector field F =

11



4. Let C be a circle centered at 4+i of radius 1. Without any calculation,
1
explain why dz = 0.
C z
5. Let C be the curve defined parametrically as follows: z(t) = t(1 − t)et + [cos(2πt3 )]i,

0 ≤ t ≤ 1.

2

ez dz. Be sure to explain your reasoning!

Evaluate the integral
C

6. Let D be a simply connected region in C and let C be a simple closed curve contained in D. Let f (z) be analytic in D. Suppose that z0 is a
1
f (z) point which is not enclosed by C. What is dz? 2πi C z − z0 ez dz, where C is a circle of
C z+1 radius 4 centered at the origin (and oriented counterclockwise).

7. Use Cauchy’s formula to evaluate

8. Let C be the unit circle centered at 0 in the complex plane C and oriented counterclockwise. Evaluate each of the following integrals, and be sure that you can justify your answer by a calculation or a clear and concise explanation.
2

z 4 dz;

(a)

(b)

C

(d)
C

z 2 − 1/3 dz; z+5

(e)

e−z dz; (c) z −5 dz; z − i/2
C
C
1
e−2z dz; (f) dz. 2
C (12z − 5)
C 3z + 2

9. Let D be a simply connected region in C and let C be a simple closed curve contained in D. Let f (z) be analytic in D. Suppose that z0 is a point enclosed by C.
(a) By the usual formulas, show that d dz

f (z) z − z0

=

f (z) f (z)

. z − z0 (z − z0 )2

(b) By using the fact that the line integral of a complex function with an antiderivative is zero and the above, conclude that

C

f (z) dz = z − z0
12

C

f (z) dz. (z − z0 )2

(c) Now apply Cauchy’s formula to conclude that f (z0 ) =

1
2πi

13

C

f (z) dz. (z − z0 )2…...

Similar Documents

Premium Essay

And but of the Way

...old clients and to their new prospects as well through email. By sending their company profile, their company is made known to people. The company also has their own Multiply site: http://majestictravel.multiply.com. Their Multiply site contains almost all the information a customer would like to know about the company and the services that they offer. They also send and give away brochures that contain information about their different packages. The company also uses direct marketing. Majestic Travel gives away flyers. Flyers are a piece of printed matter that is widely distributed. Majestic Travel’s flyers include all the information about their products and services—different tours and destinations that the company offers. Another way of promoting their services is by word-of-mouth. Since there are a number of customers who are satisfied with Majestic Travel’s services, these customers refer this travel agency to their family and friends. Majestic Travel Corporation also offers different international and local packages--hotel and resort accommodations, air and ground transportation, car rental needs, and tour packages. ----------------------- Figure 1 – Official Logo of Majestic Travel Corporation...

Words: 1768 - Pages: 8

Premium Essay

Hp Way

...1) ‘HP Way ‘is a set of primary values that define how workforce and the company are to perform. These values have become the foundation of the “HP Way”. It is an objective-oriented philosophy, where each constituency can share the same principles and work toward a common goal. The primary values of the HP Way are trust and respect for individuals; high level of achievement and contribution; conducting business with uncompromising integrity; common objectives through teamwork; innovation and flexibility. The company’s founder put focus not only on to make creative products but also to create supportive corporate culture. The Company has many personnel policies and internal structures which support these values and each policy complement and support each other. The activities of HP employees are guided by a comprehensive system of management by objectives (MBO). The greatest advantage of MBO is that objectives are goals, not specific tasks handed out by management. Goals can be achieved in multiple ways and it is expected from employees to find their own best ways to meet these goals. Job autonomy encourages creativity in the workplace and increase the sense of accountability to employees. From the beginning HP instituted participative management style to foster teamwork, trust, openness and cooperation. Teamwork is practiced within divisions between R & D, manufacturing, marketing, and finance. Through participative decision-making HP......

Words: 3423 - Pages: 14

Premium Essay

Ways to

...Technology in Banking Insight and Foresight Institute for Development and Research in Banking Technology (Established by Reserve Bank of India) Foreword The Indian banking industry, almost in keeping with the deep entrepreneurial approach of the country s business, has come a long way. This report is an effort to capture some exemplary initiatives and developments so far as well as discuss the emerging trends. The insights and understanding of the technology trends and ground-level work being done by the banks has been culled from the nominations received from banks for the IDRBT Banking Technology Excellence Awards 2010. The transformation of Indian banks in the last decade has been phenomenal from local branch banking to global presence and anywhere-anytime banking. Most of the regular banking transactions can today be carried out from mobile phones. Sustained reforms and information technology (IT) have played a pivotal role since the initiation of the second phase of reforms post 1998. The benefits of technology such as scale, speed and low error rate are also reflecting in the performance, productivity and profitability of banks, which have improved tremendously in the past decade. Regulatory initiatives from the Central Bank have also played a large role in the banking sector. Robust technology-enabled organizations have now become the mainstay of the industry. Initiatives such as electronic clearing service (ECS), national electronic funds transfer (NEFT),......

Words: 24716 - Pages: 99

Free Essay

No Way

...Feminism can be roughly defined as a movement that seeks to enhance the quality of women’s lives by impacting the norms and moves of a society based on male dominance and subsequent female subordination. The means of change in the work place, politically, and domestically. Women have come a long way since the 19th century. Women have been trying to prove to the male dominant world that they are equal. They can perform and complete any tasks equal, or in some cases better than man. Feminism has changed the definition of men in many ways. Women in the work place have transposed dramatically since the 19th and mid 20th century. Even if women had any education in the 19th century they were not allow to manifest any of it. It just was not proper for women to give any signs of intelligence and a brain of their own. They were to prepare themselves to become wives and mothers, which were the extent of their entire lives. In the early and mid 20th century some women were starting to be brave and take a stand for themselves. The beginnings of feminism were starting to take its massive role in society. More and more women were getting educated and looking for employment opportunities that had power. Men no longer can be in control of everything. Men in the work place started to feel impotent. But women fed off each other and gave each other strength. They were not looking for just the secretarial jobs; they were taking some men’s jobs and being good at it. They were becoming police......

Words: 451 - Pages: 2

Premium Essay

Can Be Done This Way or That Way

...Either you can do it this way or that way “Learn from your own mistakes, or find a master to guide you along”. Everyone in this world is granted only one life and the decisions we make have a great impact on our lives. But the best lesson we learn is from a bad decision we make. At that crucial point in time some people decide to follow the path of learning through their own experiences; while others prefer that a guide or a teacher could help them live a smooth life. Time and experience can be excellent teachers when you actually learn a lesson from your poor decisions. Experience comes from our way of living, understanding and the adjustments we make. It also comes from suffering, agony and the ordeals we are afflicted by. As one of the famous writer puts it together: “Good judgment comes from experience and experience comes from poor judgment.” Growth starts as soon as you recognize your mistake and the way to prevent it from happening again. Every human being is bound to make mistakes in life, this is normal; but how you learn from them is really the important factor. The only way to prevent oneself from making a mistake the second time is to learn. If you don’t, you will be making that same error again and again until you are forced to learn. I’ll give you an example of my own life. I started playing Table Tennis in 2008. Back then our school had recently bought the equipment for this game and there were only few good players of this game in the whole school.......

Words: 626 - Pages: 3

Premium Essay

The Toyota Way

...The Toyota Way I have learned about “the Toyota Way”, which leads Toyota to success. “The Toyota way” is the core value and culture of Toyota. I learned a lot from “the Toyota way”. The following are 5 basic principles of “the Toyota Way”: First, challenge. Challenge means that a corporation should always challenge itself in terms of challenging the process. A company should not say that we are doing very well and we need not make any improvements. Unless a company always challenges the process, it cannot have constant competitive. Secondly, improvement An organization should keep improving itself. As the CEO of Toyota said, there is no perfect car; we are improving our cars base on customer’s needs. We should never think we are the best; we should keep identifying where we can still work on and improve. Consistent improvement is one of the traits of a great company. Thirdly, go and see All managers and employees should walk around to observe and find problems. This is basically a concept of managing by walking around, however, Toyota requires that all the employees have the ability to identify the potential problems. This ability needs a lot of experiences and knowledge. Fourthly, respect We should respect our employees and customers, as well as partners. We respect others and then they are going to respect us. Then we can a very good relationship with employees, customers and partners, which are very good for us. Fifthly, teamwork Teamwork is very......

Words: 300 - Pages: 2

Premium Essay

On the Way

...if he doesn’t matter that he is just another dollar sign. The doctor makes it his point at all time to make it known that he is a doctor in the hospital and should be treated with much more respect than with the mediocrity of a patient. He slowly comes to terms with the fact that he may have been mistreating people for a long period of time. As the movie begins rock 'n' roll music is playing in his operating room while literally holding patients hearts in his hands. He leads life being the father of two sons and a husband to a loving wife. Dr. Hurt expresses his feeling openly when he tells his interns that personal feelings have nothing to do with science of medicine. When the tables turn he discovers that he really doesn’t feel that way. In the beginning his ailments starts as a small, persistent cough. On one occasion he begins coughing and in the end coughs up blood. He goes to an otolaryngologist and discovers that there is a tumor and it doesn't work, he may need surgery. In that case, it's impossible to predict how his vocal cords will respond. He could lose the power of speech. This is devastating news, the doctor is left in a state of shock and disbelief as his treatment progresses, and he doesn't like how his own hospital treats him, as he wastes time in waiting rooms, tangles with the bureaucracy and is repelled by his doctor’s frigid bedside manner. For the first time, he grows close to a patient who has been diagnosed with a brain tumor, on a daily......

Words: 737 - Pages: 3

Free Essay

The Navajo Way

...The Navajo Way Lucious Davis ANT:101 Introduction to Cultural Anthropology Instructor: Jessie Cohen March 10, 2012 The story of the Navajo is one that is filled with triumph, tragedy, and hope. The Navajo are a pastoral people originating in North America. The culture of the Navajo’s is a one filled with traditions that have been passed down from one generation to the next. Their culture is what defines them and it is a major factor in the way they live their lives- including their social organization, beliefs and the way they heal their sick. Background Few cultures have left their imprint on North America like the Navajo. With over 300,000 members, the Navajo are the largest federally recognized tribe in the Unites States. Originating in northwest Canada and eastern Alaska, the Navajo, along with other groups like the Apache migrated to their more commonly known territory- the southwestern United States. Accounts have dated the occupation of the southwestern United States by the Navajo to stretch as far back as 1400. Throughout history, the Navajo have expanded their territory through raiding and commerce, now are mostly confined to a small area that is called the “Four Corners”. This is the area of the southwest United States that is comprised of: Arizona, Colorado, New Mexico, and Utah. The Navajo can best be described as seminomadic- they tend to move according to the seasons. Jett (1978) stated that the actual movement patterns can vary greatly from...

Words: 2448 - Pages: 10

Free Essay

Mckinsey Ways

...line on changes well in advance of the meeting. * PREWIRE EVERYTHING To ensure no dispute during the meeting, consultants should let all the relevant players in the client organization know what they found in private before they hold a presentation or progress review. 2. DISPLAYING DATA WITH CHARTS * Keep it simple — one message per chart Following principle of simplicity, the more complex chart is less effective. Chart is used to convey the primary information. A chart should include a good lead which can express the key point of the chart in one simple sentence and a source attribution which would help people review the data at future time. * Use a waterfall chart to show the flow The waterfall chart is an outstanding way to illustrate quantitative flows. Through mixing the negative and positive item, it can depict vividly subtotal of cash flow and net income. 3. MANAGING INTERNAL COMMUNICATIONS * Keep the information flowing Information flowing can help your teammates understand how their work is contributing to the final goal, how their efforts are worthwhile. In addition, good information flow can help you spot emerging problems (or opportunities) faster because people in different areas will acquire some useful information which you don’t realize. There are two basic methods of internal communication: the message, such as voice mail, e-mail, or memo, and the meeting. Team meetings give a chance to exchange information in all directions......

Words: 779 - Pages: 4

Premium Essay

There Is No Way Like the American Way

...There is no way like the American way What does the slogan actually mean? An American would probably say that the American way of living is the best way of living and with high standards. But is it how the Americans really live? Fifty years ago, that might have been the answer from a white American, but for a colored American, an African American for instance, it is a whole other story. Back then, around the sixties, there weren’t much you could call laws that supported the blacks. In fact, they didn’t really have any rights at all. They weren’t allowed to sit on the same benches as the whites, nor the same buses as them. Women weren’t even allowed to vote! The females’ rights to vote happened in the twenties, but only for white women. Thinking that black women aren’t worthy enough to vote is absurd. That is what a country is about though, right? It is the people who are to decide how they want their country to be formed - All of them. Not just men. Not just the whites. They can’t just decide what other people want. The injustice does not stop there. Even as a child they grew up, believing that the black Americans were less worth than the whites. At school the black and white children went to different schools, as if they were different from each other. White children, and youth too, would refuse to go to the same school as “negroes”. Why and for what reason? Were the white youngsters smarter than the black? I doubt it. Did the blacks have more “problems”......

Words: 532 - Pages: 3

Free Essay

The Cowgirl Way

... Victoria Ivey Professor Preston English Composition I 28 Sep. 2015 The Life of a World Class Cowgirl “The Cowgirl Way,” an article written by Lonn Taylor and published by Texas Monthly (http://www.texasmonthly.com/the-culture/the-cowgirl-way/), tells the story of a young woman who grew up to become a great rodeo icon for females around the world. Barbara Inez “Tadpole” Barnes, “Tad” for short grew up to be a great trick rider and all-around cowgirl. In this article, Lonn Taylor quotes Dan Fox’s writing in the rodeo trade paper Hoofs and Horns, “Tad has always been admired by everyone who had the good fortune to meet her. She is considered the world’s greatest woman rider.” Taylor relates incidences in Tad’s background, that shaped Tad’s ambition and aspirations that helped established her as an icon for women, not only in the world of rodeo but also for women in all walks of life in that era. In paragraph one Lonn Taylor writes, “She was born in Cody, Nebraska in 1902, she later settled in Fort Worth and considered herself a Texan.” When reading this article, there is an understanding of Tad’s life and how it was shaped from the different places she explored as she competed in rodeo events. Paragraph five of the article shows us how she started out in the rodeo circuit. Taylor states, “Tad Lucas was a cowgirl’s cowgirl. She left Nebraska at the age of sixteen and joined “California” Frank Hafley’s Wild West Show…. First [appearing] on the Western rodeo......

Words: 957 - Pages: 4

Premium Essay

A Long Way

...A Long Way Gone Essay Matthew Morgan Prof. Carey “On Democracy” Due: 02/27/08 For the “Everybody Reads” assignment I choose to attend the Central library book group discussion. When I first got there I was really surprised because I thought it was going to be a bigger event than what it was. There was only about 10 people total, and 5 of us were students who were there for this exact assignment. It was a really interesting discussion because half of the people that attended were my age and the other half was about two generations older, so there was a very diverse pool of perspectives and opinions. But because there was a large generation gap it was a bit more difficult for me to share my views, so I mainly listened and observed other people’s thoughts. The discussion itself was very helpful because of the different views people had about the memoir. One of the themes of A Long Way Gone that we discussed was the importance of hope. We mainly talked about how this theme was not constant throughout the memoir and that it changed with time. For example one person brought up how at first Ishmael’s only motivator was the hope of his parents being alive, then when he realized that he would never be reunited with them he had lost his hope. It was only when he remembered what his father had said about a person only lives if they have something to live for which gave him his hope back. As far as themes that’s really the only one that we discussed, but we did discuss a......

Words: 1027 - Pages: 5

Premium Essay

The Way It Is

...We confirm our Management Agreement whereby I appoint you to represent me as my sole and exclusive manager under the provisions of this agreement as follows:- 1.  This agreement relates only to my professional activities and confers the right to represent me as a solo artist or as a member of a group or duo. 2.  Both you and I warrant by our respective signatures that there are no existing restrictions that prevent either party from entering into this agreement or performing any of our obligations. 3.  This agreement relates solely to activities as a musical artist and in no way confers the right to represent or hold yourself as representing me in any other field of entertainment or area of work not connected with the music industry without prior written consent. 4.  During the Term of this agreement defined below you shall have the following obligations:  (a)  To use your best endeavours to promote and develop my career as a music artist and provide me with regular reports on your work.  (b)  To ensure all monies due to me are promptly collected and remitted directly to me by the parties from which they are due.  (c)  To refer all enquiries connected with our work in areas which you are not permitted to act directly to me.  (d)  You shall not have the right to assign or transfer obligations to any other person or company without prior written consent, any such act shall immediately and retrospectively terminate your appointment and this agreement. 5....

Words: 847 - Pages: 4

Free Essay

The Way of the World

...Literature Lenka Drbalová Comedy of Manners: William Congreve and Oscar Wilde Bachelor’s Diploma Thesis Supervisor: prof. Mgr. Milada Franková, CSc., M.A. 2014 I declare that I have worked on this thesis independently, using only the primary and secondary sources listed in the bibliography. …………………………………………….. Author’s signature Acknowledgement I would like to thank prof. Mgr Franková , CSc., M.A. and PhDr. Věra Pálenská, CSc. for their guidance, advice and kind encouragement. Table of Contents Preface ...............................................................................................2 Introduction ......................................................................................3 Chapter I – The Way of the World 1.1 In General ..................................................................................8 1.2 True Wit and False Wit ............................................................9 1.3 Courtship and Love .................................................................14 1.4 Invention vs. Reality ................................................................18 Chapter II – The Importance of Being Earnest 2.1 In General ................................................................................22 2.2 True Wit and False Wit ..........................................................23 2.3 Courtship and Love ................................................................28 2.4 Invention vs.......

Words: 13764 - Pages: 56

Free Essay

American Way

...treasure: life, liberty, and freedom. This treasure was and still is the American Dream. Now people from all over the world come to America in search of the same Dream; some even die trying. People were not as materialistic as people are now; they just wanted happiness. As time passed, people became more materialistic and began to take for granted what they were born with. However, the “American Dream” hardly ever turns out like any individuals have anticipated. America is often considered as the “best country” in the world, but behind this façade, many people struggle daily to earn enough money to survive. America has had its times of despair and advances but through it all we have learned new ways, cultures, and overall advancements in life. Throughout the American Ways book we were given to read and analyze, there were many things about American life that not only related to now but also showed us lessons learned and conflicts repeated through time. To myself and maybe others, the pieces presented to me in class had the ability to bring me into another world. In each piece, I was almost warped to a different time of America and had the ability to see many things. One thing I would like to compare is not only the trillion changes and differences we have made from those times to now but the common sense of diversity that has lurked our environment for decades and decades. Diversity can be broken down in some pieces of life including heritage, religion, race,......

Words: 1179 - Pages: 5