Steven,
"You should take notice that a derivative of a function is represented by the letter D. The reason is because taking the derivative is division. You should also notice that the symbol for integrating a function is very much like an S. The reason is because integration is taking the sum. It is addition. Addition is made simpler by memorization of multiplication tables. Division and multiplication are covered in lower mathematics as shortcuts for counting. We are still counting things. we count up and we count down."
I often write using differentials. Here is the reason why I write the mathematics as I do. If one knows algebra, then much of differential Calculus can be understood if written using those differentials. It is easy to follow. Someone reading my words might more easily connect my word meanings to my equations.
Acceleration is a change of velocity with respect to time. Since all physics empirical evidence is communicated to us via photons as measures of change of velocity with respect to time, that empirical evidence arrives in the form of incremental measures. like pieces, of acceleration, i.e., a usually very small measure of a change of velocity with respect to a unit of time. We all know that a=dv/dt. What may not be so clearly known is that dv is the differential of velocity and dt is the differential of time. The ratio of dv/dt is the derivative of velocity with respect to time.
Both of those differentials are very small changes. So small that they are said to be approaching infinitely small magnitudes. Yet their ratio does not change. Their ratio is the change of velocity with respect to time, located at some point on a curve of a plot of velocity vs time. I write it as dv/dt. Physicists might write it as Dtv. That D might also be fancy script form.
How is one to easily visualize the physical meaning of dv/dt? The answer is that the ratio of differentials comes from a right triangle. A right triangle plotted on rectangular Cartesian coordinates is the basis for the derivative of a function. The 'x' coordinate is horizontal and called the abscissa. The 'y' coordinate is vertical and called the ordinate. A right triangle has a hypotenuse. A curved line representing a change of velocity with respect to time is plotted on graph paper. The hypotenuse is called the tangent line because it is placed so that it touches the plotted curve at a point. It is the two other sides of the triangle that are of immediate interest. If velocity is plotted against time, the vertical side of the triangle represents a measure of a change of velocity; and, the bottom or base side represents a measure of a change of time.
The value of the vertical side divided by the value of the base side gives the slope of the tangent line. The slope is the rate of change of the variable 'v' with respect to the variable 't'. Having the tangent line touching only at a point means that the changes of velocity and time are as small as possible without becoming zero. Division by zero is not permitted. The extremely small change in 'v' is represented by dv and the corresponding extremely small change in time is represented by dt. These extremely small changes contribute to accuracy in the solution. The derivative is the slope of the plotted line at any point. If the line represents a plot of velocity with respect to time, then the slope (dv/dt) of the line at any point is the acceleration at that point. The steeper the slope, the greater the rate of change.
Sometimes the Greek letter Delta (A small triangle.) is used instead of the letter 'd', when the changes are not so very small. Those values of change are referred to as being incremental. Getting away from my style of math, in Calculus books you will usually not see the derivative written as dv/dt. It is customary to use the Capital letter 'D' as in Dt(v) which is saying the same thing as dv/dt. The velocity 'v' is usually a variable and its 'function' is how it varies such as v=1/2gt. The velocity 'v' is a function of the variable 't' as in 1/2gt. So, in general, all variables that are functions of 't' can be represented by a general form of a derivative written as dtf(t).
Mathematics books will be far more rigorous than I have been. My intent is to get to the basic idea of taking a derivative. You will see much more style applied to the mathematics used here by the professionals. There is good reason behind the use of all of that style. It conveys far more meanings than this simplified form that I use. However, I write so that non-professionals might also understand my work. Truth is that doesn't happen much, but, that is what I try to accomplish.
Lastly, Integration in Calculus is reversing what was done when taking a derivative. That doesn't always go smoothly. That is why Calculus books have lists of 'Integrals' that are general forms for known types of solutions. You look for the type of solution you need and let your mathematics take that form and run with it. Pay attention to lone constants. They disappear when taking a derivative. They are unknown what they were before. They have to be solved for when you reverse the process and are instead Integrating. You have to know some data, independent of the Integration process, that allows you to solve for the constant.
It all gets handled well in the end by the mathematicians. Physics is another matter. There is interpretation and invention that invades and, I think, often overpowers the mathematics causing it to serve 'interpretation and invention' rather than serve to reveal what empirical evidence is communicating to us.
James Putnam
