Woodland Park, CA
"Woodland Park, CA"

As of late, it seems Calculus is everywhere in my life. The uptick began when I ran into the tacoid a few months ago. In my first semester of college, I really enjoyed and aced calculus for a variety of reasons:

  • I solved extra problems, not just those assigned
  • I went to numerous TA office hours( hi Dunnah McMurray!)
  • Great environment/students! ( hi Sandip Roy! )

In this post I would like to focus on the Calculus Perspective, and demonstrate what an expression of Calculus is capable of:

  • The problem landscape, , can be expressed clearly without artifacts, such as those that would arise if the sequence required an index.

  • share a sudoku-like equivalence

    :alarm_clock: “ “ :coffee:

    ” “ :bus: :coffee:

    :alarm_clock: :bus: “ “

  • A profound example with succinct clarity.

    • Maxwell’s equations represent the full interaction, curl(dissimilar) and dot-product(similar), between electromagnetic fields. Magnetic field rotation, described by the right-hand rule, require a change of sign(artifact) to distinguish R/L.
    • Clifford algebra, allows built-in sign change, using a bivector for . The geometric product multiplies the wedge product terms for curl with dot product terms for divergence.
    • Thus 4 equations are expressed as one, with sign-flipping built-in.

In the examples that follow, I will highlight clear expression, the changing variable nature of the integral, vanishing infitesimals, Fundamental Law of Calculus..

The Tacoid

  • problem
    • Model a tortilla as a pliable disk radius=1, center at origin, and against surface \( z=f(x) \)
    • arc-length s of
    • is continuous, even, positive, and increasing from
    • Can the volume (convex hull) be maximized with respect to all such ?
  • construct an integral
    • Changing variables: ds; V between two surfaces
    • length along x-axis is 2x
    • width along y-axis between two surfaces is \( \sqrt{1-s^2} \)
    • height of z-axis is \( dz = \dot{z} \, ds \) \[ Volume = \int_1^1 2x \, \sqrt{1-s^2} \, \dot{z} \, \ast \, ds \]
  • substitution
    • There is then a substitution to remove the z, use of Euler-lagrange, and numerical solver rk4…

The important aspects to consider are the changing arc length along the edge of the tacods Volume of the region bounded by the taco arc length. Alternatively, consider filling the taco from the bottom to top. From this perspective, moving up the z-axis, the integral is written as \[ Volume = \int 2x \, \sqrt{1-z^2} \, \ast \,dz \].

See the link for more details.

Nicomachus Theorem

This is a case where Calculus is clearer than either visualization or induction.

Nicomachus, who was the son of Archimedis, discovered \[ 1^3 + 2^3 +…+ n^3 = (1+2+..+n)^2 \].

  • In trying to solve by induction, neither substitution using the series-sum form \( (n)(n+1)/2 )\, nor expansion of the cubes … er- lead anywhere.

  • This is a visual proof. Beyond vertigo

  • Expressing the behaviour we are trying to prove, using Calculus, relies on the limiting precision of the dx term.
    - The dx plays the role of the iterator, allowing us to express the \( \sum n^3 \) without worrying about series form or where possible substitutions, such as the sum of the series of odds \( n^2 \), might take place.
    - Next, taking the sum of values is simply the integral, which we solve on both sides to show equality

\[ \int n^3 dn = \left(\int n dn \right)^2 \] \[ \frac{n^4}{4} = \left(\frac{n^2}{2} \right)^2 \] \[ \frac{n^4}{4} = \frac{n^4}{4} \]

find notes and write up: - Max-likelihood estimator of German Tank Problem, both unbiased and mean-square error forms. - The change in boosting weights as logs, and their relation to the exponential loss function - Sorting in trees, in-place, and the binomial combinatorics. - Optimization minimax