## Div, Grad, Curl, and All That: An Informal Text on Vector Calculus

This is an excellent informal introduction. It is written clearly, uses simple but illustrative worked examples with helpful graphics. It covers the title topics and uses electrostatics as a tangible application for illustration. There a lot of problems and solutions at the end.

## What to Expect

I found this question on Mathematica Stackexhange interesting. I learned a lot from wolfies answer. I post this not as an answer to the question of closed form but to illustrate consistency of a number of approaches.

The calculations take some time but it does not seem onerous.

Insights into the expectation can be obtained from the region of of interest.

ms[p_] :=

Show[Plot3D[(1 - x)^y, {x, 0, 1}, {y, 2, 5},

PlotStyle -> {LightBlue, Opacity[0.5]}

, PerformanceGoal -> "Speed", PlotRange -> {0, 1}],

Plot3D[(1 - x)^y, {x, 0, 1}, {y, 2, 5}, MeshFunctions -> {#3 &},

Mesh -> {{1 - p}}, MeshStyle -> {Red, Thick},

RegionFunction -> Function[{x, y, z}, (1 - x)^y "Speed", PlotRange -> {0, 1}],

AxesLabel -> (Style[#, 20] & /@ {x, y, (1 - x)^y}), ImageSize -> 500]

Manipulate[ms[t], {{t, 0, "p"}, 0, 1, Appearance -> "Labeled"}]

Or more easily with:

Manipulate[

RegionPlot[(1 - x)^y < 1 - p, {x, 0, 1}, {y, 2, 5}], {p, 0, 1}]

Now to numerically calculate the expectation:

int[p_] :=

Integrate[

Boole[{x, y} \[Element]

ImplicitRegion[0 < x < 1 && 2 < y < 5, {x, y}]] x/3, {x, 0,

1}, {y, Log[1 - p]/Log[1 - x], 5}]

td = TransformedDistribution[(1 - x)^y, {x, y} \[Distributed]

UniformDistribution[{{0, 1}, {2, 5}}]];

d[p_] := Probability[z < 1 - p, z \[Distributed] td]

result[p_] := int[p]/d[p]

ex[p_] :=

Expectation[

x \[Conditioned] (1 - x)^y < 1 - p, {x, y} \[Distributed]

UniformDistribution[{{0, 1}, {2, 5}}]]

Definition[rp]

rv[p_, n_] :=

Module[{r},

r = RandomVariate[UniformDistribution[{{0, 1}, {2, 5}}], n];

Mean[Pick[r, ((1 - #1)^#2 < 1 - p) & @@@ r][[All, 1]]]]

rv1000[p_] := rv[p, 1000];

The above firstly uses wolfies method, then just asks *Mathematica* to calculate explicitly and the simulates.

Tabulating:

grid = Prepend[

Through[{Identity, result, ex, rv1000}[#]] & /@

Range[0.1, 0.9, 0.1],

Style[#, Bold] & /@ {"p", "Derived", "Expectation",

"Estimated (N=1000)"}];

gr = Grid[grid, Frame -> All,

Background -> {None, {{White, LightBlue}}}]

Plotting (using `Table`

to minimize computation time):

plt = ListLinePlot[Table[ex[t], {t, 0.1, 0.9, 0.1}], Frame -> True,

FrameLabel -> {"p", "Conditional Expectation x"}, BaseStyle -> 16,

ImageSize -> 600]

This does little to answer the closed form question but I found it fun.

## How Not to Be Wrong

.

This is a wonderful book. I thoroughly enjoyed it. I learned Mathematical, historical and cultural jewels that I very much look forward to contemplating further or delving into. The author is refreshingly forthright in exploring the impediments to Mathematics Education and its appreciation in society at large. He does this from the position of a working Mathematician and educator. A broad array of concepts are covered and their exposition are an intricate weaving of historical events and individuals with current events and instructive toy examples. Wald, Fisher, Pascal, Shannon, Galton, Hilbert and many more appear at varying times. These insights into these individuals was remarkably more complex for the amount of words: strengths and weaknesses of their contributions (and the positions, e.g Galton and eugenics).

One of the greatest joys of this book is the discussions are neither superficial nor simplistic. The author starts from the simple and when you think you have digested what is at hand, he add another wrinke and another and enriches your understanding. He supports his case that Mathematics is the extension of common sense. Particularly informative were the discussions of the law of large numbers, lotteries, the concept of regression to the mean. The importance of the role of uncertaintuy in our lives, acknowledging its existence and the value of quantifying marries well with Nate Silvers “The Signal and The Noise” (Silver also discussed in the book).

The author discusses the “cult of genius” and the obstacle it creates to recruitment and retention of Mathematicians and the enrichment of other disciplines by Mathematics. This was compelling and struck me and derived from the insights of one who had negotiated the journey (though admittedly a prodigy) as well as tasked with guiding and supporting others through it.

There is a lot in this book. It is, however, more than content. It is a challenge to think more deeply about everyday life, about culture, about history and the seeing the “unreasonable effectiveness of Mathematics”.

## Money money

I am thoroughly enjoying Jordan Ellenberg’s “How not to be wrong”. This small post is some fun looking at Buffon’s coin puzzle. This shows a sample of twenty coins of varying radius relative to the underlying square. The fair game arises when the radius to length of square side is . The fraction in the boxes are the observed number of coins in the square and the predicted fraction. The value below the box is the ratio of radius to square length.

## Joy in Small Things

I have learned a lot from the Mathematica Stackexchane community. I just crossed the 10 k…(it could easily go down). It has been a trying, stressful period in the everyday life, the small thing brings me some joy. I am not an expert nor a professional, dwell on the simpler level of the question spectrum and I have made some major gaffes (great opportunities to learn). I take each question as an opportunity to learn but learn the most from the creative, sometimes amazing but always inspirational other answers.

Peace to all, and solace to fellow sufferers of the ‘black dog’.

## Real Mathematical Analysis

This is another excellent Springer undergraduate Mathematics textbook. This book is clearly written. The book goes systematically progressively deeper. It covers continuty, differntiability, integration, introduces differential forms and Lebesgue theory. In addition to showing the results, the author shows the “pathological” functions, spaces and points to the wonderful unexpected results, e.g. unmeasurable sets and Banach-Traski paradox. I learned a lot from the Cantor sets.

I should have taken the time to dive into this book but even with my quick read, this book explains the strong foundations of real analysis and calculus that we use in tame, smooth, well behaved domains and provides insights into the world beyond the tame shores I usually play in…”there be dragons out there”.

## A Student’s Guide to Entropy

I have always found Entropy to be a difficult concept. This book is useful in improving understanding. I builds and explores the concept from classical, quantum physical and information theory perspectives. There is very useful discussion of statistical mechanicsand the derivation of the Maxwell-Boltzmann distribution. The relationship between classical and quantum physics results are well illustrated.

## Unknown Quantity

I enjoyed this book. The author has a very entertaining writing style. The book is historical journey from Mesopotamia to modern times tracing the development of algebra within the broader history of Mathematics. The author puts the developments in a historical context and provides insights into the characteristics/personalities (from what is available…little for some in the remote past) of the key figures in the development of algebra.

I am once again inspired to diminish my ignorance of Algebra.

## Probably Approximately Correct

This is a very interesting book. The author presents the case for ecorithms (algorithms, heuristics perhaps) that could explain and ultimately allow quantitative assessment and testable predictions of the mechanisms (and timescale) of evolution and one of its most “mysterious” byproducts consciousness/cognition (I should perhaps not conflate these two).

The author looks at the central problem of evaluating and decision making based on incomplete information, small empirical samples and within biological and physical constraints and be successful. The linear/polynomial time algorithms (using the generalized concept of computation: universal Turing machines) for learning from inputs from external environments in a “theory-less” context could lead to “probably approximately correct” classifications, decisions and actions and be explanatory for evolution and perhaps human learning and human cultural evolution (with the latter having Lamarckian as well Darwinian aspects).

The book explores these matters through the lens of computer science (the author’s expertise). This is a very interesting and instructive perspective. The limits, and similarities and contrasts between computer systems and algorithms was well presented.

I think this book fits nicely with Penrose “Emperor’s New Mind” (which argues for non-algorithmic apsects to consciousness and learning), Kahneman’s “Thinking Fast and Slow” which explores the limitations of human reason (our hard wiring for making fast decisions with limited information but our limitations in statistical and probabilistic reasoning) and Silver’s “The Signal and the Noise”.

I had (and continue) to think hard about the concepts in the book. It is, however, I believe a very refreshing viewpoint to seek to explain the gaps in evolution, cognition and learning. The author ends by looking at issues of artificial intelligence and why this has been more challenging than anticipated and the authors appeal to reason in relation to fears about a ‘Sky-net’ future was very interesting. The integration of external inputs, the central role of learning, the power of inductive reasoning in and the need for composite induction and deductive reasoning (the latter indispensible for what the author calls theory-ful contexts) are all part of the authors rich explanation…it seemed to me “probably approximately correct”.