Perhaps William could upload a copy of the file here, via the ‘Attachments’ option which is presented when you use the Post reply feature.

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 am using the Firefox browser on a desktop  all in one computer by HP, if that helps.

Alfred wrote:

I can open the file if I click on the error message, I get this:

Edition    Windows 10 Home
Version    21H2
Installed on    18/06/2020
OS build    19044.1586
Experience    Windows Feature Experience Pack 120.2212.4170.0

I hope this helps.

Did you open it and was it alright?

The file in my webspace should be fine, I exported it from PagePlus X7. The webspace is located on a server run by Plusnet PLC.

The only slightly unusual thing about the PDF document, though it should not be a problem as far as I am aware, is that one of the fonts that I use in the PDF document, subsetted, is one that I made using the High-Logic FontCreator program. Yet FontCreator is a high quality program and I did validate the font. So I suppose it could perhaps pick up that the font was not produced by a mainstream font manufacturer, though I am not sure if that field of the font properties (Vendor ID) is encoded in the PDF document.

The font carries HL

which is

High-Logic / Made with FontCreator

William

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

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

Of course, none of the topic maths, calculus, belongs in this thread. I opened it to show William some interesting stuff on emoji. It has gone very much astray from the OP topic, and could frighten off anyone interested in emoji etc.

In fairness, you included something about emoji, so although not congruently a story, it was about emoji and it was on BBC TV. So you were not very much off-topic if at all.

It was me, who having watched the programme, mentioned the symbol that was not explained. And I used it to include a link to a chapter of my first novel.

And before that, having watched all of the three episodes of the programme of which Jack had drawn attention to just one part of ne episode, I rambled on about Bruges and Mainz.

But going back to Jack's comment about the thread having gone off-topic.

Yes it has, but does it matter?

The thing is, I think of this forum as much like a staff room at tea break or lunch time, someone raises a topic and then discussion goes where it goes.

Nevertheless, whilst I did that in a thread that starts "Specially for you, William" and indeed I might have done it in another thread, I would not do it in some threads. It depends. For example, in the Serif forum Jack sometimes posted about a religous festival. I never posted in those, and if Jack posts similarly in this forum, if I did post in such a thread it would be on-topic and regarded in its religous context.

I find that threads going off-topic is often interesting and welcome.

As I mentioned, I think of a thread like this as like chatting in a staff room, the topic can meander here and there. It is not like being in a formal lecture with a discussion section.

BUT if someone posted something of a crisis nature I would remain tightly on-topic.

William

Trouble with going off topic is that people not interested in the OP thread won't read the off topic material, which might interest them if they knew  it was there.

I mean, lots of people are not interested in telling stories in any kind of code, including emoji, while they may be interested in the off topic maths.  but being off topic, it isn't flagged.