- #1

phys_student1

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I find my professor doing this a lot of times, here is it:

∫{ ∂(f[x])/∂x } dx = ∫d(f[x])

How is that possible?

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- Thread starter phys_student1
- Start date

In summary, the professor is taking the derivative of a function and turning it into an integral by using the 'antiderivative' or integral operator. This is possible because integration is the inverse of differentiation and vice versa.

- #1

phys_student1

- 106

- 0

I find my professor doing this a lot of times, here is it:

∫{ ∂(f[x])/∂x } dx = ∫d(f[x])

How is that possible?

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- #2

bigfooted

Gold Member

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Suppose you have a function f(x).

Take the derivative of the function: d/dx(f(x)) = df(x)/dx.

By taking the 'antiderivative' or the integral, you end up with f(x) again, which is exactly what your professor is doing.

For the rest, there is not really a difference between a partial and an ordinary differentiation operator, except that the ordinary differentiation just means that there is only one independent variable x, and with partial differentiation you can have more.

- #3

phys_student1

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That's exactly my question. You see, the idea of having a partial derivative in itself means

that f is not only a function of (x) but also of another variable.

That's why I am wondering how that operator was changed to ordinary derivative.

Is it possibly because only in this integral we are interested in f as a function of x only, and

for that reason we can, only here, change the partial to ordinary derivative ?

- #4

HallsofIvy

Science Advisor

Homework Helper

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[tex]\int \frac{\partial f(x,y,z,...)}{\partial x}dx= \int df(x,y,z,...)[/tex]

follows from

[tex]\int \frac{df}{dx}dx= \int df[/tex]

for functions of a single variable.

- #5

phys_student1

- 106

- 0

HallsofIvy said:

[tex]\int \frac{\partial f(x,y,z,...)}{\partial x}dx= \int df(x,y,z,...)[/tex]

follows from

[tex]\int \frac{df}{dx}dx= \int df[/tex]

for functions of a single variable.

Thanks,

So, because we are interested only in x, then the partial, in this case, is equivalent to the ordinary derivative, hence the change.

Thanks everyone for your help, appreciated :)

The process of converting partial derivative to ordinary in an integral involves integrating a function with respect to one variable while treating all other variables as constants. This allows us to find the relationship between the variables and determine how changes in one variable affect the function.

Converting partial derivative to ordinary in an integral is necessary because it allows us to evaluate how a function changes with respect to one variable, while keeping all other variables constant. It also helps us to solve problems involving multiple variables and understand the relationship between them.

A partial derivative measures the rate of change of a function with respect to one of its variables, while treating all other variables as constants. An ordinary derivative, on the other hand, measures the rate of change of a function with respect to a single variable.

To convert a partial derivative to an ordinary derivative in an integral, we simply integrate the function with respect to the specified variable, while treating all other variables as constants. This allows us to eliminate the partial derivative and evaluate the integral using standard integration techniques.

We would need to convert a partial derivative to an ordinary derivative in an integral when solving problems involving multiple variables, such as optimization or finding the relationship between variables. This process is also commonly used in physics and engineering to analyze how changes in one variable affect a system.

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