Ballad on the Divergence Theorem

A Story of Love and Integrals

by by Jeremy Wagner


The divergence theorem is employed in many branches of science. It states that the integral taken over a closed surface of the dot product of a vector-valued or tensor- valued function with the surface normal is equal to the integral of the divergence of that function taken over the volume bounded by that surface. In other words, it allows one to say something definite about the behavior of a function inside a volume based only on how it behaves on the surface surrounding that volume. Thus, an unknown interior can be characterized by a known exterior with the application of the divergence theorem. It makes, however, no predictions about the heights of folly reached by a young man in love. In fact, though the problem is still open, it is widely supposed that such a function is entirely unbounded.


There was a lovelorn mechanician,

facing Cupid’s dread perdition,

daily sending vain petition

to the object of his love.

Daily sending, daily writing,

everyday on dates inviting,

losing battles always fighting,

for her heart he could not move.


Said she, “I cannot know what lies

beneath a suitor’s tender guise.

His gentle hands and lovely eyes

conceal the contents of his heart.

The features on his surface tell

me nothing of his volume; sell

me not the looks I know so well,

love is not won by lover’s art.


“I will not be your Ariadne,

cast aside when once you’ve had me,

bride in mourning tatters clad. Be

sure of what I tell you now:

the poems that you write today,

the kisses on my cheeks you lay

mean nothing; inner truth will sway

me only; you must find out how.”


She gave the hand she held a squeeze,

and parted, leaving on his knees

our hero, failing to unfreeze

the heart of his beloved one.

This mechanician wandered home;

his thoughts began to widely roam

of how for once he should intone

that he loves her as he loves none.


His dinner went untouched that night,

he only had an appetite

for teasing out how yet he might

relieve his hopeless suit’s travail.

But leaden-lidded Somnus came

and called him by his very name.

When he or else his brother claim

a soul, no protest will avail.


So fast asleep, he fell to dreaming

(easier to bear than scheming

ways to show that more than seeming

was his promised love to her).

But ev’n in dreams his troubles goosed

him, hounded him, and still refused

him comfort.  Oh, a man abused

was he as no men ever were!

Then suddenly, an apparition

woke our sleeping mechanician,

following the grand tradition

of more famous bedside ghosts.

“Lagrange I was in life,” it said,

“since then I’ve wandered cold and dead,

with Green and Gauss (of whom you’ve read)

on Hades’ dark and gloomy coasts.


“You see, for lack of hobbies there,

we shades take up a special care

for mechanicians everywhere,

and rise to give advice when needed.

The cry goes ‘round the underworld

“A Mechanician Seeks His Girl!”

So back up onto Earth I’m hurled:

now let my words be duly heeded.


“I’ll tell you of a lonely lad

who loved a lady, but sat sad

and wept, because the poor boy had

too little to endear him.

The lady scorned the bright-eyed youth

for want of syllogistic proof

of love. Yet lacking her, in truth,

the boy had naught to cheer him.

But finding some mathematics book

he chanced to take a lucky look

and felt a shock! The whole earth shook!

His answer now was near him!

The proof was on the printed page,

the calculus would set the stage:

young lovers two would get engaged


Along his surface, head to toe,

he’d integrate his outward show

of love with normal dotted (so

the theorem always goes.)

This integral’s equivalence

to that of true love’s divergence

within him stood as fact. And hence,

his heart he could expose!

The loving surface features hold

a loving volume, theorem told.

And with a ring of solid gold

she married in a year him.

They revel, in regular intervals,

while lovingly solving their integrals

recalling that wonderful principle,



The ghost an end of speaking made,

“I’ll say no more, son, I’m afraid.

The answer is before you laid;

no longer can I linger here.

Two married mechanicians fight

o’er frame indifference every night -

And I must set the two aright!

My duty calls me towards their care.

“A phantom’s work is never done.

Take care that you might not be one!

Stay living while you’re able, son.”

And suddenly the ghost was gone.

Our mechanician, startled by

the haunting, breathed a heavy sigh.

The ghost’s advice he vowed he’d try

tomorrow at the rosy Dawn.


And when the rosy Dawn got up,

he’d long since drained his breakfast cup.

He couldn’t wait for her to sup,

so anxious was he on that day.

He tied his best and brightest tie;

he pulled his argyle socks up high;

to Bachelorhood he waved “Good-bye,”

(at least he hoped things went that way.)


To his Beloved’s door he came,

announced his suit in true love’s name.

He promised not the tired, same

old tale he brought, but novel proof!

With patience and with hand-drawn graphs

he taught her all his vector maths,

professed his love, but she with laughs

responded to the earnest youth.


“Dear boy, you are much denser than

a stone to think a tensor can

prove love. Would any censor ban

your mad address? I’d say he should!

“Depart, without my pity, now

You’ll never have my love, I vow,

Nor will my sire a fool allow

to take my hand in his for good.”


Our mechanician never saw

that girl again; he took in law-

ful wedlock someone else. Now “Pa”

four daughters call him and two sons.

And he, when met with apparitions

(‘specially those of mechanicians)

holds no tactful inhibitions:

their every word he duly shuns.