5.3 Leibnitz's Calculus

The systems of beginning level studies of physics and mechanics are typically some solid, or statically constrained liquid or gas. Motions of systems that are, and remain solid or constrained, are easier studies than are motions of liquids and gases in general. Here, beginning with that simple vantage of matter, a techniques of calculus is explained then used to describe more complicated motion of matter - motion of liquids and gases.

Derivative of an Integral with a Variable Limit:

Given:

(1)1

then:

(2)1

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5.3 Leibnitz's Calculus

Matter and its motion as treated in physics and mechanics is at the beginning level some solid, or statically constrained liquid or gas. This is an easy beginning point. Motions of systems that are and remain solid are easier studies than are motions of liquids and gases. Here, beginning with that simple vantage of matter, techniques of calculus are explained then used to describe more complicated motion of matter - motion of liquids and gases.

Derivative of an Integral with a Variable Limit:

If one has something expressed as an integral, what is the time derivative? A theorem from of calculus states the derivative with respect to x of the integral of a function of "f(u)," ("u" being a dummy variable) equals the differential of the integrand of the integral as a function of "x." Thus given:

(1) 1

Then:

(2) 2

While this result is commonly understood there is a tendency to forget how the result is obtained. Beginning with the following equation, correct steps are taken to arrive at the result.

(3) 3

The derivative of the constant is zero. Inspecting the next term we observe that it is a derivative of an integral that has a variable upper limit. Leibnitz has shown that this derivative equals another integral plus a term for the upper limit (term_upper) minus a term for the lower limit (term_lower). Thus

(4) 4

Again the integral is zero. The "upper" and "lower" limit terms are evaluated in a similar fashion. The term for the upper limit is the derivative of the upper limit with respect to "x" times the integrand, f(u) evaluated at the upper limit. Thus:

(5) 5

Consequently,

(6) 6

which was to be proved.