# Is Leibniz integral rule allowed in this potential improper integral?

• I
In summary, the electric potential at a point inside a charge distribution can be calculated by taking the limit of an integral over a small volume surrounding that point. This integral takes into account the density of charge distribution at various points and the distance between those points and the field point. Taking the gradient of this potential yields the electric field at that point. However, there is some debate about the validity of this technique for improper integrals, but it is still considered valid for points inside the source region. The book provides a proof for this argument, but it may be difficult to understand without a deeper understanding of potential theory.
Electric potential at a point inside the charge distribution is:

##\displaystyle \psi (\mathbf{r})=\lim\limits_{\delta \to 0} \int_{V'-\delta}

\dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'##

where:

##\delta## is a small volume around point ##\mathbf{r}=\mathbf{r'}##
##\mathbf{r}## is coordinates of field point
##\mathbf{r'}## is coordinates of source point
##\rho (\mathbf{r'})## is the density of charge distribution

##\displaystyle \nabla \psi (\mathbf{r}) =\nabla\ \left[ \lim\limits_{\delta \to 0} \int_{V'-\delta} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV' \right] =\lim\limits_{\delta \to 0} \int_{V'-\delta} \rho (\mathbf{r'})\ \nabla \left( \dfrac{1}{|\mathbf{r}-\mathbf{r'}|} \right) dV'##

In the last step, we have applied Leibniz integral rule (basic form).

The validity of this technique for improper integrals is discussed below:

The following passage from the book "Foundations of Potential Theory page 151" says the technique is not valid. But it says the equation ##\mathbf{E}=-\nabla \psi## still holds at points inside source regions ##V'##. It also gives a "little" proof of the argument.

(continued below)

Proof:

Unfortunately I am not well versed in potential theory to understand this little proof that the book offers. Can anybody explain the proof in a way in which a Physics graduating student can understand?

## 1. What is Leibniz integral rule?

Leibniz integral rule, also known as the generalized Leibniz rule or the Leibniz differentiation under the integral sign, is a mathematical theorem that allows for the differentiation of an integral with respect to a parameter under certain conditions.

## 2. Is Leibniz integral rule applicable to all types of integrals?

No, Leibniz integral rule is only applicable to integrals with a parameter in the limits of integration, known as definite integrals, and not to indefinite integrals.

## 3. What is meant by a potential improper integral?

A potential improper integral is an integral that does not have a finite value due to either the function being integrated or the limits of integration being infinite or undefined.

## 4. Can Leibniz integral rule be used to evaluate potential improper integrals?

Yes, Leibniz integral rule can be used to evaluate certain types of potential improper integrals, specifically those that are convergent and have a parameter in the limits of integration.

## 5. Are there any limitations to using Leibniz integral rule in evaluating integrals?

Yes, there are certain conditions that must be met for Leibniz integral rule to be applicable, such as the integrand being continuous and the integral being convergent. Additionally, the order of differentiation and integration must be interchangeable for the rule to be valid.

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