What Are Infinitesimals of Order Higher Than Δx, Δy, and Δz?

[koustav]

in a certain problem it was written Δθ=∂θ/∂x*Δx + ∂θ/∂y*Δy + ∂θ/∂z*Δz + infinitesimals of order higher than Δx,Δy and Δz.can anyone tell me what is "infinitesimals of order higher than Δx,Δy and Δz?"

 

[arildno]

Squares, cubes and so on.

For "a" tiny, a^2 is even tinier.

 

[koustav]

yes but there are no square or cube terms inthe expression

 

[D H]

Sure there are. It's the "infinitesimals of order higher than Δx,Δy and Δz". Expanding that Δθ to incorporate some of the second order terms yields

Δθ = (∂θ/∂x)Δx + (∂θ/∂y)Δy + (∂θ/∂z)Δz
+ (1/2)((∂²θ/∂x²)Δx² + (∂²θ/∂x∂y²)ΔxΔy + (∂²θ/∂x∂z²)ΔxΔz + (∂²θ/∂y∂x²)ΔyΔx + ∂²θ/∂y²)Δy² + ···)
+ terms of order three and higher.