Differentials and Measurement Error

[mikegonz00]

Hello there,

I'm reviewing differentials from calculus, and I'm a bit confused about how we can interpret them as measurement errors.

For example, the volume of a sphere is given by V=4/3πr³. The differential form: dV=4πr²dr. Let dr=r₂-r₁. On a graph of V with respect to r, dV is the (vertical) length corresponding to the difference y(r₂)-y(r₁), where y is the linear approximation at r₁. I'm just confused about how that relates to the measurement error

 

[Dr.Peterson]

The ratio (y(x₂)-y(x₁))/(x₂-x₁) approximates dy/dx. One is an average slope, and the other is an instantaneous slope.

In your example, dV/dr approximates (V(x₂)-V(x₁))/(r₂-r₁), so if you calculate dV, using a given dr, the result is an approximate error in V caused by the error in r.

 

[mikegonz00]

Thanks, I think that helped. I'm probably leaving the realm of posts meant for this discussion board, but, as a follow-up, it looks like this method of calculating measurement error is typically only useful when our instruments are accurate. Otherwise, dV is more likely to underestimate or overestimate the actual value of V(r₂)-V(r₁)=(Error in V) by a significant amount.

 

[Dr.Peterson]

View attachment 25900
Thanks, I think that helped. I'm probably leaving the realm of posts meant for this discussion board, but, as a follow-up, it looks like this method of calculating measurement error is typically only useful when our instruments are accurate. Otherwise, dV is more likely to underestimate or overestimate the actual value of V(r₂)-V(r₁)=(Error in V) by a significant amount.

Yes, differentials are assumed to be "small", the meaning of which depends on the function.