Compact connected R^m

In mathematics, as in most human pursuits, we draw pictures not only as a means to represent reality but often as a means to explain reality. The distinction I make is that one may draw a picture of a building to be beautifully exact or one may draw a schematic that encodes relative size, angles, materials etc. without paying attention to ornamental details. We call the latter ones doodles, cartoons, diagrams, or sketches. 

Above are doodles from Michael Spivak’s Calculus on Manifolds. It’s a great little introduction to differential geometry for anyone with solid calculus and point-set topology. These aren’t exactly life changing pictures, but the book was published in 65 and I just dig these groovy blue drawings.

These images should be treated as copyrighted by Spivak.

The first picture is a representation of a smooth function sending the blue square on the left to the interior of the black squiggle on the right. We like to think of smooth functions as simply pushing  the points in a plane around a bit. This figure was used in the proof of the inverse function theorem, which basically tells us that a small enough area around a point (point “a” in this case) smooth functions don’t really do anything.

The next picture gives an example of changing of variables . In reality we might like to think about cardinal directions (the blue square on the left), but then be able to switch to interstate directions (the squiggly one in the middle), but then be able to switch to state route directions (the squigglier one on the right). 

The third picture is a drawing of the definition of manifold. Basically if you look closely enough at a manifold it looks flat. 

The next picture is used in the discussion of surface area. Where for curves (bits of string) we know that the length (if you pull it taught) is just the upper bound on the length of line segments that approximate it (using a short measuring stick). This is called rectification. This picture is meant to show that no such similar thing can be done with surfaces (sheets of paper). That is, all polygonal (flat) approximations can be increasing in surface area. 

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{n | !E x,y s.t 4x+7y=n}

f(x+ε) = f(x) + f'(x) . ε + O(ε)^2

whats r? euclidean norm?

Correlation coefficient!  This is a thing that works if your function is almost a straight line, so maybe we’re asking how much of the variation is explained by the line of best fit?

its the product of all distinct eigenvalues… no because that doesn’t includ emultiplicity

so each eigenvalue, you put it to the power of the dimension of the associated eigenspace?

Π λ^dimension(nullspace(M - λ I))

That definition works if the eigenvectors span the underlying space. But sometimes you have a matrix like [ [1,1], [1,0]] where the only eigenspace is 1-dimensional, so the eigenvectors don’t span.

To handle this we define something called a generalized eigenvector. A vector v is a generalized eigenvector of L with eigenvalue λ if it’s in the nullspace of (L - λ I)^n. It’s a theorem that (over the complex numbers) the generalized eigenvectors span, and thus the sum of the dimensions of the generalized eigenspaces is n.

Then we can define the determinant to be

Π λ^dim(ker(L - λ I)^n).

And this is actually what I take the definition of the determinant to be; the formula recursive formula in terms of minors is then a theorem, not a definition.

The gravity we experience here on earth is *mostly* due to curvature of time

The aspect ratio of a polygon is a measure for “circlishness”: it is the ratio of the diameter of the smallest circumscribing circle to the diameter of the largest inscribed circle. The closer this number is to 1, the more the shape looks like a circle. For instance, a perfect square has an aspect ratio of √2.