Late this afternoon I wondered

Late this afternoon I pondered

Late this afternoon I discovered

The double integral

With domains of 2 dimensions

With the chance of 2 integrations

With the prospect of volume calculations

The double integral

I can tell Carol the center of mass

I can tell Daryl the volume between surfaces

I can tell Cheryl the average value

The double integral

Early next morning I wondered

Early next morning I pondered

Early next morning I remembered

The double integral

There once was a family of variables who lived deep in the textbook of Calculus 3

The parents liked to be called F, and the children were x, y, and sometimes z

The children never liked to do anything together, even though they had so much in common

Because of this everything was done separately, giving respect to each child

One day at the school of FUNCTIONS

X and Y both decided to play in coloring Math and Arts

They were supposed to work together, but of course this didn’t happen

In coloring math and Arts the children picked a picture to color

And at the end of Coloring Math and Arts the children have to take derivatives

Derivatives is when all the children show how the picture has changed after coloring it

X and Y did not want to present together

So two derivatives had to be done so each child could feel special

X went first and Y did not want to participate at all

So when it was Y’s turn X didn’t want to play

Finally when both children did there derivatives a strange image was made

Together they were able to make the slope of the curves on the picture

The children were so happy about their accomplishments

And since then they have been making partial derivative history

3D surfaces are solid, not flat.

Look at a cone; it is like a pointy hat.

Even a cylinder looks like a can.

Of course, a sphere is like a ball man.

However, if it loses air it looks more like an ellipsoid.

Even an hourglass is shaped like a one sheet hyperboloid.

Nevertheless, a paraboloid is more like a bowl.

With a 2- sheet hyperboloid, you get two for one bowl goal.

3D surfaces can be so much fun.

Haiku Poems on Vectors

Two dimensional,

Sometimes three dimensional,

Vectors are unique.

Two points in the plane,

An initial point P and

Terminal point Q.

Length or magnitude,

The distance from P to Q,

Is easy to find.

Distance formula,

Square-root of a²+b²,

Length of the vector.

Compute the dot product,

Multiply corresponding

Components and add.

Calculus three has

Vector geometry and

Many more concepts.

Based on "How Will I Know" by Whitney Houston

https://www.youtube.com/watch?v=LzgaJpp5edM

There’s some things I know, all in Chapter Four-Teen

In the end you’ll see, how it will all come to be

Here comes the first one, just for several variables

It’s partial derivatives, and it measures rate of change

[Chorus #1]

What do you do? (With Partial Derivatives)

What do you do?

What do you do? (Given a nice Function)

What do you do?

You have to single out each variable

Then take their distinct derivative

Fx for “x” and fy for “y”

Well, it depends on what dimension I’m in

Treat the other variable as a constant

And if there are some given points (Keep Colm)

All you have to do is plug them in

Now you’re done what else could there be?

Oh, of course, higher order, partial derivatives

Said there’s no mistaking, it’s the second-order deriatives

[Chorus #2]

Ooh There’s more: What do you do? (for a Mixed Partial)

What do you do?

What do you do? (Relation to another)

What do you do?

Look at the first partial derivatives,

Take the rate of change of one nice variable

In accordance of the other variable

Find mixed partials in respect of both variables

And just remember mixed partials must match

If you try it and it fails you (replay)

Solving it out is all very easy

If you listen to me, credits to Mulcahy

Now you know too, all that this includes (3x)

What do you do?

What do you do?

What do you do?

[Chorus #3]

What do you do? (With Partial Derivatives)

What do you do?

What do you do? (Given a nice Function)

What do you do?

You have to single out each variable

Then take their distinct derivative

Fx for “x” and fy for “y”

Well, it depends on what dimension I’m in

Treat the other variable as a constant

And if there are some given points (Keep Colm)

Now all you do is continue through

This short process, and now you it too!

Volume of Solids.

Make them iterated then,

Fubini's Theorem.

What is the cross product?

Can you sketch part of the plane?

Is the equation of a 2-sheet hyperboloid x²-y²-z²=1?

Ummm,

Note to self, Calculus 3 is not a game!

Gradients and Derivatives and Integrals OH MY,

Gradients and Derivatives and Integrals OH MY,

Gradients and Derivatives and Integrals OH MYYYYYYY!

Is it a min?

Is it a max?

Can you find the derivative?

What is the directional derivative?

What is the partial derivative?

Ummm,

Reality Check, Calculus 3 is still not a game!

XY-Planes to XYZ-Planes,

XY Coordinates to Cylindrical Coordinates to Spherical Coordinates,

Integration of one variable to Integration in two variables,

Single Integrals to Double Integrals to Triple integrals.

Calculus 1 and Calculus 2 and Calculus 3, OH MY,

Calculus 1 and Calculus 2 and Calculus 3, OH MY,

Calculus 1 and Calculus 2 and Calculus 3, OH MYYYYYYY!

None of it is a game!

The Best of Both Worlds: Rendition of Hannah Montana lyrics

Oh yea, come on

You got the vector in the plane

Any length, two dimensions, two points

Yea when you’re a vector, it can get confusing

It’s really simple but no one ever discovers

In some ways vectors represent direction

But it is just a line

You get the best of both worlds

Three dimensions

Then you have three coordinates now

You get the best of both worlds

Switch it up, same concept

Then you know it is the best of both dimensions

Best of both dimensions

Vectors are in physics

See your applications

Living two dimensions is a little weird

But it’s cool because it is the same

You start with an initial point

But end with a terminal one

You get the best of both worlds

The length is magnitude

And is from P to Q

You get the best of worlds

Switch it up, same concept

I Got a Feelin’ Remix (Black-Eyed Peas)

Some Words of Wisdom

Some words of wisdom, I’ll share a good few theorems,

I’ll share a few good theorems

I’ll share a few good the-e-orems

Some definitions, remember definitions

Remember definitions

These really important definitions

Todays the day! Lets turn it up!

I got my Reals notes, lets learn it up!

Just attempt it, its not so hard

Contradiction! Round of applause!

I know that we’ll have a ball

Let’s start with the order properties of R

It says, that there of three of them

P is closed under addition and multiplication!

Law of Trichotemy! (shout)

Let’s move on, lets talk closed balls

With center x, and radius r

Represented, just like this Br(x)={y Î R | ||y-x||£ r}

Remember tis equation for other things

Lets do it, let’s do it, lets do it, lets do it

And do it, and do do it, (do it again)

Lets do it, let’s do it, lets do it, lets do it

Lets do it, let’s do it, lets do it, lets do it

Some words of wisdom, I’ll share a good few theorems,

I’ll share a few good theorems

I’ll share a few good the-e-orems

Some definitions, remember definitions

Remember definitions

These really important definitions

Another topic, lets talk about points

Interior, exterior, boundary points

All together, they make the set

But x can only be, one of them

Here it comes, Heine-Borel

It makes this easy, we’ll use it well

A set is compact if its bounded and closed

So what’s this set [0,1] (compact)! Round of applause

Now guess what we’re at the end

Another topic, sup and inf

They describe bounds greatest and least

They’re from a property called Complete!

Lets do it, let’s do it, lets do it, lets do it

And do it, and do do it, (do it again)

Lets do it, let’s do it, lets do it, lets do it

Lets do it, let’s do it, lets do it, lets do it

A Conversation between the Number Groups

Starring: The Rational and Irrational Numbers

Rational: Oh Real Numbers, how do I love thee? Let me count the ways…

Irrational: Dear brother, you cannot, do not lead my love astray.

Because someone like you cannot comprehend the power that she holds.

A number group so uncountable, so unbounded, and so bold.

Rational: But can you not see how different she is?

How unafraid?

She is my hero.

The way she carries herself so gracefully while including the number zero.

Irrational: Are you sure you really love her if that is the only quality you see?

If so, you can gladly love the Integers and leave the Real Numbers to me.

Integer: So am I just disposable to two number groups like you?

Because I am exclusive and never include you two?

I have a certain way of doing things so don’t sneer about me.

Zero, positive, and negative numbers are my friends.

Something you all will never be!

Rational: I thought I came here all alone to court my lady love.

Irrational: Your lady love?

Rational: My lady love!

Real: Must we always have this argument?

I’m simply sick of this.

You two are both a part of me!

This quarrel must be dismissed!

I refuse to choose between you two.

This decision is making us all tense.

Just know, I love you both the same.

In the heart of me, so dense.

Rational and Irrational: *blank stares*

Real: *triumphant exit*

Integer: *wondering why she was mentioned in the first place*

When you hear the word math, what comes to mind?

Most people say numbers, both composite and prime.

My hardest math course thus far has been Reals,

While taking this class, I struggled a great deal.

A’s and B’s in Reals took effort to earn,

But I came out smarter and this is what I learned.

The first thing you should know is how to write a proof,

Using all the rules and theorems to show something is true.

If a set has an upper bound, it has one below,

And it has an inf and supremum fa’sho!

If the complement of a set is open then it’s closed,

if a point is interior in the set it’s enclosed!

Exterior points are in the complement of a set,

And since they aren’t included they always are upset!

Cluster points are cool when talking about sets,

Every open ball contains a point that isn’t x!

Compact sets are, closed, bounded, and covered,

And we proved that in class, word to yo’ mother!

Sequences and limits were my least favorite topics,

The stressful subjects almost turned me into an alcoholic

If a limit of a sequence is zero it converges,

But if it’s limit is infinite then you know it diverges!

When proving convergence, there’s a few things you must know,

Like seek an epsilon that is greater than zero!

Even though this the end, there’s much more I could say,

But if I talked about it all we’d be here all day!

My relationship with Reals is made up of love and hate,

I am thankful for the knowledge plus my teacher was great.

Will I see these theorems again? Only time will tell.

But for now goodbye "Reals I" I bid you farewell!

Based on www.youtube.com/watch?v=gJLIiF15wjQ

by the Spice Girls

“Wanna Learn about Reals”

CHORUS:

Yo, I'll tell you what I know, what I really really know,
So tell me what you know , what you really really know (x2)

I gonna, I gonna, I gonna, I gonna, I gonna really really really teach you some reals.

Verse 1:

Reals is the study of functions and continuity.

Whole numbers, Integers, and rationals alike

Real numbers are these sets combined

Plus the irrationals like e and π.

CHORUS

Verse 2:
If you wanna learn about reals, a couple things you should understand

Make the reals a field, you can order it,
If you have bounded sets, you need sups and infs,
It sounds too easy, but that's the way it is.

Bridge:
What you think about that? now you know some reals,
Can you handle this math? Prove it for real
Does root 2 exist? Lets give it a try.
If the sequence diverges, then I’ll say goodbye.

CHORUS

Real…

Real analysis is equal to

Proofs measured by time spent,

And the names attached to them

Equal to the open balls of endless opportunities

Analysis of mainstream minds

In search of the challenge of real life

Solutions uncovered by any means necessary

Real time multiplied by answers

Only meant to detract from the

Real analysis of

Life.