daily 03/12/2014

Calculus In A Few Minutes  Calculus BetterExplained

We usually take shapes, formulas, and situations at face value. Calculus gives us two superpowers to dig deeper:

XRay Vision: You see the hidden pieces inside a pattern. You don’t just see the tree, you know it’s made of rings, with another growing as we speak.

TimeLapse Vision: You see the future path of an object laid out before you (cool, right?). “Hey, there’s the moon. For the next few days it’ll be white, but on the sixth it’ll be low in the sky, in a color I like. I’ll take a photo then.

Calculus In 10 Minutes: See Patterns StepByStep

Calculus to the rescue. Let’s use our XRay vision to realize a disc is really just a bunch of rings put together. Similar to a tree trunk, here’s a “stepbystep” view of a filledin circle:

The height of each ring depends on its original distance from the center; the ring 3 inches from the center has a height of 2π⋅3 inches

Our XRay vision revealed a simple, easytomeasure structure within a curvy shape. We realized a circle and a set of gluedtogether rings were really the same. From another perspective, a filledin disc is the “time lapse” of a ring that got larger and larger.

Remember learning arithmetic? We learned a few things to do with numbers (and equations): add/subtract, multiply/divide, and use exponents/roots. Not a bad start.
Calculus gives us two new options: split apart and glue together. A key epiphany of calculus is that our existing patterns can be seen as a bunch of gluedtogether pieces. It’s like staring at a building and knowing it was made brickbybrick.

Our primary goal is to feel what a calculus perspective is like (What Would Archimedes Do?). Over time, we’ll work up to using the specific rules ourselves.


About BetterExplained.com  BetterExplained

Ideas start hard and finish simple.

any topic can become intuitive, after overcoming the initial complexity.

You, 10 minutes after learning a new idea, are the perfect tutor for your current self.


An Intuitive Guide to Linear Algebra  BetterExplained

Linear algebra gives you minispreadsheets for your math equations.

We can take a table of data (a matrix) and create updated tables from the original. It’s the power of a spreadsheet written as an equation.

“Algebra” means, roughly, “relationships”.

“Linear Algebra” means, roughly, “linelike relationships”. Let’s clarify a bit.

Contrast this with climbing a dome: each horizontal foot forward raises you a different amount.

In math terms, an operation F is linear if scaling inputs scales the output, and adding inputs adds the outputs:

. Many problems, though not obviously geometric, can be shoved into some geometric form and more easily tackled thus.

They have oddly nice properties as well. For example, no matter how weird the vector space, it has a well defined dimension (though it may be infinite).

So matrices and vectors of numbers are nice, but they’re barely the tip of the ice berg of linear algebra.

Every linear map can be represented by a matrix.

We work with matrices because they completely characterize the functions we care about.

“However, linear algebra is mainly about matrix transformations, not solving large sets of equations (It’d be like using Excel for your shopping list).”

Matrix/vector multiplication never made any sense to me, until I realized it’s just projecting the vector onto the original identity basis, and then reconstituting it using the new basis instead. You can discover and draw this process entirely visually.

Did the rotation equations then the whole concept of matrix maths ‘clicked’ when I realised its nothing special it’s just a neat way of doing the maths!

In fact the whole mystery of maths clicked – maths is nothing more than a human language and tool for describing how things interact. Maths doesn’t have rules it simply implements observations of physical reality in a convenient way. For example complex numbers in Electrical engineering integration etc.

Matrixes can definitely go deeper (to any linear operation) but it’s a crawl/walk/run thing.

Yep, matrixes started off as bookkeeping for equations. And math is definitely a tool/language for communication. If we’re using math, but missing the ideas, we’re not doing math!

OMG in a matter of just a few lines you’ve completely demystified the notion of eigenvector. Thanks!

I’d like to point out that an eigenvector is a vector whose direction is unchanged or invariant under a transformation.

This is not giving the correct intuition for linear algebra.
See gilbert strang’s first couple of lectures. They give an intuitive feel and are presented by someone who really understands linear algebra.


Breaking Down the X, Z, and Slot Receiver – Field Gulls

Millen broke down, in detail, the different receiver positions: X, Z, and the Slotreceiver.


Texas Longhorns offensive spring position battles – Burnt Orange Nation

Onyegbule is working at quarterback some in an effort to build better depth at the position with Jerrod Heard the only 2014 signee set to join the group this fall, although USC graduate transfer Max Wittek is reportedly leaning towards Texas.



While I’ll forever lament Smith’s failure to get the film going with Jason Lee as Irwin Fletcher, Sudeikis seems like a solid choice. With his easygoing, lackadaisical attitude mixed with a habit of lying through a smile, this isn’t the first time he’s been compared to Chase, and he should slip into his basketball shoes just fine.


Marginally Interesting: Data Analysis: The Hard Parts

Here are a few books I can recommend for learning the basics about data analysis. These are no practical books, but require some knowledge of math, in particular linear algebra, multivariate calculus, and probability theory


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Here’s the whole thing. “Unlike most woman…” http://t.co/GpMgSzARsH
Posted from Diigo. The rest of my favorite links are here.