To my knowledge, Neumann gave the formula in 1845.It is said that Faraday did not provide any mathematical formulas. The law of electromagnetic induction was obtained by Neumann in 1845 by calculating the energy between two coils based on the Ampere force formula.
$$Energy=-\frac{I_{1}I_{2}}{2}\oint_{C_{2}}\oint_{C_{1}}\frac{d\boldsymbol{l}_{1}\cdot d\boldsymbol{l}_{2}}{r}\ \ \ \ (1)$$
From this we can obtained that
$$\mathcal{E}_{2,1}=-\frac{\mu_{0}}{4\pi}\oint_{C_{2}}\oint_{C_{1}}\frac{dI_{1}}{dt}\frac{d\boldsymbol{l}_{1}\cdot d\boldsymbol{l}_{2}}{r}\ \ \ \ (2)$$
I do not clear how can we from (1) to Derived (2). But when we obtained (2) it is easy to obtained the Faraday law.
First define Induced electromotive force and magnetic vector potential
$$\mathcal{E}_{2,1}\triangleq\oint_{C_{2}}\boldsymbol{E}_{1}\cdot d\boldsymbol{l}_{2}$$
$$\boldsymbol{A}_{1}\triangleq\frac{\mu_{0}}{4\pi}\oint_{C_{1}}\frac{I_{1}d\boldsymbol{l}_{1}}{r}$$
Hence we obtained,
$$\oint_{C_{2}}\boldsymbol{E}_{1}\cdot d\boldsymbol{l}_{2}=-\oint_{C_{2}}\frac{\partial}{\partial t}\boldsymbol{A}_{1}d\boldsymbol{l}$$
or
$$\oint_{C_{2}}(\boldsymbol{E}_{1}+\frac{\partial}{\partial t}\boldsymbol{A}_{1})\cdot d\boldsymbol{l}_{2}=0$$
or
$$\iint_{\Gamma}\nabla\times(\boldsymbol{E}_{1}+\frac{\partial}{\partial t}\boldsymbol{A}_{1})\cdot\hat{n}d\Gamma=0$$
or
$$\nabla\times(\boldsymbol{E}_{1}+\frac{\partial}{\partial t}\boldsymbol{A}_{1})=0$$
or
$$\boldsymbol{E}_{1}+\frac{\partial}{\partial t}\boldsymbol{A}_{1}=-\nabla\phi_{1}$$
or
$$\boldsymbol{E}_{1}=-\nabla\phi_{1}-\frac{\partial}{\partial t}\boldsymbol{A}_{1}$$
or omit the subscript,
$$\boldsymbol{E}=-\nabla\phi-\frac{\partial}{\partial t}\boldsymbol{A}$$
This is Faraday's law proposed by Maxwell himself. Maxwell's descendants further rewrote this formula as,
$$\nabla\times\boldsymbol{E}=-\nabla\times\nabla\phi-\frac{\partial}{\partial t}\nabla\times\boldsymbol{A}$$
or
$$\nabla\times\boldsymbol{E}=-\frac{\partial}{\partial t}\nabla\times\boldsymbol{A}$$
define
$$\boldsymbol{B}\triangleq\nabla\times\boldsymbol{A}$$
we obtained
$$\nabla\times\boldsymbol{E}=-\frac{\partial}{\partial t}\boldsymbol{B}$$
We see that the Faraday's law of electromagnetic induction derived from Neumann's electromagnetic induction formula is mediocre, which we are using in today's textbook. Therefore, this credit belongs to Neumann The vector potential we also defined above is also known as Neumann vector potential.
Considering,
$$\oint_{C}\cdots Idl\rightarrow\int_{V}\cdots JdV$$
Obtain,
$$\boldsymbol{A}\triangleq\frac{\mu_{0}}{4\pi}\int_{V}\frac{\boldsymbol{J}}{r}dV$$
It is worth mentioning that Weber also proposed the law of electromagnetic induction in 1846, and the vector corresponding to this law of electromagnetic induction is,
$$\boldsymbol{A}_{W}\triangleq\frac{\mu_{0}}{4\pi}\int_{V}\frac{(\boldsymbol{J}\cdot\boldsymbol{r})\boldsymbol{r}}{r^{3}}dV$$
If the coil is a closed loop, these two magnetic vector potentials are equivalent, that is, the induced electromotive force calculated by them is equal. Therefore, Weber is also another contributor to the law of electromagnetic induction.
It is worth mentioning that Maxwell defined a magnetic field based on the curl of a magnetic vector under quasi-static conditions
$$\boldsymbol{A}\triangleq\frac{\mu_{0}}{4\pi}\int_{V}\frac{\boldsymbol{J}}{r}dV\ \ \ \ \ （3）$$
Because,
$$\nabla\times\boldsymbol{A}=\frac{\mu_{0}}{4\pi}\int_{V}\nabla(\frac{1}{r})\times\boldsymbol{J}dV$$
or
$$\nabla\times\boldsymbol{A}=\frac{\mu_{0}}{4\pi}\int_{V}\boldsymbol{J}\times\frac{\boldsymbol{r}}{r^{3}}dV\ \ \ \ (4)$$
From Biota's law
$$\boldsymbol{B}=\frac{\mu_{0}}{4\pi}\int_{V}\boldsymbol{J}\times\frac{\boldsymbol{r}}{r^{3}}dV\ \ \ \ (5)$$
Compare (4) and (5) to obtain the definition,
$$\boldsymbol{B}\triangleq\nabla\times\boldsymbol{A}$$
