Suppose that y is the height of some plant, say a particular sunflower growing in a garden, and x is time.

I don't know the exact figures, so these are just any numbers, just to illustrate the idea.

So one could say, for example, in 20 days the sunflower has grown 40 inches.

So, one could say, ah, that is a rate of change of two inches per day.

However, the growth might have been more rapid on some days than on others, so 2 inches per day is an average, and on any particular day, the growth might be 2 inches, it could be more, it could be less, or it could be exactly 2 inches.

So if, say, we looked at the growth over some chosen 5 day period, the sunflower has grown 11 inches in thoise five days, so 2.2 inches per day on average during that time period.

Now, if we reduce the time period to, say. half a day, we might find that in that paeticular half day it has grown 1.5 inches, so that is an average of 3 inches per day.

On a different half a day, it may have grown 0.4 inches, so that is a average of 0.8 inches per day.

Now if we take the length of time over which we take the change of height down smaller and smaller, then the average is being found more frequently.

Now in the limit as that time gap for the sample tends to zero, that average growth rate over the tme period becomes more accurate as to what is happening at that instant.

That rate of change at that instant is what is designated by dy/dx in the mathematics.

William

Basically,

you mention dy/dx as in dy/dx = something. The process of calculating dy/dx is called differentiation.

Well the symbol, ∫ which is like an old fashioned long s, which looks like a lowercase f, but does not have a horizontal part to the right side of the vertical, is used in mathematics to mean "integral". So ∫ is the integral sign. The old fashioned long s in upright form did have a horizontal line to the left of the vertical, but the ∫ sign is based on the italic version, so no line at the left. So ∫ is really like a stylised letter s and stands for sum or summation or similar. But it is not called summation as summation is the word used for something else that need not be mentioned here.

Integration, as the process is called, is the reverse of differentiation.

So, for example,

Say there is a car travelling at x miles per hour but that x can vary in time.

So dx/dt is the rate at which x changes with time, so dx/dt is called the acceleration if dx/dt is postive and deceleration if dx/dt is negative. If the car is travelling at a constant speed, dx/dt is zero.

So the car is travelling at x miles per hour.

Integration with respect to time, ∫x.dt is the total distance travelled by the car, and one needs to add what are called "limits of the integration" to get an answer. So, for example, the limits could be 8.00 am for the lower limit to 9.00 am for the upper limit for someone driving to work.

At 8:00 am the value of x might be 0 as the person has not left for work. Then the person drives, with x gradually increasing within the built up area near home, then x increases outside the 30 miles per hour limit, then slows down when near the workplace and goes to 0 when the person gets to the car park near work at, say, 8:52 am, then the car is stationary until 9:00 am (and in fact until 5:05 pm, but 9:00 am is the upper limit of the integration in this example).

So there can be a graph of x shown vertically with time shown horizontally.

So at each point on the graph, dx/dt is the slope of the graph, showing the rate at which x is changing with time, which could be positive, negative or zero at any particular time.

The integral is the area under the curve, and that is the distance of the journey travelled, because x at any time is the miles per hour and the time at any time is, well, the time, and miles per hour times hours is miles.

So, please note how the distance travelled gradually increases throughout the journey, but at a variety of rates of increase, because the rate of increase of the distance is the speed of the car. Please note how the familiar everday knowledge that, well, yes, the distance travelled increases, ties in with the way that it is expressed in mathematics. They are both about the same thing. The mathematics is the precise way to express it to be able to get precise numerical results.

So, the odometer of a car has integrated the speed of the car with respect to time from when the car ws first run until the present time.

So, "What's its recorded mileage?" is like asking "What is the value of the integral of the speed of the car with respect to time with limits of from the first use of the car to now?"

I hope this helps.

You are welcome to ask for clarifcation if any part of that is unclear.

William

Thank you.

Wlliam

Here an image of the symbol being used as it is used in my research project in a way that is very different from the way that the symbol is used in mathematics.

William

I have just watched that programme.

Looking at the credits at the end, there is an amazing number of people listed.

I notice that he didn't explain the mathematical symbol in the upper left corner!

Edith recognizes it though.

http://www.users.globalnet.co.uk/~ngo/l … er_019.pdf

(Yes Alfred, I know, I took on board what you wrote)

William

Oh. I should have checked that, just in case, for provenance.

I wonder if the presently rare word in the following linked document will one day be in the dictionary.

http://www.users.globalnet.co.uk/~ngo/l … er_057.pdf

William

Um ... possibly.

William

I have just completed watching the third episode.

Thank you, Jack, for drawing my attention to this.

I have learned much.

A couple of things stand out.

The people painting characters using water. That is interesting.

I once, many years ago, saw a pavement artist produce a large picture.

It was in Stratford-upon-Avon.

This was in an area where the public had access but there was a notice on a wall something like "This route is not dedicated to the public."

I don't know the details but it was then a new development.

It seemed that where there were two roads of shops at right angles to each other, with perhaps back yards behind them, two of the shops had been converted to walk through routes, one on each road, and they both led to a large open paved area around which there were lots of small shops. I don't know if the walk ways into the area were open to the sky but perhaps not, perhaps just the ground floor of a not very wide shop having its front window and door removed and the back wall opened so as to provide a tunnel.

Perhaps I can find out from Google street view.

The large central area was open to the sky and covered with what were just basic plain paving slabs.

So it was not public pavement.

I don't know by what authority (if any) the pavement artist was there.

Another thing in the programme was that for Chinese chracters there are books that give the order (and the directions?) of the strokes.

It would be interesting to know the order of the strokes and their direction when people write Latin letters.

For example, is a letter o written clockwise or counterclockwise? Does it vary from person to person? Does each person always write a letter o in the same direction? Is a capital O written in the same direction as a lowercase o?

What about a lowercase letter d? Which order of strokes, which directions?

William

I have just watched episode 2.

https://www.bbc.co.uk/iplayer/episode/m … -on-a-page

I feel that I have learned much from it. About papyrus, about paper, about book production in Bruges.

It is due to go unavailable at 2:30 am Friday morning, British Summer Time. I note for the archive that today is Thursday.

If you can access it from where you are, and can find time to watch it while it is still available,I highly recommend viewing it.

William