Two kinds of vector rates

This post builds on the previous one here. Vector rates rates of change are of two kinds. An ordinary rate for the vector change of f relative to a unit of x is defined as: The reciprocal vector rate is the vector reciprocal of an ordinary rate with a vector change of g relative to …

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Two kinds of rates

Rates of change are of two kinds. An ordinary rate for the change of f relative to a unit of x is defined as: The reciprocal rate is the reciprocal of an ordinary rate with a change of g relative to a unit of x is defined as: An ordinary rate has its independent variable …

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From ratios to quotients

Ratios and proportions are symmetric. A:B = B:A and A:B :: C:D = C:D :: A:B. But when ratios are converted to quotients or fractions, they are no longer symmetric. There must be a convention as to which is the denominator and which is the numerator. In an ordinary fraction or quotient or rate the …

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Space Time Speed Pace and Mean

Legend: x-axis coordinate, x; length interval, Δx; unit of length, x̂; t-axis coordinate, t; time interval, Δt; unit of time, t̂. For scalars: Rates Speed/Pace Symbols Space Time Ordinary Time speed Δx(t)/t̂ measure length/distance/ displacement given period/unit Alternate Space speed x̂/Δt(x) given length/distance/unit measure interval/period /dischronment Inverse Space pace Δt(x)/x̂ given length/distance/unit measure interval/period /dischronment …

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Duality of subject and object

This post reflects a previous one here. Color (or colour) is both subjective and objective. Objectively, the rays of color light from a glass prism are different wavelengths (or frequencies) of light. The colors we see are those that reflect from objects; the others are absorbed. Colors are additive. Primary colors are red, green, blue; …

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Instantaneous time and space speed

This post relates to the previous one here. Speed is the time rate of distance traversed. Pace is the space rate of elapsed time. The time speed of a body is the distance traversed per unit of independent time without regard to direction, Δx/t̂. The instantaneous speed is the time speed at a point in …

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Reciprocal derivative

The reciprocal difference quotient is The reciprocal derivative of f(x), symbolized by a reversed prime, is the limit of the reciprocal difference quotient as h approaches zero: An alternate limit form of the reciprocal derivative definition of f(a) is The reciprocal derivative of a linear function, f(x) = ax + b, is The reciprocal derivative …

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Adding and averaging rates

A rate is the quotient of two quantities with different, but related, units. A unit rate is a rate with a unit quantity, usually the denominator. A vector rate is a rate with a vector quantity, usually the numerator. Rates with the same units may be added, subtracted, and averaged. Addition Rates with the same …

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Harmonic vector realm

This post expands on Harmonic Algebra posted here. A vector space, or better a vector realm, to avoid connecting it with physical space, is a set V on which two operations + and · are defined, called vector addition and scalar multiplication. The operation + (vector addition) must satisfy the following conditions: Closure: If u …

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Vector inverse and mean

This post is based on research papers by Anderson and Trapp, Berlinet, and the post on Harmonic algebra. The vector inverse x−1 is defined as with positive norm. For a non-zero scalar k, The harmonic (or parallel) sum is usually symbolized by a colon (:), but I prefer a circle plus to maintain its relation with …

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