Can Differentials and Errors be Split in Calculus?

In summary: If you want to do higher order derivatives, you need to use more sophisticated methods.In summary, differentials are a way to describe how one variable changes with another when the variables are not represented directly. Differentials are invariant, which means that they remain the same if the coordinates are changed. Differentials can be used to understand how one variable changes with another when the variables are not represented directly. Differentials are used to approximate a curve near a point. Differentials are only accurate for first order derivatives.
  • #1
KingBigness
96
0
I asked a question a few weeks ago about 'splitting' the derivative. The thread can be found https://www.physicsforums.com/showthread.php?p=3581188#post3581188"

The answer to why it can not be split is because dx does not exist, it is simple a notation and not a fraction.

However, I just started Differentials and Errors, and the paper I read said.

The ratio of the two derivatives is actually the derivative of the function.

[itex]f'(x)=\frac{dy}{dx}[/itex]

and the relationship between the two differentials can be given by

[itex]dy=f'(x)dx[/itex]

Is this not 'splitting' the derivative?

Thank you
 
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  • #2
dx does exist and dy=f′(x)dx is fine

The actual definitions in use can very and sometimes confusion can arise. Often the implicit function theorem and its conditions are in play.
 
  • #3
KingBigness said:
The answer to why it can not be split is because dx does not exist, it is simple a notation and not a fraction.The ratio of the two derivatives is actually the derivative of the function.
[itex]f'(x)=\frac{dy}{dx}[/itex]

and the relationship between the two differentials can be given by
[itex]dy=f'(x)dx[/itex]

Is this not 'splitting' the derivative?
I think you mean the ratio of two differentials , yes you are right , dy and dx are merely notations , and don't exist seperately but when you write [itex]dy=f'(x)dx[/itex] , dy becomes function of variables x and dx , and meaningful when used together as in the case of dy/dx , also see the following link
http://en.wikipedia.org/wiki/Differential_of_a_function" [Broken]
 
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  • #4
That is like saying 3/4 is merely a notation and 3 and 4 do not exist separately.
The advantage of differentials is when we have several variable we can understand how each changes with the others independent of the representation as differentials are invariant. dy/dx means the same thing if we have
y=f(x)
x=g(y)
h(x,y)=0
w(u(x,y),v(x,y))=0
and so on
it would be quite limiting to say dy/dx=f'(x) the end, if we change variables we will start from scratch.
 
  • #5
lurflurf said:
That is like saying 3/4 is merely a notation and 3 and 4 do not exist separately.
The advantage of differentials is when we have several variable we can understand how each changes with the others independent of the representation as differentials are invariant. dy/dx means the same thing if we have
y=f(x)
x=g(y)
h(x,y)=0
w(u(x,y),v(x,y))=0
and so on
it would be quite limiting to say dy/dx=f'(x) the end, if we change variables we will start from scratch.
well said ! , But i don't say dy/dx is merely a notation , tell me what's dx ? ( without talking anything about dy)
 
  • #6
Again different definitions of differential are used in different contexts so this is for the purpose of regular calculus
dx is just some variable for an arbitrary change in x
which is not the most interesting part
given some sufficiently well-behaved curve or relationship of variables we can define a tangent space that depends upon the point and the curve at that point, but in no way depends upon specific coordinate choices. The projection of the curve is a linear function of the differentials and approximates the curve well near the point.
thus
f(x+dx)-f(x) is some function of x and dx
and
f'(x)dx is some function of x and dx
they do not agree in general only approximately near x

This only works so well for total differentials of first order.
 

1. What is a differential?

A differential is a mathematical concept that describes the instantaneous rate of change of a function at a specific point. It is represented by the symbol "d" followed by the variable with respect to which the change is being measured.

2. How do you calculate differentials?

To calculate a differential, you can use the derivative of the function. This involves finding the derivative of the function with respect to the variable, and then substituting the specific point into the derivative. The resulting value is the differential.

3. What is the difference between absolute and relative error?

Absolute error is the difference between the measured value and the true value of a quantity. Relative error, on the other hand, is the ratio of the absolute error to the true value. It is often expressed as a percentage.

4. How do you calculate absolute and relative error?

Absolute error can be calculated by subtracting the measured value from the true value. To calculate relative error, divide the absolute error by the true value and multiply by 100 to get a percentage.

5. What is the significance of errors in scientific experiments?

Errors are an inevitable part of scientific experiments and measurements. They can occur due to a variety of factors such as equipment limitations, human error, and natural variation. Understanding and minimizing these errors is crucial in ensuring the accuracy and reliability of scientific results.

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