Proving Integration Notation for Smooth Function f

In summary, the conversation discusses proving a statement involving a smooth function f:R^n -> R on a neighbourhood of a in R^n, using the integral symbol I and the partial derivative symbol D. The conversation also delves into the meaning of (D/Dt)f(a+t(x-a)), and justifies the steps taken in the original solution. The conclusion is that the original solution is correct, and the expression (D/Dt)f(a+t(x-a)) can be written as (d/dt)f(a+t(x-a)) within the integral.
  • #1
mathboy
182
0
Notation: I = integral sign from 0 to 1, D= partial derivative symbol.

Please help me prove that for any smooth function f:R^n -> R defined on a neighbourhood of a in R^n,

f(x) = f(a) + I{(D/Dt)f(a+t(x-a))dt}

Here's my attempt:
(D/Dt)f(a+t(x-a))dt = d[f(a+t(x-a)] (justification needed?)
so

I [(D/Dt)f(a+t(x-a))dt] = I d[f(a+t(x-a)]
= f(a+1(x-a)) - f(a+0(x-a)) (Fundamental theorem of calculus, right?)
= f(x)-f(a).

Am I right, or am I making many unjustified steps here?
 
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  • #2
Looks fine to me.

After all, set [tex]y=a+t(x-a),t=1\to{y}=x,t=0\to{y}=a,\frac{dy}{dt}=(x-a)[/tex]
Thereby, your integral is readily converted to:
[tex]I=f(a)+\int_{a}^{x}f'(y)dy=f(x)[/tex]
 
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  • #3
Thanks arildno. But there is one major confusion here. Your dy/dt is supposed to have the partial derivative symbols, right? Because y is a function of x=(x_1,...,x_n) and t, i.e. a function of n+1 variables and not just a function of t alone.

Also, y= a + t(x-a) is a function that has n components, because x and a are elements of R^n. So what exactly is the meaning of f'(y)dy when your y is not a real number but a variable in R^n?
 
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  • #4
The only thing I can make out of your f'(y) is a Jacobian matrix, in which case the antiderivative cannot be defined.
 
  • #5
You can regard "a" and "x" as arbitrary constants.
 
  • #6
arildno said:
You can regard "a" and "x" as arbitrary constants.

Ok, I think I'm onto something. Please tell me if my explanation below is correct. Remember, I'm using D for the partial derivative symbol.

Let’s first understand what (D/Dt)f(a+t(x-a)) means: Since x belongs to R^n, then f is a function of the n variables x=(x_1,...,x_n). Now with the introduction of the new independent variable t (which is totally independent of x and vice versa), the expression
f(a+t(x-a)) is now a function of n+1 variables, and hence the partial derivative symbol D/Dt.

Having said that, f(a+t(x-a)) also appears as the integrand in I{(D/Dt)f(a+t(x-a))dt} (I'm using I as the integral symbol from 0 to 1), and here we are integrating with respect to t only. Thus we are treating x as a constant within the expression f(a+t(x-a)) during the process of integration, since the integration is with respect to t only (if it was a double integral where we are also integrating with respect to x, then x is certainly no longer treated as a constant). Thus (within the integral only) we can write (D/Dt)f(a+t(x-a) as (d/dt)f(a+t(x-a)) , whereby we get

I{(D/Dt)f(a+t(x-a))dt} = I{(d/dt)f(a+t(x-a))dt} = I{d[f(a+t(x-a))]},

which is the formal justification of the first line in my original solution. Am I right?
 
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1. What is integration notation for a smooth function?

Integration notation for a smooth function is a mathematical representation of the process of finding the area under the curve of a smooth function. It is denoted by the integral symbol (∫) and involves a function, an integration variable, and the limits of integration.

2. Why is proving integration notation important for smooth functions?

Proving integration notation for smooth functions is important because it allows us to accurately calculate the area under a curve, which has many real-world applications in fields such as physics, engineering, and economics. It also helps us to better understand the behavior of functions and their relationship to calculus.

3. What are the steps involved in proving integration notation for a smooth function?

The steps involved in proving integration notation for a smooth function are as follows:

  • 1. Identify the function and the limits of integration.
  • 2. Write the integral using the integration symbol (∫).
  • 3. Substitute the limits of integration into the integral.
  • 4. Simplify the integral using algebraic techniques.
  • 5. Evaluate the integral using integration rules or techniques.

4. How do I know if a function is smooth?

A function is considered smooth if it is continuous and differentiable on the interval of integration. This means that the function has no breaks or sharp turns and can be represented by a single smooth curve.

5. Can integration notation be used for non-smooth functions?

Yes, integration notation can also be used for non-smooth functions, but it may involve more complex techniques such as using multiple integrals or breaking the function into smaller, smooth intervals. However, in order to use integration notation, the function must still be continuous on the interval of integration.

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