Propagation of uncertainties with calculus

In summary, the partial derivative of y in the equation F = x/y comes out negative. There is no need to take the absolute value of the partial of y, but when adding the two partials gives zero uncertainty, the uncertainty for that particular measurement would be zero. To get uncertainty for two independent variables, you need to take the square root of the sum of squares. Statistically uncorrelated errors add in quadrature.
  • #1
Darkmisc
204
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Suppose F = x/y

dF= [itex]\frac{\partialF}{\partialx}[/itex][itex]\delta[/itex]x+[itex]\frac{\partialF}{\partialy}[/itex][itex]\delta[/itex]y

This gives

dF=[itex]\frac{\deltax}{Y}[/itex]-[itex]\frac{x}{y^2}[/itex][itex]\delta[/itex]y


That is, the partial derivative of y comes out negative. Should i leave it as a negative?

I see no reason to take the absolute value of the partial of y, but what happens when adding the two partials gives zero uncertainty? Would the uncertainty for that particular measurement just be zero?
 
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  • #2
Your expression for dF is correct. However the interpretation as far as uncertainty is concerned is flawed. To get uncertainty for 2 independent variables you need square root of sum of squares. This is what you need unless there is some relationship between x and y.
 
  • #3
Statistically uncorrelated errors add in quadrature:

[tex] dF^2=\left( \frac{\partial F}{\partial x}\delta x \right)^2 +\left( \frac{\partial F}{\partial y}\delta y \right)^2 [/tex]
 

1. What is the purpose of using calculus in propagating uncertainties?

Calculus is used to calculate the propagation of uncertainties in order to determine the overall uncertainty in a final measurement or result. It allows for a more accurate and precise estimation of uncertainty by taking into account the potential errors in each individual measurement or variable.

2. How does calculus account for uncertainties in measurements?

Calculus accounts for uncertainties by using the concept of derivatives and partial derivatives. These mathematical tools are used to find the rate of change of a function with respect to a variable, which can then be used to calculate the uncertainty in the final result.

3. What is the difference between absolute and relative uncertainties?

Absolute uncertainty is a numerical value that represents the range of possible values for a measurement, while relative uncertainty is a percentage or ratio that compares the absolute uncertainty to the actual measurement. Calculus can be used to propagate both types of uncertainties.

4. Can calculus be used to determine uncertainties in non-linear equations?

Yes, calculus can be used to determine uncertainties in non-linear equations through the use of Taylor series expansion. This allows for the estimation of uncertainties in more complex equations that cannot be solved using basic algebraic methods.

5. Are there any limitations to using calculus in propagating uncertainties?

One limitation is that calculus assumes that uncertainties are normally distributed, which may not always be the case. Additionally, it may not account for all sources of uncertainty, such as human error or systematic errors. It is important to consider these factors when using calculus to propagate uncertainties.

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