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