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, $x$, can be expressed clearly without artifacts, such as those that would arise if the sequence required an $x_i$ index.

• $\int dxdz = \int dydz = \int dxdy$ share a sudoku-like equivalence

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• 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 $M$. 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. $\nabla F = \frac{1}{c\epsilon_0}J$

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 $f_{(-a,a)} \ge 0$
• $f$ is continuous, even, positive, and increasing from $[0,a]$
• Can the volume (convex hull) be maximized with respect to all such $f$ ?
• 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 $\ast$ 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