- Definition of Derivatives using big O notation
- Line Tangent to a Curve
- Exponential Function
- Tangent Lines as Linear Approximations
- Some Useful Trigonometric Limits
- Differentiating Trigonometric Functions
- Inverse Function Theorem
- The Natural Logarithm, the inverse function to the exponential mapping.
- Optimization: its motivation, the First Derivative Test, an example optimizing a rectangle’s area
- Implicit Differentiation
- Curve Sketching
- Applications
- Techniques

- The Antiderivative part 1, part 2
- Finite Series
- Riemann Summation a first step towards definite integration
- Example Riemann Sum working with
- Fundamental Theorem of Calculus
- Properties of the Integral
- Not all functions are (Riemann) integrable!
- The Natural Logarithm Revisited!
- Integration Techniques:
- Integration Technique #1: Substitution
- Integration Technique #2: Integration By Parts
- Differentiation Under the Integral Sign
- Applications of Integrals
- Calculating Arc-Length of Curves
- Calculating the Area for a Surface of Revolution (Part 1) when we revolve about the axis
- Optimization
- Calculus isnâ€™t useless: When Velociraptors Attack, calculus solving matters of life and death!
- Thinking “Infinitesimally”

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I got hired to program in Clojure, a peculiar member of the Java family and distant relation to the LISP clan.

Nevertheless, I am still writing mathematical notes, and will try to post them when time allows. Do not expect them to be as regular as they used to be!

I am going to finish up vector calculus, beginning with line integrals. After vector calculus, linear algebra will commence!

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(1)

The first thing we do: consider the rectangle and form a partition of into segments and into segments. This gives us a mesh of rectangles as specified by the following diagram:

Observe the area of is

(2)

We can approximate the volume of by a sort of Riemann sum, picking points in and taking

(3)

However, as with Riemann sums, we recover the exact volume when we take the limits :

(4)

if the limit exists.

**2. Definition.** We define the **“Double Integral”** of over the integral as

(5)

if this limit exists.

Note this sum is called a *double Riemann sum*, just as a for the single integral we had a *Riemann Sum*.

**3. Properties of Integrals.** We won’t prove, but note, there are three important properties double integrals satisfy. Two can be grouped together as linearity:

(6)

and

(7)

where and are continuous on , and is a constant.

If for all , then

(8)

**4. Problem:** How do we compute ?

**5. Fubini’s Theorem.** If is continuous on the integral , then

(9)

We won’t prove it (that’s what real analysis discusses!), but the parenthetic terms are functions of a single variable and describe the area of a “sheet”. By stacking “sheets” we can compute the volume of the region.

**6.** This is fine for rectangular regions, but over arbitrary regions what do we do? We bound the region by a pair of curves :

In these situations, we write

(10)

Please note the order of integration, and corresponding boundary of integration!

**7.** We can likewise bound the region by curves , for example:

The trick lies with writing these integrals as

(11)

Again, note the order of integration!

Example 1.Lets consider the domain

(12)

Evaluate the integral

(13)

Solution:\quad\ignorespaces We first doodle this domain, shade it in red:

Now we use Fubini’s theorem writing

(14)

Observe the order of integration, and the bounds of integration are determined by specifying and for the quantity. Similarly, since , we determine the bounds of integration for the integral.Using linearity, we can write

(15)

where

(16)

and

(17)

Now we can evaluate each of these separately.We see for , there are no expressions, so we have

(18)

Performing this inner integral

(19)

We plug this back in:

(20)

This can be solved quickly as

(21)

Thus

(22)

Moreover this implies .

What to do about ? Recall

We first perform the inner integral

(23)

We plug this back in

(24)

We can calculate this out directly as

(25)

But we forgot the coefficient in front of the integral! Thus our integral is

(26)

giving us our final result

(27)

which concludes our example.

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**1.** I’m going to be posting my double integral notes, but I’d like to discuss my strategy when writing “more rigorous” mathematics. Modern mathematics consists of definitions, theorems, and proofs…so I’ll discuss the idiosyncrasies of each. (I’ll probably revise this several times…)

**2. Definitions.** A definition introduces a new gadget (e.g., vector spaces), some new structure (e.g., a linear operator, a norm, etc.), or some new property (e.g., invertibility).

So the typical definition would look like the following:

An

“Object”consists of“Stuff”equipped with“Structure”such that these equations — called“Properties”(orAxioms) — hold.

A typological remark: sometimes axioms are given in bold. For example, consider the following snippet of a definition

…satisfying the following properties:

Disjoint Union:[snip]

Sewing:[snip]

Normalization:[snip]

This has the added bonus of allowing us to write “By sewing, this gadget then transforms Equation (blah) into Equation (snort).”

Some people prefer giving as few axioms as possible, then proving the rest. This seems pathological to me, and confuses the reader. I prefer being frank and honest, telling the reader “This gadget has the following useful properties” and in a remark note when some imply others.

**3. Theorems.** Euclid had a systematic manner of presenting theorems, as Proclus noted.

Knuth suggests (in rule 11) we should “try to state things twice, in complementary ways…” which should be done with theorems as well. Personally, I think writing the theorem without symbols — if possible — would be best. This would correspond to Euclid’s “Enunciation”.

After the theorem statement, we should note the proof strategy and/or key moments of the proof.

Continuous functions on a closed interval are uniformly continuous.

Theorem.If is continuous on every , then is uniformly continuous on .We will prove this directly.

Proof.

Let be a continuous function on the closed interval , then… [the rest of the proof is omitted].

We should make the theorem be precise, boring, and cookie-cutter in its format. But we should say, either before or after, what it really says in a frank way: “Continuous functions on a closed interval are uniformly continuous.”

**4. Proofs.** We should begin introducing all the necessary variables and quantities. Constantly ask yourself “What does the reader know?” Don’t be afraid to reiterate what has been done “We’ve just shown (blah), but we need to prove (oink).”

**5. Examples.** Examples should be worked out exercises. So it consists of two parts: (1) statement of the problem, and (2) its solution.

In physics, it consists of 4 parts, which we will not discuss now [google “Hugh and Young IDEE”].

Now the problem lies in *how do we pick good examples?* It’s a very hard thing to do. It should be clean, i.e. free of excessive calculations; it should be inviting, interesting to the reader; and it should be challenging, i.e. not as simple as it seems.

Harrington has Ten Commandments for mathematical writing, which I more or less agree with, they are:

**Rule 1. Organize in segments.** Harrington’s “segments” are what I call chunks (in my blog entries, they are the numbered collection of paragraphs, possibly with a title of some sort). It can be read comfortably from beginning to end without pausing.

So what’s a good segment? An example, a definition, a theorem and its proof. These are typical. What else? Well, motivating a concept, reflections, etc.

**Rule 2. Write segments linearly.** We should organize each chunk to do one thing, and only one thing. Give us an example, or a result and its proof, or a concept.

The chunk should flow linearly, making it easy to read. How can we accomplish this? As Halmos once said: organize, organize, organize. Write up an outline, with only the highlights written down. For example:

1. Definition.Continuity…

Example 1.Polynomials.

Example 2.Trig Functions.

Non-Example 3.Step function at the jump.

2. Definition.Derivative…

Example 1.Polynomials.

Example 2.Trig Functions.

Non-Example 3.Absolute value at zero.

3. Theorem.Chain rule.

Example 1.Apply it to .

4. Theorem.Product rule.

Example 1.Apply it inductively to to recover the product rule.

**Rule 3. Consider a hierarchical development.** Think about how the segments depend on each other. This affects how we organize our outline.

Don’t be afraid to reorganize.

**Rule 4. Use consistent notation and nomenclature.** Again, I don’t think I have to say anything about this, but if you use LaTeX you might want to consider using macros to specify notation. I usually write things like `\def\CC{\mathbb{C}}`

and so on, just so I have consistent typography.

**Rule 5. State results consistently.** Be sure to write theorems consistently. Usually it’s of the form “If *P*, then *Q*.”

Similarly, with definitions, be consistent “A gadget consists of…equipped with…such that…”.

**Rule 6. Don’t underexplain but don’t overexplain.** But if in doubt, overexplain. Harrington demands you write for an audience (Halmos says “Pick someone you know, and pretend you’re explaining it to them”)…I usually write for myself, knowing I’ll forget the material and have to re-learn it quick.

Consequently, I write specifically so if someone says “I need to learn everything you know about [blank]” I can say “Give me five minutes” (so I can read my notes, and recall everything about the subject). That’s how I wrote my Feynman diagram notes.

**Rule 7. Tell them what you’ll tell them.** Tell the reader where you’re going. Also consider motivating the problem appropriately.

**Rule 8. Use suggestive references.** Don’t say “By theorem blah, we transform Equation (snort) into Equation (oink).” The reader has to flip back to find “theorem blah”, then consider carefully how to apply it. How to revise it?

One should write “Since all continuous functions on closed intervals are uniformly continuous” instead of “By theorem blah”: i.e., write the contents of the theorem instead of referring to it.

One should then write “…we obtain [from our previous equation] (oink)”.

Don’t fear repetition, but whenever you repeat yourself…reword it differently, just to break the monotony.

**Rule 9. Consider examples and counterexamples.** Examples should demonstrate the point, or demonstrate the bounds of some definition. Count-examples are useful for showing when some algorithm fails.

**Rule 10. Use visualization when possible.** You know, a picture is worth a thousand words…

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I think I will ultimately get to discussing the Standard Model in mathematical detail. It would be awesome to discuss the *classical* Standard Model, then the *quantum* Standard Model.

I have already posted some (updated) Notes on Lie Groups, some notes on Fourier analysis, and some notes on Relativistic Quantum Mechanics.

Not to mention my Notes on Feynman Diagrams, focusing on the φ^{4} model and QED.

But in the short run (i.e., next couple weeks), I will be discussing:

(a) double and triple integrals,

(b) curls and divergences,

(c) changing coordinates and Jacobians,

(d) line integrals,

(e) Green’s theorem,

(f) Surface integrals,

and much, much more!

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(1)

How do we find extrema for on the unit circle?

**2.** What can we do? First we can consider the level curves . These are precisely the curves . The gradient vector for is precisely

(2)

This points in the direction of increasing values of .

**3.** We want to consider the situation when , i.e., when the gradient of is precisely a scaled tangent of . Why? Because we want the gradient of to point in the direction of a tangent to our surface. We draw the circle, the gradient vector in red, and in blue. Remember, the red vectors point in the direction of increasing values of , and we restrict our movement along the circle:

Observe when the red and blue vectors are perpendicular, . But when they overlap as a purple vector or point in completely opposite direction, what happens?

This happens when

(3)

or equivalently

(4)

Solving for , we find

(5)

which implies

(6)

But we’re not quite done!

**4.** We must remain on the circle, so we also must demand that . This equivalently implies

(7)

**5.** Is this optimal? Lets try approaching the problem differently. We are working on the circle, which is the parametric curve

(8)

Thus the function we are optimizing becomes

(9)

We see

(10)

We need to solve for , to do so we rearrange terms

(11)

and divide through by , getting

(12)

But this implies where , , , or . Look: that’s precisely describing .

**6. Lagrange Multipliers, Constraints.** One way to consider this situation “Optimize subject to the constraint ” is to say *Okay, so suppose , then wouldn’t we have*

(13)

The term vanishes anyways, so intuitively it seems “equal-ish”.

Example 1 (Minimizing Surface Area).Find the dimensions of the cylinder with smallest surface area whose volume is fixed at .

Solution: The outline takes several steps, namely, (1) construct the function, (2) take the derivatives, (3) solve.

Step One: Construct the Functions. We first write

(14)

for the surface area, and

(15)

describes the area. The constraint is

(16)

Thus we construct the function

(17)

This concludes the first step.

Step Two: Take the Derivatives. We find

(18)

Observe

(19)

We also have

(20)

The derivative with respect to the Lagrange multiplier gives us

(21)

So we can set up our equations as

(22)

That concludes our second step.

Step Three: Solve. We see immediately from the equation that

(23)

We plug this into the equation, we get

(24)

and subtracting from both sides yields

(25)

Now we use the constraint

(26)

substituting we get

(27)

Thus when we take , we minimize the surface area.

**Exercise 1.** Find the extrema of subject to the constraint .

**Exercise 2.** Find the extrema of subject to the constraint .

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(1)

What’s the multivariable analog to this notion?

**2. Definition.** If , we say is a **“Critical Point”** of .

Example 1.Consider . What are its critical points?

Solution: We find its gradient first

(2)

Next we need to set each component to vanish. This implies

(3)

is the only critical point.

**3. Problem:** How do we determine if a critical point describes a maxima or minima?

Lets consider the critical point for . We Taylor expand to second order about writing

(4)

where we use the matrix

(5)

Each row is precisely , and each column likewise is ; the intuition is .

Now, since we are Taylor expanding about a critical point, our approximation becomes

(6)

We can consider the behaviour of near by studying the properties of . Specifically, the signs of the eigenvalues tells us whether the critical point is a local maxima (all eigenvalues are positive) or a minima (all are negative) or some saddle point (mixture having both positive and negative eigenvalues). If there exists at least one eigenvalue that vanishes, this test is inconclusive.

**4. Parting Thoughts:** What if we want to optimize a function constrained to live on a surface?

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(1)

Let be some unit vector. How does change in the direction?

We can consider this quantity as a function

(2)

What does this look like?

**2.** Lets restrict our attention to the smallest non-boring case: . Then we write . We have

(3)

Expanding this to first order in lets us write

(4)

But what does this look like? It’s simply

(5)

Example 1.What is the derivative of in the direction of at ?

Solution: We find the directional derivative is

(6)

We compute

(7)

and

(8)

Thus we have

(9)

We plug in ,

(10)

Then we evaluate to get

(11)

This gives us the directional derivative of .

**3.** We denote

(12)

and call it the **“Gradient”**. Note we will write interchangeably with the vector arrow , and they mean the same thing. The vector arrow doesn’t add anything semantically, it’s just different syntax.

The directional derivative is then

(13)

Note that the gradient acting on a scalar function produces a vector-valued function of several variables, but we can also take the dot product of the gradient with such a monstrosity.

**4. Question:** What is a vector-valued function of several variables?

For us, in practice, we think of this as a *Vector Field*: a “function” which assigns to each point a vector. Each vector-component is a function, usually smooth (i.e., infinitely differentiable).

(Again, just as we warned the reader with vectors, this too is a lie. A vector field is a bit more than just a function , it’s a more complicated beast which is studied further in differential geometry.)

**5. Meaning of Gradient.** Consider a family of level curves . The gradient points towards the direction of increasing . How can we see this? Well, consider the function

(14)

We see its gradient is

(15)

Lets draw a few level-curves and see what the vectors point to:

We see the vectors point towards .

**Exercise 1.** Consider the function defined by . What is its gradient?

**Exercise 2.** Let be defined by . Find its gradient.

**Exercise 3.** Let . Find its derivative in the direction at the point .

**Exercise 4.** Let . Find its gradient.

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**1. Problem.** Consider a function where we parametrize

(1)

If , how does change? We will need to use partial derivatives and Taylor expansion…

**2.** We first note

(2)

Similarly

(3)

Thus we find

(4)

But specifically, we are interested in

(5)

and

(6)

Plugging this in allows us to write

(7)

as

(8)

Observe under our substitution, we have the and other big O terms be gathered into the term.

So what? Observe

(9)

This is precisely the chain rule. Similarly, we find

(10)

**3. Implicit Differentiation Revisited.** Recall implicit differentiation required us to find from some complicated expression like

(11)

What to do? First we write

(12)

Next we say . So we find

(13)

where we set the derivative of to be zero since it’s equal to the derivative of zero. We can then write (taking )

(14)

and divide both sides by to get

(15)

But this is precisely what implicit differentiation gives us!

**4. Warning for Physicists.** Physicists often use partial derivative notation slightly differently. If is the position of a particle, and is its momentum, physicists consider arbitrary functions of the form

(16)

and write

(17)

This is strictly speaking not quite true. The error committed lies in treating and as functions of time: really they are variables whom we are trying to express as functions of time.

**Exercise 1.** Let where , , and . Find .

**Exercise 2.** Let where and . Find and .

**Exercise 3** [Math.SE]**.** Consider the equation . What is ?

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First we approximate the car as a point. The point-like car moves in time, so the value of its components are functions of time. More precisely, the position of my car is

(1)

where the functions , , and are sometimes called *component functions*. Another way to think about this is writing

(2)

where .

Classical mechanics studies such curves under various circumstances. We will discuss some notions of kinematics, and study what it means to differentiate curves.

Example 1.Consider a point traveling in circular motion in the -plane. What does this look like?

Well, it’s a paramteric curve, using trigonometric functions we write

(3)

This descrivbes an anti-clockwise circular motion with radius 1, lying in the -plane.

Example 2.Suppose a particle travels along a parabolic curve, what does the curve look like? We can write it explicitly as

(4)

This is precisely aa parabola.

**2. Calculus with Vector-Valued Functions.** We should recall the construction of the tangent line to a curve at a point had us write

(5)

When we consider the situation when we work with instead of a function . We have be the base point for the tangent to the curve, then we have

(6)

The problem: what exactly is ?

**3.** We can let (or more generally the codomain can be for any positive integer ). We have

(7)

describe the rate of change of the position vector with respect to time. What does this look like? Well, writiing out

(8)

we have

(9)

where primes denote differentiation with respect to time.

Remark 1.

We can keep iterating this procedure to obtain higher order derivatives of a curve.

**4. Kinematics.** We have describe the position of a particle. The velocity of the particle is a vector-valued function

(10)

However, we also can consider the *speed* or the magnitude of the velocity

(11)

Observe the speed is a scalar quantity: it’s just some function of time. The velocity is a vector-valued function of time.

We have one last kinematical quantity to consider: the acceleration. This is just the rate of change of velocity with respect to time:

(12)

Observe we can reconstruct the position from the velocity by considering

(13)

which when we consider we have the integral evaluated “component-wise”:

(14)

We can similarly reconstruct velocity from acceleration.

Example 3.Consider the curve describing circular motion

(15)

What is its velocity vector, acceleration vector, and speed?

Solution: We find its velocity

(16)

From this we can compute its speed as

(17)

The acceleration is precisely the derivative of the velocity vector

(18)

That concludes our example.

**Exercise 1.** Find the velocity vector, speed, and acceleration of the parabolic curve

**Exercise 2.** Let and be differentiable vector-valued functions of time. Prove or find a counter-example that

(19)

**Exercise 3.** Calculate

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