Should I Use Total or Partial Differentiation in Absolute Error Calculation?

In summary, the conversation discusses calculating the absolute error of the quantity p, which is defined as p = a*b where a and b are two variables. The conversation also mentions using logarithms and differentiating to find the absolute error. The speaker then asks which method is most correct, to which the other person responds that using d is the correct method. The conversation also briefly touches on different methods for calculating absolute error, with the preferred method being E[f(x_1,x_2,...,x_n)]=\sqrt{\sum_{k=1}^n (\frac{\partial f}{\partial x_k} dx_k)^2}.
  • #1
Chemist@
115
1
The absolute error of the quantity p should be calculated.
p=a*b where a and b are two variables.
By taking ln on both sides:
lnp=lna+lnb, then differentiating:
dp/p=da/a+db/b
dp=abolute error
Should I have used total or partial differential in the differentiation step?
 
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  • #2
Doesn't matter. [itex] da [/itex] or [itex] \partial a [/itex] or [itex] \Delta a [/itex], they all represent the error in measuring a.
 
  • #3
Which is the most correct to use when I differentiate a function of two variables? Partial derivative?
 
  • #4
Well, now that I think, [itex] d [/itex] is the correct one. Because you're actually differentiating [itex] \ln p [/itex] w.r.t. p regardless of what p depends on! The same for other two logarithms.
Anyway, I've heard several methods for calculating absolute errors. But the one I prefer, is the following:
[itex]E[f(x_1,x_2,...,x_n)]=\sqrt{\sum_{k=1}^n (\frac{\partial f}{\partial x_k} dx_k)^2}[/itex]
Because it obviously gives the maximum change in f for given changes in its arguments.
 
  • #5


It is appropriate to use partial differentiation in this case since we are only considering the effects of changes in a and b on the overall quantity p. Total differentiation would be used if we were considering the effects of changes in all variables on p.
 

FAQ: Should I Use Total or Partial Differentiation in Absolute Error Calculation?

What is absolute error calculation?

Absolute error calculation is a method used to determine the difference between the measured value and the true value of a quantity. It provides a numerical value that represents the amount of error present in a measurement.

How is absolute error calculated?

To calculate absolute error, you first need to know the measured value and the true value of the quantity. Then, you subtract the true value from the measured value and take the absolute value of the result. The final value represents the absolute error.

What is the difference between absolute error and relative error?

The main difference between absolute error and relative error is that absolute error is a numerical value that represents the difference between the measured and true value, while relative error is expressed as a percentage and compares the absolute error to the true value.

Why is absolute error important in scientific experiments?

Absolute error is important in scientific experiments because it allows us to assess the accuracy of our measurements. By knowing the amount of error present, we can make adjustments to improve the precision and reliability of our results.

What are some sources of absolute error in scientific experiments?

There are several sources of absolute error in scientific experiments, such as limitations of measuring instruments, human error, and experimental conditions. It is important to identify and minimize these sources of error to ensure the accuracy of our measurements.

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