What is the "Engine of Calculus" in Differential Equations?

[Substance D]

"engine of calculus"

hey

hope this is the correct section, but I am just starting to learn differential equations and in our book there is a quote about how differentials are the "engines of calculus". Could someone explain that a bit better to me?

Thanks!

 

[HallsofIvy]

Darned if I know what is meant! Certainly, basic calculus concerns itself with two basic concepts- the integral and the derivative. Most modern calculus books start with the derivative and then introduce the integral in terms of the "anti- derivative". Perhaps that is what is meant.

 

[Substance D]

but isn't there a sleight difference between derivatives and differentials?

 

[symbolipoint]

hey

hope this is the correct section, but I am just starting to learn differential equations and in our book there is a quote about how differentials are the "engines of calculus". Could someone explain that a bit better to me?

Thanks!

The word, "engine", might not be the best vocabulary choice; but differentials are an essential detail constituting Calculus. A differential is an infinitessimally small increment.

 

[Substance D]

ah, ok I see what you mean. when I hear "engine" I picture the driving force behind something, which is why it didnt make sense to me or I thought I was missing something fundamental to calculus.

 

[z-component]

The differential determines which variable is affected by the differentiation and/or integration, so we can say that they are the "engines" for differentiation and integration, I suppose. I've come across this when students in introductory calculus forget to write their differentials, and don't realize how important they are to the problem.

 

[Substance D]

by "differentials" it would be like dy = 2x dx (if y = x^2), then dx would be the differential? so what's the difference between that and differential equations, sorry if this sounds like a really confused question. thanks for all the replies

 

[HallsofIvy]

If y= x², then dy/dx= 2x is the "derivative of y with respect to x". In dy= 2xdx, both dx and dy are "differentials". A "differential equation" is an equation that involves either derivatives or differentials of the unknown function.

 

[Substance D]

and so when the text asks you to solve the differential equation, you are working with the differentials and going back to get the original function that has those differentials, correct?

 

[Gib Z]

Yup :) [ 10 character limit ]

 

[Substance D]

perfect!

thanks for all the replies!