#490 AP book of science-- We have Instantaneous Velocity in calculus, and now we have Instantaneous Energy

[Archimedes Plutonium]

The units of velocity is distance / time, which is seen as the derivative of a cell in calculus.

The units of action, angular momentum, energy are distance^2 / time, which is seen as integral of a cell under the function graph.

This is the Least Action Principle in physics which is the calculus integral versus derivative.

AP, King of Science

#490 AP book of science-- We have Instantaneous Velocity in calculus, and now we have Instantaneous Energy

[Archimedes Plutonium]

Now my favorite Old Math calculus textbook is Stewart, 5th edition 2003, Calculus.

And he discusses Instantaneous Velocity on pages 90, 153 and 199.

He talks of a Average Velocity but says that in a limit, as time intervals get smaller, the velocity is instantaneous.

He gives a chart of this.

time interval        average velocity

5<= t <=6            53.9

5<= t <=5.1           49.49

5<= t <=5.05            49.245

5<= t <=5.01            49.049

5<= t <=5.001            49.0049

And Stewart remarks: "It appears that as we shorten the time period, the average velocity is becoming closer to 49 m/s. the instantaneous velocity when t=5 is defined to be the limiting value of these average velocities over shorter and shorter time periods that start at t=5."

Of course, no mathematician before AP had the concept of Instantaneous Energy as the integral to be the reverse of Instantaneous Velocity.

So in New Math, Calculus comes from Decimal Grid Numbers with gaps and holes and empty space between one number and its successor Number.

We have a far easier time of explaining what Instantaneous Velocity is in New Math because it is part of the calculus itself when the true numbers are discrete.

The function x -> Y in 10 Grid looks like this where the slanted line, or diagonal, splits the graph quadrant in half: 

| 

|    / 

|  /___ 

Let me draw x -> Y again with coordinate numbers. 

y-axis 

^ 

| 

| 

.3  .    .    d 

.2  .    d    . 

.1   d    .    .         > x-axis 

0   .1  .2 

You see, there are no numbers between 0 and .1 nor between .1 and .2. And the first cell in 10 Grid is 0 to .1 and the derivative is the in each individual cell is a instantaneous velocity, and the integral in that specific individual cell is the intantaneous energy.

In Old Math, with their Reals as a continuum, they have a horrible time of explaining Instantaneous velocity because there are no cells, but a continuum. And because there are no empty space from one number to the next number, Old Math cannot even form a Calculus, for the derivative is NOT the tangent line to a point on the function graph but that the derivative is actually what connects the last point of (x_1,y_1) to the next point of (x_2,y_2). The integral is the area below the function graph per each cell.

You see, in New True Math we are able to easily explain Instantaneous Velocity and Instantaneous Energy as integral all because Numbers are discrete and not the phony baloney Reals continuum.

AP, King of Science

[Archimedes Plutonium]

Yes, so now, it is clear to see that Stewart, far above all other authors of Old Math Calculus, far above, has the clearest explanation of Average Velocity versus Instantaneous Velocity. 

The Instantaneous velocity is when I read the speedometer in the car while traveling on the Expressway. The Average velocity is the when I bobb up and down between different speeds but manage to make it to the city after 60 minutes. Sometimes my speed was fast sometimes slow.

But on page 154, Stewart talks of the Secant line between x_1 and x_2 for Average Velocity. But for Instantaneous Velocity, he talks about a tangent line at the specific point x_1. That was Old Math Calculus which is now antique and muddle headed and wrong. Not all wrong but mostly wrong, so much so, that all Old Math Calculus books are ready for the dumpster and then land-fill.

When you have the Wrong Numbers of Math--- Reals with their continuum. You cannot have empty space from one number to the next number. You cannot have a calculus at all in a continuum, for you need that empty space. Say we are talking about the point 5.6 in Decimal 10 Grid. The next number is 5.7 with empty space between 5.6 and 5.7. So the speed of my car on the expressway, say at point 5.6 then comes 5.7, and the dx is 5.7-5.6 =0.1 and say the y-value at 5.6 was 4 and the y value at 5.6 was 8 for a difference of 8-4=4  and when we divide 4 / 0.1 we end up with a speed of 40. In New Math, Calculus has Cells, individual cells from 0 to 0.1, then from 0.1 to 0.2, then from 0.2 to 0.3 all the way along the x-axis until we reach 10. The Derivative is ____not a tangent_____ line but is the actual function that is a straight line segment that connects up (5.6, 4) to that of (5.7, 8). The Instantaneous velocity, is not a tangent line to the graph but actually, the line that joins the left wall of each and every cell to the right wall of each and every cell.

Now we still can have a Average Velocity and that can be a secant line.

The mistake Old Math made is they have Reals with a continuum that cannot give a proof of Fundamental Theorem of Calculus, for Calculus needs and requires Discrete numbers with empty space from one number to the next number.

And once we have Instantaneous Velocity, we need a Instantaneous Energy to create that velocity. this is the area inside that cell.

And this is why AP calls all Old Calculus classrooms as Torture Chambers. When you have the wrong numbers, you have millions of stupid kinds of functions making the textbook be over 1,000 pages. When you have the true numbers of Mathematics-- Decimal Grid Numbers, you have only One Valid Function--- Polynomials. In New Math we need less than 300 pages for a complete study of calculus. Stewart needs 1,300 in his masterpiece of Old Math.

I can write a 200 page textbook of Calculus and teach it to Junior High School with ease.

AP, King of Science

[Archimedes Plutonium]

I better be more accurate for Energy is frequency of Angular Momentum

We can write velocity as the frequency of distance and so the frequency of area is energy.

Instantaneous Velocity is thus the reverse of Intantaneous Angular Momentum. Angular Momentum requires energy to support it.

velocity = LT^-1  or distance/time

acceleration = LT^-2


energy = ML^2T^-2

action = ML^2 T^-1

force = MLT^-2

frequency = T^-1

linear momentum  = MLT^-1

Angular momentum = ML^2T^-1

Force = MLT^-2

Energy = ML^2T^-2

Pressure = ML^-1T^-2

Power = ML^2T^-3

Entropy = ML^2T^-2


Angular momentum L =  kg*m^2/s

Action = kg*m^2/s where angular momentum = action