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\title{Poor Polly}
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{\centering 18.02---Fall 1985}

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AMUSEMENT SET I (Due to the concept of reality).

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Note:  If, after reading this, you feel for Polly and her curvilinear
coordinates, honor her memory by saying ``Poor Polly!'' each time you
hear the professor say curvilinear coordinates.

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Once upon a time (1/T), pretty little Polly Nomial was strolling across
a field of vectors, when she came to the edge of a singularly large
matrix.

Now, Polly was convergent,and her mother had made it an absolute
condition that she must never enter such an array without her brackets
on.  Polly, however, who had changed her variables that morning and was
feeling particularly badly behaved, ignored this condition on the
grounds that it was insufficient and made her way in amongst the complex
elements.  

Rows and columns enveloped her on all sides.  Tangents approached her
surface.  She became tensor and tensor.  Quite suddenly, three branches
of a hyperbola touched her at a single point.  She oscillated violently,
lost all sense of directrix and went completely divergent.  As she
reached a turning point, she tripped over a square root which was
protruding from the erf and plunged headlong down a steep gradient.
When she was differentiated once more she found herself, apparently
alone, in a non-Euclidean space.

She was being watched, however.  That smooth operator, Curly Pi was
lurking inner product.  As his eyes devoured her curvilinear
coordinates, a singular expression crossed his face.  Was she still
convergent, he wondered.  He decided to integrate improperly at once.

Hearing a vulgar fraction behind her, Polly turned round and saw Curly
Pi approaching with his power series extrapolated.  She could see at
once, by his degenerate conic and his dissipative terms, that he was
bent on no good.

``Eureka,'' she gasped.

``Ho, ho,'' he said.  ``What a symmetric little Polynomial you are.  I
can see you're bubbling over with secs.''

``O, Sir,'' she protested, ``keep away from me.  I haven't got my
brackets on.''

``Calm yourself, my dear,'' said our suave operator, ``your fears are
purely imaginary.''

``I, I,'' she thought.  ``Perhaps he's homogenous then.''

``What order are you?'' the brute demanded.

``Seventeen,'' replied Polly.

Curly leered.  ``I suppose you've never been operated on yet,'' he
asked.

``Of course not,'' Polly cried indignantly.  ``I'm absolutely
convergent.''

``Come, come,'' said Curly.  ``Let's off to a decimal place I know and
I'll take you to the limit.''

``Never,'' gasped Polly.

``Exchlf,'' he swore, using the vilest oath he knew.  His patience was
gradually Coshing her over the coefficient with a log until she was
powerless, Curly removed her discontinuities.  He stared at her
significant places and began smoothing her points of inflexion.  Poor
Polly.  All was up.  She felt his hand tending to the asymptotic limit.
Her convergence would soon be gone forever.

There was no mercy, for Curly was a heavyside operator.  He integrated
by parts.  He integrated by partial fractions.  The complex beast even
went all the way around and did a counter integration.  What an
indignity.  To be multiply connected on her first integration.  Curly
went on operating until he was absolutely and completely orthogonal.

When Polly got home that evening, her mother noticed that she had been
truncated in several places.   But it was too late to differentiate now.
As the months went by, Polly increased monotonically.  Finally she
generated a small but pathological function which left surds all over
the place until she was driven to distraction.

The moral of our sad story is this:  If you want to keep your
expressions convergent, never allow them a single degree of freedom.

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