An interesting method of making derivatives rigorous as fractions that I have found is by introducing a non-associative algebra (related to the dual numbers), by having a value ε such that ε^2=0, and it’s reciprocal ω such that ε*ω=1 (which is non-associative in some situations so be careful due to ε*ε*ω being either ε or 0), we can define the differential as df(x)=f(x+ε)-f(x), then we can see that dx=x+ε-x=ε, and therefore the ratio of differentials df(x)/dx is precisely the derivative. This can be proven using Taylor series. For the multi-variable case we can introduce more ε like elements such that for A and B that are epsilon like, we have A/B=0 if A≠B, and A/B=1 if A=B, and the product of epsilon like values is non-zero if the epsilon like elements are different whereas if the epsilon like values are the same, the product is zero. we can generalise the ratio formula for products of epsilon likes by cancelling out the values, and if they can’t cancel, the ratio is zero. The new differential operator is d_{n}f(x,y,z)=f(x+n,y,z)-f(x,y,z) where n is the epsilon like value associated with x, then df(x,y,z) is the sum over each epsilon like value “t” of d_{t}f(x,y,z), this can be extended to any amount of independent variables. We end up with the multi-variable chain rule and other nice results.
Nice thank you soooo much!!! Yes when I wrote the article I was a bit confused about differentials but now I would have also added a section when the differential of a function is the linear increment of the function at that point df=f'(x_0)Δx and if the function is f(x)=x (the identity function) we would get dx=Δx and so being dx=Δx df=f'(x_0)dx and the ratio of the differentials equals the derivative of the function at that point. Then by picking an arbitrary point, df is a function of both the point where the derivative is taken and the linear increment dx.
@@JonnyMath I really liked the essay, it was pretty comprehensive about the main approaches to differentials and infinitesimals. The main approach I have seen for differentials is differential forms, which is pretty interesting due to how it represents the fundamental theorem of calculus in higher dimensions. I would also like to clarify a few things about my approach for infinitesimals: the approach I provided is actually mostly my own discovery and therefore there is basically no info about it online as I haven’t published it yet. I was inspired by the Wikipedia article of automatic differentiation where it says that before the limit definition was invented, mathematicians just let dx^2=0 and then cancel out somewhat unrigorously, and I was also a bit annoyed about the limit approach and the hyper real approach both requiring an additional function to remove a factor of dx, so I just invented a new number system to get rid of the problems and make the dx^2=0 approach rigorous. To make the approach make more sense (also because I didn’t like how function notation was interacting with calculus) I invented some new notation which I call expression notation (which I have seenpeople use it implicitly but never explicitly, as they always change the notation back to function notation after doing algebraic manipulations with it), we can define the dependent variable f as f=x^2, instead of f(x)=x^2, this method also make coordinate transforms easy notation wise, as we can equally define x as x=f^{1/2}, this also makes the notation for my differential approach easier, making df=(f with x at x+ε)-f, which may seem like more work, however the notation makes more sense because now instead of f(x)=x^2 and f(y)=y^2 being equivalent definitions of the function f, making df/dx ambiguous as we don’t have f being dependent of x or y due to f being a function. To take the derivative of a function, using partial derivatives makes more sense, the partial derivative notation that I personally use is (\partial_{1}f)(x)=df(x)/dx, which in general \partial_{n} is the derivative of the function f with respect to the nth component.
The advantage of Leibnizian differentials is that one quantity does not have to be a function of another quantity. Also dx, dt, etc are not necessarily constants. There is a person named Jonathan Bartlett who wrote about that.
Guys if you want to you can download and read the essay as well as other essays at the link in the description!!!😊🤩
An interesting method of making derivatives rigorous as fractions that I have found is by introducing a non-associative algebra (related to the dual numbers), by having a value ε such that ε^2=0, and it’s reciprocal ω such that ε*ω=1 (which is non-associative in some situations so be careful due to ε*ε*ω being either ε or 0), we can define the differential as df(x)=f(x+ε)-f(x), then we can see that dx=x+ε-x=ε, and therefore the ratio of differentials df(x)/dx is precisely the derivative. This can be proven using Taylor series. For the multi-variable case we can introduce more ε like elements such that for A and B that are epsilon like, we have A/B=0 if A≠B, and A/B=1 if A=B, and the product of epsilon like values is non-zero if the epsilon like elements are different whereas if the epsilon like values are the same, the product is zero. we can generalise the ratio formula for products of epsilon likes by cancelling out the values, and if they can’t cancel, the ratio is zero. The new differential operator is d_{n}f(x,y,z)=f(x+n,y,z)-f(x,y,z) where n is the epsilon like value associated with x, then df(x,y,z) is the sum over each epsilon like value “t” of d_{t}f(x,y,z), this can be extended to any amount of independent variables. We end up with the multi-variable chain rule and other nice results.
Nice thank you soooo much!!! Yes when I wrote the article I was a bit confused about differentials but now I would have also added a section when the differential of a function is the linear increment of the function at that point df=f'(x_0)Δx and if the function is f(x)=x (the identity function) we would get dx=Δx and so being dx=Δx df=f'(x_0)dx and the ratio of the differentials equals the derivative of the function at that point. Then by picking an arbitrary point, df is a function of both the point where the derivative is taken and the linear increment dx.
Thanks for your comment!!!🤗🤗🤗
@@JonnyMath I really liked the essay, it was pretty comprehensive about the main approaches to differentials and infinitesimals. The main approach I have seen for differentials is differential forms, which is pretty interesting due to how it represents the fundamental theorem of calculus in higher dimensions. I would also like to clarify a few things about my approach for infinitesimals: the approach I provided is actually mostly my own discovery and therefore there is basically no info about it online as I haven’t published it yet. I was inspired by the Wikipedia article of automatic differentiation where it says that before the limit definition was invented, mathematicians just let dx^2=0 and then cancel out somewhat unrigorously, and I was also a bit annoyed about the limit approach and the hyper real approach both requiring an additional function to remove a factor of dx, so I just invented a new number system to get rid of the problems and make the dx^2=0 approach rigorous. To make the approach make more sense (also because I didn’t like how function notation was interacting with calculus) I invented some new notation which I call expression notation (which I have seenpeople use it implicitly but never explicitly, as they always change the notation back to function notation after doing algebraic manipulations with it), we can define the dependent variable f as f=x^2, instead of f(x)=x^2, this method also make coordinate transforms easy notation wise, as we can equally define x as x=f^{1/2}, this also makes the notation for my differential approach easier, making df=(f with x at x+ε)-f, which may seem like more work, however the notation makes more sense because now instead of f(x)=x^2 and f(y)=y^2 being equivalent definitions of the function f, making df/dx ambiguous as we don’t have f being dependent of x or y due to f being a function. To take the derivative of a function, using partial derivatives makes more sense, the partial derivative notation that I personally use is (\partial_{1}f)(x)=df(x)/dx, which in general \partial_{n} is the derivative of the function f with respect to the nth component.
Thanks!!!😉🤗
The advantage of Leibnizian differentials is that one quantity does not have to be a function of another quantity. Also dx, dt, etc are not necessarily constants. There is a person named Jonathan Bartlett who wrote about that.
I read through your essay! It’s very fun haha. Well done!
Thanks!!!!
in which app did you write your essay
Overleaf, it's an online latex editor!!! There are a lot of videos on UA-cam about how to use it!!! It's worth learning it!!!😉🤗
@@JonnyMath can i have a link?
Type overleaf latex and open the first website
Then you create your account and the job is done!!!
@@JonnyMath thanks