# Converting partial derivative to ordinary in an integral

• phys_student1
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.
phys_student1
Hi,

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

∫{ ∂(f[x])/∂x } dx = ∫d(f[x])

How is that possible?

Integration is the inverse of differentiation and vice versa.
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.

Thanks, but...

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 ?

The partial derivative of a function of several variables treats the other variables as constants so
$$\int \frac{\partial f(x,y,z,...)}{\partial x}dx= \int df(x,y,z,...)$$
follows from
$$\int \frac{df}{dx}dx= \int df$$
for functions of a single variable.

HallsofIvy said:
The partial derivative of a function of several variables treats the other variables as constants so
$$\int \frac{\partial f(x,y,z,...)}{\partial x}dx= \int df(x,y,z,...)$$
follows from
$$\int \frac{df}{dx}dx= \int df$$
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 :)

## 1. Can you explain the concept of converting partial derivative to ordinary in an integral?

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.

## 2. Why is it necessary to convert partial derivative to ordinary in an integral?

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.

## 3. What is the difference between a partial derivative and an ordinary derivative?

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.

## 4. How do you convert a partial derivative to an ordinary derivative in an integral?

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.

## 5. In what situations would you need to convert a partial derivative to an ordinary derivative in an integral?

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.

Replies
2
Views
875
Replies
3
Views
2K
Replies
1
Views
2K
Replies
5
Views
599
Replies
4
Views
1K
Replies
1
Views
2K
Replies
3
Views
1K
Replies
1
Views
2K
Replies
5
Views
2K
Replies
2
Views
1K