Calculus 101
Notation
- d means "a little bit"
- dx means "a little bit of x"
- ∫ means "the sum of"
- ∫ dx means "the sum of all the little bits of x"
Integration
x = ∫ dx — x is the sum of all the little bits of x, its "integral", it's "the whole"
Small Quantities
In our processes of calculation we have to deal with small quantities of various degrees of smallness.
We shall also have to learn under what circumstances we may consider small quantities to be so minute that we may omit them from consideration. Everything depends upon relative minuteness.
Functions
Explicit function
Expresses one variable directly in terms of the other, such as y = f(x) or x = g(y).
Implicit function
One where the relationship between x and y is mixed, e.g. x² + y² = l².
Dependent variable
A variable whose value depends on another (e.g. y = x + 1, y is dependent).
Independent variable
A variable whose value does not depend on another (e.g. y = x + 1, x is independent).
Differentiation
dy/dx
The differential coefficient of y with respect to x, a.k.a the ratio of dy/dx.
Finding the ratio
Finding the ratio of dy/dx is called differentiating. But specially the value of this ratio when both dy and dx are themselves indefinitely small. The true value of the differential coefficient is that to which it approximates in the limiting case when each of them is considered as infinitesimally minute.
Notation for derivatives
If y = F(x), which means that y is some unspecified function of x, we may write F′(x) instead of d(F(x))/dx.
Similarly, F″(x) will mean that the original function F(x) has been differentiated twice over with respect to x.
When doing several differentiations we might also note it as d²y/dx² since d(dy/dx)/dx. This is not algebra but symbolic representation.
Historical Notation
Leibniz notation
dy/dx — the notation we commonly use today.
Newton notation (fluxion)
y with a dot on top (ẏ). Called "fluxion" because it looked at how values were flowing.
The problem is that this notation doesn't give you the independent variable to which the differentiation has been effected. This works if the independent variable is always the same one such as time (t). Since you can always assume it will be that.