Error Propagation Index Of Refraction
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with: (1) Functions of several variables. (2) Evaluation of partial derivatives, and the chain rules of differentiation. (3) Manipulation of summations in refractive index calculation algebraic context. At this mathematical level our presentation can be briefer.
Snell's Law
We can dispense with the tedious explanations and elaborations of previous chapters. 6.2 THE CHAIN RULE derivative calculator AND DETERMINATE ERRORS If a result R = R(x,y,z) is calculated from a number of data quantities, x, y and z, then the relation: [6-1] ∂R ∂R
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∂R dR = —— dx + —— dy + —— dz ∂x ∂y ∂z
holds. This is one of the "chain rules" of calculus. This equation has as many terms as there are variables. Then, if the fractional errors are small, the differentials dR, dx, dy and dz may be replaced by the absolute errors ΔR, Δx, Δy, and Δz, and written: [6-2] ∂R ∂R ∂R ΔR ≈ —— Δx + —— Δy + —— Δz ∂x ∂y ∂z Strictly this is no longer an equality, but an approximation to DR, since the higher order terms in the Taylor expansion have been neglected. So long as the errors are of the order of a few percent or less, this will not matter. This equation is now an error propagation equation. [6-3] Finally, divide equation (6.2) by R: ΔR x ∂R Δx y ∂R Δy z ∂R Δz —— = —————+——— ——+————— R R ∂x x R ∂y y R ∂z z The factors of the form Δx/x, Δy/y, etc are relative (fractional) errors. This equation shows how the errors in the result depend on the errors in the data. Eq. 6.2 and 6.3 are called the standard form error equations. They are also called determinate error equations, because they are strictly valid for determinatephysical quantities such as length and time are measured. These are then analysed in order to compare with the predictions of theory and/or with the results of related experiments. However, no measurement of a physical quantity is exact. There is always some uncertainty in the value obtained. To decide whether a measurement agrees with another, or with a theoretical prediction, we are interested in whether or not the measurement agrees with the prediction "within the uncertainty"; that is, whether or not the predicted value falls https://www.lhup.edu/~dsimanek/scenario/errorman/calculus.htm in the range covered by the measurement plus or minus its uncertainty. For example, in an experiment to measure the value of the acceleration due to gravity, g, you measured it to be 9.6 m/s2. Does this agree with the theoretically predicted value of 9.8 m/s2. We can't say until we know what the uncertainties on the measured http://www.physics.mcmaster.ca/undergrad/Uncertainties/U.htm value are. If the measured value is 9.6 ± 0.3m/s2, then our measurement does agree "within the errors". If it is 9.6±0.1m/s2, then it does not (since the uncertainty says that the maximum value of our measurement that we believe is 9.7 m/s2). If we want to compare two different measurements to see if they agree, then we need to look at the errors on each measurement and see if the ranges overlap. Systematic Errors and Random Uncertainties There are two classes of uncertainties, which arise under different circumstances in an experiment. Systematic errors are due to biases inherent in the experiment. They are always a concern as the experimenter may be unaware of their presence. Experiments must be carefully designed to eliminate systematic errors. A simple example of the source of such an error is the use of a voltmeter that does not read zero when the voltage is zero - all the readings made with it will be in error by a fixed amount. A second example is the neglect of fr
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