Re: revamping the teaching of the Calculus by using a geometrical explanati

> I am guessing it was Fermat who came up with the
> chord concept to
> obtain the slope of a arbitrary point on a smooth
> curve. Anyway, most
> textbooks of Calculus omit the Chord concept and that
> is bad news
> because the Chord concept is better than any
> algebraic concept to
> understanding the derivative and how it is linked to
> the integral.

My calculus textbook doesn't mention the chord concept directly, although it uses the word "secant", which is a line extension of a chord. (a line segment connecting two points on a curve). The concept of a secant is directly involved in the definiton of the derivative, since given
some h > 0, the number

s(x;h) = 1/h [f(x + h) - f(x)]

is nothing more than the slope of the extension of the line segment between the points ( x, f(x)) and (x+h, f(x+h)). The derivative is defined to be

f'(x) = lim_{h --> 0} s(x;h)

whenever it exists. (the secant approaches a tangent)

> The old thread was getting too long and so this new
> thread.
>
> I been meaning to ask a historical question about
> Newton and Leibniz
> concerning the discovery of the Calculus. In Newton's
> Principia, if my
> memory serves me, Newton made a mistake by calling
> energy as m*v, not
> m*v^2. The v^2 is area and the integral is an area.
> And it was Leibniz
> who claimed energy is m*v^2. So my question is, does
> this historical
> facts indicate that Leibniz had a better grasp of the
> Calculus rather
> than Newton's fluxions. Did Leibniz get the energy
> correct because of
> Calculus?

I can't really answer that as I know almost nothing of the history here, although I would like to point out that the quantity m*v is usually called momentum and energy is 1/2 * m * v^2, not m*v^2.

> So, now, what about 3rd dimension where a summation
> of cross-sections
> of a object in 3rd dimension would be the volume of
> the object and so
> the integral in 3rd dimension should be easy and
> strong, but now what
> about derivative in 3rd dimension? If the integral is
> summation of
> cross-sections what is the derivative? What is the
> generalization of a
> chord in 3rd dimension.

A chord is a line segment between two points on a curve. Why do you see a problem with "generalizing" this to dimension 3, or any dimension for that matter? Also, the relationship between integral and derivative (actually, there are several types) is not so simple in higher dimensions, as it is in 1 dimension where you have the fundamental theorem of calculus

int_{a}^{b} f'(x) dx = f(b) - f(a)

> So we seem to see Calculus falling apart in 3rd
> dimension. So why is
> Calculus so weak as an enterprise that it is not able
> to generalize
> into the 3rd dimension, and we only piecewise compute
> integration and
> differentiation in 3rd dimension?

Calculus is alive and well in higher dimensions, even for maps between arbitrary manifolds. You've never taken a course in multivariable calculus, have you?

> Is it because in Physics, velocity is linked to
> energy, but volume is
> not linked to anything of physical forces. Can we say
> a Force is a
> volume measure?

How? It is generally a vector quantity, where its norm has units of Newtons. How can that be a measure of volume, which has units of m^3?

> So when we have F = m*a that the
> integral in 3rd
> dimension is m*a and the derivative is acceleration?

Integral of what? Acceleration is the time derivative of velocity. Multiply it by the scalar quantity mass, and you've got yourself another vector quantity called force. Alternatively, force is the (time) derivative of momentum,
m*v.

> Archimedes Plutonium
> www.iw.net/~a_plutonium
> whole entire Universe is just one big atom
> where dots of the electron-dot-cloud are galaxies
>
.

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