Algebra in Calculus: Understanding Integration & Differentials

[elementbrdr]

My question is very basic, and maybe I'm just having a brain malfunction, but I'm curious why it's ok to integrate both sides of an equation. It's pretty easy to integrate a lot of functions, but there are a lot of operations implicitly being performed in the background of any integration. So I'm just wondering how we know that, when integrating both sides of an equation, the equality of the sides is preserved.

In a related vein, I'm having trouble understanding the types of algebraic manipulation that can be performed on differentials. For example, I've viewed a number of Khan Academy videos where Sal pretty casually multiplies both sides of an equation by dy or dx. This makes sense to me. However, I have read that this is an abuse of notation.

Thank you.

 

[clanijos]

separation of dy/dx is not considered to be an abuse of notation because it is considered to be a ratio equal to 1. It works in this case, because it's not an operator, it's a "number", the differential.

 

[AlephZero]

It's not really an abuse of notation, even if it looks like one.

Suppose you separate the variables to get
$$f(y)\frac{dy}{dx} = g(x)$$
for some functions f and g.
Now integrate both sides with respect to the same variable, x
$$\int f(y)\frac{dy}{dx}\,dx = \int g(x)\,dx$$
Now change the variable in left hand integral using the chain rule
$$\int f(y)\,dy = \int g(x)\,dx$$
And integrate ...

The only "abuse of notation" is that in "real life" nobody writes this out in full.

 

[DivisionByZro]

separation of dy/dx is not considered to be an abuse of notation because it is considered to be a ratio equal to 1. It works in this case, because it's not an operator, it's a "number", the differential.

That's not true at all. dy/dx is the rate of change of y with respect to x (i.e. a small change in y and a small change in x). It can sometimes be equal to 1, but in general it is not. Some people consider it to be an abuse, others do not.

dx and dy by themselves are called differentials. "dy/dx" is a derivative. These two concepts are very different things and are not to be confused.

This is what dy/dx is:
$$\frac{dy}{dx}= \lim_{h\rightarrow 0}\frac{f(x+h)- f(x)}{h}$$

It is a limit. So it does not make sense to separate dy from dx as it is the limit of two quantities (Not a conventional fraction). However, it does work when using regular first-order derivatives. It fails to work for higher derivatives.

 

[elementbrdr]

It's not really an abuse of notation, even if it looks like one.

Suppose you separate the variables to get
$$f(y)\frac{dy}{dx} = g(x)$$
for some functions f and g.
Now integrate both sides with respect to the same variable, x
$$\int f(y)\frac{dy}{dx}\,dx = \int g(x)\,dx$$
Now change the variable in left hand integral using the chain rule
$$\int f(y)\,dy = \int g(x)\,dx$$
And integrate ...

The only "abuse of notation" is that in "real life" nobody writes this out in full.

Are you applying the chain rule to a differential in your second to last step? I didn't realize you could do that. I thought the chain rule applied to functions. Sorry for the confusion.

 

[clanijos]

Regarding this:

That's not true at all

First of all, Thank you good sir for ever so eloquently regurgitating the definition of the derivative.

Secondly, Leibniz describes the derivative as "the quotient an infinitesimal increment of y by an infinitesimal increment of x", which makes it perfectly acceptable to split up in a differential equation.

I do admit, my statement earlier in this thread was not correctly phrased. What I meant was (dy/dx):1. A ratio.

Thanks for the reply

 

[mathwonk]

when you integrate both sides of an equation, they results may differ by a constant (of integration).

 

[jgens]

Secondly, Leibniz describes the derivative as "the quotient an infinitesimal increment of y by an infinitesimal increment of x", which makes it perfectly acceptable to split up in a differential equation.

This would only be a sufficient argument if you believe Lebiniz's formulation of calculus was rigorous. Most mathematicians do not quite believe that. Even in non-standard analysis, where you actually have infinitesimals, the derivative still is not quite a quotient of infinitesimals; in that context, it is the standard part of a quotient of infinitesimals.

 

[clanijos]

Indeed his calculus may not be rigorous, but it is generally accepted that it can be used to make things easier. Regardless of all the things mentioned here, and unsurprisingly, solving differential equations is not an abuse of notation. If that were to be true, the world would be chaos! CHAOS I tell you! Cheers everyone.

 

[Stephen Tashi]

Not surprisingly, the question of whether Leibnitz-like manipulations is an "abuse of notation" can't be settled unless we agree on a rigorous definition of "abuse of notation".

If we agree not to bother to agree on a rigorous definition then we can agree to disagree, which might be the more amusing and natural course of action.