[Index] Calculus in a Single Variable

  1. Differential Calculus in a Single Variable
    1. Definition of Derivatives using big O notation
    2. Line Tangent to a Curve
    3. Exponential Function
    4. Tangent Lines as Linear Approximations
    5. Some Useful Trigonometric Limits
    6. Differentiating Trigonometric Functions
    7. Inverse Function Theorem
    8. The Natural Logarithm, the inverse function to the exponential mapping.
    9. Optimization: its motivation, the First Derivative Test, an example optimizing a rectangle’s area
    10. Implicit Differentiation
    11. Curve Sketching
      1. Curve Sketching (General Scheme)
    12. Applications
      1. L’Hopital’s Rule for Limits
    13. Techniques
      1. Differentiation Technique #1: Logarithmic Differentiation
  2. Integral Calculus in a Single Variable
    1. The Antiderivative part 1, part 2
    2. Finite Series
    3. Riemann Summation a first step towards definite integration
    4. Example Riemann Sum working with f(x)=x^{2}+1
    5. Fundamental Theorem of Calculus
    6. Properties of the Integral
    7. Not all functions are (Riemann) integrable!
    8. The Natural Logarithm Revisited!
    9. Integration Techniques:
      1. Integration Technique #1: Substitution
      2. Integration Technique #2: Integration By Parts
      3. Differentiation Under the Integral Sign
    10. Applications of Integrals
      1. Calculating Arc-Length of Curves
      2. Calculating the Area for a Surface of Revolution (Part 1) when we revolve about the x axis
    11. Optimization
      1. Calculus isn’t useless: When Velociraptors Attack, calculus solving matters of life and death!
    12. Thinking “Infinitesimally”
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Just to keep readers in the loop: I have not forsaken you! I just have a job now…

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!

Posted on by Alex Nelson | Leave a comment

Introduction to Double Integration

1. Consider {z=f(x,y)}. What is the volume of the region
(1)\displaystyle  S = \{ (x,y,z)\in{\mathbb R}^{3}\mid z=f(x,y),\; a\leq x\leq b,\; c\leq y\leq d\}
The first thing we do: consider the rectangle {R=[a,b]\times[c,d]} and form a partition of {[a,b]} into {M-1} segments and {[c,d]} into {N-1} segments. This gives us a mesh of rectangles {R_{ij} = [x_{i-1},x_{i}]\times[y_{j-1},y_{j}]} as specified by the following diagram:

Observe the area of {R_{ij}} is
(2)\displaystyle  \Delta A = \Delta x\Delta y.
We can approximate the volume {V} of {S} by a sort of Riemann sum, picking points {(x_{ij}^{*}, y_{ij}^{*})} in {R_{ij}} and taking
(3)\displaystyle  \sum^{M}_{i=1}\sum^{N}_{j=1}f(x^{*}_{ij},y^{*}_{ij})\Delta A\approx V.
However, as with Riemann sums, we recover the exact volume when we take the limits {M,N\rightarrow\infty}:
(4)\displaystyle  V = \lim_{M,N\rightarrow\infty}\sum^{M}_{i=1}\sum^{N}_{j=1}f(x^{*}_{ij},y^{*}_{ij})\Delta A
if the limit exists. Continue reading

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Notes on Mathematical Writing

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…) Continue reading

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So, they discovered the Higgs Boson the other day.

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 φ4 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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Lagrange Multipliers

1. So, last time we considered finding extrema for some function {f\colon{\mathbb R}^{n}\rightarrow{\mathbb R}}, but what if we constrain our focus to some surface {g\colon{\mathbb R}^{n}\rightarrow{\mathbb R}}? For example, the unit circle would have
(1)\displaystyle  g(x,y) = x^{2}+y^{2} - 1=0
How do we find extrema for {f(x,y)=xy} on the unit circle? Continue reading

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Finding Extrema of Multivariable Functions

1. Remember for a curve {y=f(x)}, we have maxima and minima occur whenever
(1)\displaystyle  f'(x_{0}) = 0
What’s the multivariable analog to this notion? Continue reading

Posted in Calculus, Gradient, Optimization, Vector Calculus | Tagged | 1 Comment

Directional Derivative, Gradient

1. Suppose we have a scalar function of several variables
(1)\displaystyle  f\colon{\mathbb R}^{3}\rightarrow{\mathbb R}
Let {\widehat{u}} be some unit vector. How does {f} change in the {\widehat{u}} direction?

We can consider this quantity as a function
(2)\displaystyle  g(\vec{x}) = \lim_{h\rightarrow0}\frac{f(\vec{x}+h\widehat{u})-f(\vec{x})}{h}
What does this look like? Continue reading

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Chain Rule for Partial Derivatives

1. Problem. Consider a function {f(x,y)} where we parametrize
(1)\displaystyle  x=x(t,u),\quad\mbox{and}\quad y=y(t,u).
If {t\rightarrow t+\Delta t}, how does {f\rightarrow f+\Delta f} change? We will need to use partial derivatives and Taylor expansionContinue reading

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Curves, Velocity, “Classical Kinematics”

1. Curves. We are interested in describing the motion of my car. Well, everyone is interested in the motion of my car. How can we describe it mathematically?

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)\displaystyle  \vec{r}(t) = \langle f(t),g(t),h(t)\rangle = f(t)\widehat{\textbf{\i}} + g(t)\widehat{\textbf{\j}} + h(t)\widehat{\textbf{k}}
where the functions {f(t)}, {g(t)}, and {h(t)} are sometimes called component functions. Another way to think about this is writing
(2)\displaystyle  \vec{r}\colon[0,1]\rightarrow{\mathbb R}^{3}
where {0\leq t\leq1}.

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

Posted in Calculus, Curves, Geometry, Vector Calculus | Leave a comment