![]() ![]() to construct a logically satisfactory definition of a "differential", as was done by Cauchy in particular. \frac=0$.ĭifferential calculus has many subtleties which require quite a bit of machinery to elucidate in a rigorous manner (the basic language of differential calculus: tangent spaces, cotangent spaces, differential forms, vector fields, etc.).D f ( x ) = f ′ ( x ) d x. Neither of these, standing alone, have a real meaning. Small quantity of the first order small enough in itself. N all cases we are justified in neglecting the small quantities of the second - or third (or higher) - orders, if only we take the SECTION 2.2 defines continuity and discusses removable discontinuities. SECTION 2.1 introduces the concept of function and discusses arithmetic operations on functions, limits, one-sided limits, limits at ±, and monotonic functions. But $dx \cdot dx$ would be negligible,īeing a small quantity of the second order. IN THIS CHAPTER we study the differential calculus of functions of one variable. Not follow that such quantities as $x \cdot dx$, or $x^2 \cdot dx$, or If $dx$ be a small bit of $x$, and relatively small of itself, it does In hopes of further clarifying the question, I'm including a little more of what Thompson said: So I'm wondering, is there a difference between $d^2x$ and $(dx)^2$? Should I read $d^2x$ as "a little bit of a little bit of $x$" and $(dx)^2$ as "a little bit of a little bit of $x^2$?" Am I missing the point entirely? Have I gone completely mad? From the diagram, it actually seems more significant than $dx$, being $dx$ times greater than $dx$. ![]() Yet, I don't see how this second description is making the case for $(dx)^2$ being any less significant than $dx$. ![]() While not necessarily appealing to my intuition, this description does seem to be more in line with his description of $dx$.Ī little bit $\cdot$ $x$ $\cdot$ a little bit $\cdot$ $x$ $=$ a little bit $\cdot$ a little bit $\cdot$ $x^2$ And in doing so, he backs up his statement with a figure of a square with sides of length $x + dx$ and notes that any one of the corners of the square represents the magnitude $(dx)^2$. However, a little later he describes $(dx)^2$ as "a little bit of a little bit of $x^2$. Now, at this time it seems as though Thompson is making the point that $(dx)^2$ may be considered to be "a little bit of $dx$" or "a little bit of a little bit of $x$," which seems intuitive, but perhaps not consistent with the language. small quantities of the second order of minuteness). Smaller parts, which, in Queen Elizabeth's days, they called "second Smaller subdivisions of time, they divided each minute into 60 still With an hour, and called it "one minute," meaning a minute fraction. Indeed, our forefathers considered it small as compared Obviously 1 minute is a very small quantity of time compared with a Thompson starts his discussion of degrees of smallness with an appeal to time. At times his explanation seems to make sense, and then again it does not. In Chapter 2 he begins discussing the various degrees of "smallness" and this is where I begin to lose track. In Chapter 1 Thompson describes a differential $dx$ as "a little bit of $x$" or "an element of $x$" and then proceeds to describe $\int dx$ as "the sum of all the little bits of $x$." I'm okay upto this point. I've just started reading through Calculus Made Easy by Silvanus Thompson and am trying to solidify the concept of differentials in my mind before progressing too far through the text. ![]()
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