Basic Thermodynamics ~ J. Pohl © | www.THERMOspokenhere.com (A415) - (A416)) |
Time is relevent to every physical analysis. Description of the event of a physical system requires preciseness in description of time. This a partial review of calculus at its basic, definitional, level.
Scenario: Suppose at a time we designate as t = 0+, a BODY is located at the position, P0+. Suppose further that at t = 0+ and for times thereafter the velocity of the BODY is know by the implicit function, V(t). By this information we can write the position of the BODY as:
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(1) (1) |
What is described mathematically is the position in space of a our system modeled as a BODY. Position (in the equation)is superscripted "0XYZ" to convey that a cartesian space has been defined (origin, coordinate axes and a vector basis) and subscripted "BODY" to designate the system. Position is a vector; vectors are written with and "over-bar."
Determine the derivative of P(t).
♦ Differentiation, a mathematical process, has precise steps. The first step is to apply the differentiation operator, "d/dt" to the entire equation.
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(2) |
The derivative of an equation equals the equation with the derivative taken of each of its terms (Eqn 3). The derivative of the position, P, of a BODY is defined by Newton to be its velocity. Also the derivative of the constant initial position, P0 equals zero (Eqn 4).
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(3) |
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(4) |
We arrive at the mathematical statement:
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(4a) |
To proceed we will apply the classical definition and operation of derivative to the term right of equality. Momentarily let: The definition is:
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(4b) |
The group within brackets is called the "difference quotient." To construct that, we notice immediately that P(t) is known (above left). So construct P(t + Δt):
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(6) |
Then, constructing the quotient we have for the derivative:
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(7) |
Upon inspection of Eqn-7, those who know calculus it is wrong. This writing walked us into the mistake as an easier way of explaining how to avoid it.
♦ ♦ Understanding, attention to detail supported by good notational habits is helpful. To start, we rewrite the initial equation as:
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(8) (1) |
Notation of the above equation is enhanced:
i) Since the equation applies to a "BODY," the subscript "BODY"
is appropriate.
ii) To write a vector requires a space be defined to include an origin,
orthogonal coordinates and a vector basis. The superscript identifies the 0XYZ space.
iii) The distinction, "vector entity" is made by placing an "over-arrow" above
each vector.
iv) In the integral, the lower limit of time is written as: "t = 0+." A subtle distinction is intended. The usual time mark of commencement of an event is simply, at time, t = 0. But so stated, is the event in progress or about to start. This writing uses the notation, "t = 0+" to mean "at the start time but started."
v) By the writing of Eqn-8, it appears that the upper limit t, the of the integrand, t and the integral differential, dt, are related. More of this below.
A further enhancement to Eqn-8 (by one who knows calculus) would be to place a prime notation ( ' ) on differential of the integral and on that variable within the integrand to have the equation below:
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(9) (1) |
Our path is a correct derivative of the above. We write the difference quotient:
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(10) (1) |
By properety of integrals, the above equals:
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(11) (1) |
The Mean Value Theorem of calculus is very useful. For the integral above:
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(12) (1) |
Hence we have:
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(13) (1) |
Notice within the limit t added to zero then Δt divided out. Also for the limit, α is squeezed between t and t + Δ and as Δt vanishes, α becomes t.
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(14) (1) |