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- Calculus made easy is a book by Silvanus P Thompson on infinitesimal analysis that I have a question regarding.

Hello everyone!

I have quite a bit of experience with standard calculus methods of differentiation and integration, but after seeing some of Walter Lewin's lectures I noticed in his derivation of change in momentum for a rocket ejecting a mass dm, with a change in velocity of the rockey dv, he gets a term of dv*dm, and says we can ignore this term in the overall summation, as it's the product of two very small numbers, it's very small relative to the change of momentum. I believe this is a method of infinitesimal analysis, and I've seen the method discussed in Silvanus P Thompson's book "Calculus Made Easy." I feel I have some intuition of the method, but am lacking a rigorous understanding of our treatment of infinitesimals in calculus and physics. When computing volumes, I sometimes see dV = dxdydz, and people treat this as a differential quantity, but isn't this a product of three incredibly numbers? When can we safely ignore a differential term as being too small to matter in the overall equation?

In the following I write an example of the derivation of the derivative of the product uv with respect to x

$$y = uv$$$$y = (u + du)(v + dv) = uv + u{dv}+ v{du} + {du}{dv}$$Based on the methods of infinitesimal calculus we can ignore ##{du}{dv}##, so we get$$y = uv$$$$y = (u + du)(v + dv) = uv + u{dv} + v{du}$$And we can subtract the first equation and divide by dx to get$$\frac {dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$$In summary, when can we safely ignore a differential term? Is there a way to show when we can safely ignore a specific term as we take the change in the independent variable arbitrarily close to 0 to make a perfect approximation?

Let me know what you guys think, thank you!

I have quite a bit of experience with standard calculus methods of differentiation and integration, but after seeing some of Walter Lewin's lectures I noticed in his derivation of change in momentum for a rocket ejecting a mass dm, with a change in velocity of the rockey dv, he gets a term of dv*dm, and says we can ignore this term in the overall summation, as it's the product of two very small numbers, it's very small relative to the change of momentum. I believe this is a method of infinitesimal analysis, and I've seen the method discussed in Silvanus P Thompson's book "Calculus Made Easy." I feel I have some intuition of the method, but am lacking a rigorous understanding of our treatment of infinitesimals in calculus and physics. When computing volumes, I sometimes see dV = dxdydz, and people treat this as a differential quantity, but isn't this a product of three incredibly numbers? When can we safely ignore a differential term as being too small to matter in the overall equation?

In the following I write an example of the derivation of the derivative of the product uv with respect to x

$$y = uv$$$$y = (u + du)(v + dv) = uv + u{dv}+ v{du} + {du}{dv}$$Based on the methods of infinitesimal calculus we can ignore ##{du}{dv}##, so we get$$y = uv$$$$y = (u + du)(v + dv) = uv + u{dv} + v{du}$$And we can subtract the first equation and divide by dx to get$$\frac {dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$$In summary, when can we safely ignore a differential term? Is there a way to show when we can safely ignore a specific term as we take the change in the independent variable arbitrarily close to 0 to make a perfect approximation?

Let me know what you guys think, thank you!