Functions having the same integral are equal

In summary, the conversation is discussing the relationship between two functions, f and g, and the integral and differential forms of Gauss's law. The statement "for any solid region D, it is true that the integral of f over D is equal to the integral of g over D" is being considered, and the question is whether this means that f and g are equal. The idea of "almost everywhere" is brought up as a possible answer, with an example given for further clarification.
  • #1
Bipolarity
776
2
Suppose that for any solid region D, it is true that
[tex] \int\int\int_{D}f(x,y,z)dV = \int\int\int_{D}g(x,y,z)dV [/tex]

Then is it the case that f(x,y,z) is g(x,y,z). I am not sure if it's true but I seem to need it to equate the integral and differential forms of Gauss's law.

Any thoughts?

BiP
 
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  • #2
Hint: consider f-g.
 
  • #3
To be more precise: it is true almost everywhere.

If you have learned measure theory, then the words "almost everywhere" have a very precisely defined meaning. Otherwise, you can just take it very loosely, and think of examples like
$$f(x) = x^2; g(x) = \begin{cases} x^2 & \text{if } x \neq 0 \\ -1 & \text{ if } x = 0 \end{cases}$$
 

1. What does it mean for two functions to have the same integral?

When two functions have the same integral, it means that the area under the curve of both functions is equal, even though the functions may have different shapes and equations. This is also known as the fundamental theorem of calculus.

2. Does having the same integral guarantee that two functions are equal?

Having the same integral does not necessarily guarantee that two functions are equal. There may be cases where two functions have the same integral, but have different values at certain points. However, if the two functions have the same integral over a closed interval, then they are considered equal.

3. Are there any limitations to the statement "functions having the same integral are equal"?

Yes, there are limitations to this statement. It only applies to continuous functions, which means that the functions have no breaks or gaps in their graphs. It also only applies to functions with a finite domain and range.

4. How does the concept of "Functions having the same integral are equal" relate to the concept of antiderivatives?

The concept of "Functions having the same integral are equal" is closely related to the concept of antiderivatives. An antiderivative is a function whose derivative is equal to the original function. This means that if two functions have the same integral, their antiderivatives will be equal as well.

5. Can this statement be extended to multiple functions having the same integral?

Yes, this statement can be extended to multiple functions having the same integral. If a set of functions have the same integral over a closed interval, then they are considered equal over that interval. This can be useful in solving problems involving multiple functions with the same integral.

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