SECTION A : The example in Feynman's Lectures
Let a body P (Planet or Particle or whatever) moving in orbit around a center of attraction called $\:\rm{SUN}$, as in above Figure. Suppose that the attractive force $\:\mathbf{f}\left(r\right)\:$ depends continuously only on the distance $\:r\:$ of the body P from the center $\:\rm{SUN}$. Here it's not necessary this force to obey an inverse square law or to be any special function of $\:r\:$. In other words we would say that $\:\mathbf{f}\left(r\right)\:$ is something like that
\begin{equation}
\mathbf{f}\left(r\right)=\;-\;f\left(r\right)\dfrac{\mathbf{r}}{r}=\;-\;f\left(r\right)\mathbf{n}_{r}
\tag{01}
\end{equation}
where $\:f\left(r\right)\left(>0\right)$ the magnitude of $\:\mathbf{f}\left(r\right)\:$, continuous and integrable, and $\:\mathbf{n}_{r}\:$ the unit vector along $\:\mathbf{r}\:$. In the case of Newton's gravitation or Coulomb's electrostatic force
\begin{equation}
f\left(r\right)= \dfrac{C}{r^{2}} \;, \quad C=\text{positive real}
\tag{02}
\end{equation}
Now, let the body $\:\rm{P}\:$, moving on its orbit, is located at an instant at point $\:\rm{P}_{1}\:$ with velocity $\:\mathbf{v}_{1}\:$ and at distance $\:r_{1}\:$. Later on the body $\:\rm{P}\:$ is located on its orbit at point $\:\rm{P}_{2}\:$ with velocity $\:\mathbf{v}_{2}\:$ of greater magnitude and at a less distance $\:r_{2}\:$.
A twin body $\:\rm{P}^{\boldsymbol{\prime}}\:$, exact copy of $\:\rm{P}\:$, starts from $\:\rm{P}_{1}\:$ at this same distance $\:r_{1}\:$with velocity $\:\mathbf{w}_{1}\:$, of equal to $\:\mathbf{v}_{1}\:$ magnitude ($\:{w}_{1}=\Vert\mathbf{w}_{1}\Vert=\Vert\mathbf{v}_{1}\Vert={v}_{1}\:$), travelling radially and reaching at point $\:\rm{P}_{2}^{\boldsymbol{\prime}}\:$ with velocity $\:\mathbf{w}_{2}\:$ at this same distance $\:r_{2}\:$. The result to be proved is that the velocity $\:\mathbf{w}_{2}\:$ is of equal magnitude to $\:\mathbf{v}_{2}\:$ : $\:{w}_{2}=\Vert\mathbf{w}_{2}\Vert=\Vert\mathbf{v}_{2}\Vert={v}_{2}\:$.
We'll apply the well-known principle :
\begin{equation}
\textbf{Change of kinetic energy = Work done by forces}
\tag{03}
\end{equation}
For the body $\:\rm{P}\:$ on its orbit between points $\:\rm{P}_{1}\:$ and $\:\rm{P}_{2}\:$ above principle yields
\begin{equation}
\tfrac{1}{2}m\left( v_{2}^{2}- v_{1}^{2}\right)=\int_{\rm{P}_{1}}^{\rm{P}_{2}}\mathbf{f}\left(r\right)\circ d\mathbf{r}=\int_{\rm{P}_{1}}^{\rm{P}_{2}}\left[\;-\;f\left(r\right)\dfrac{\mathbf{r}}{r}\right]\circ d\mathbf{r}= -
\int_{r_{1}}^{r_{2}}f\left(r\right)dr
\tag{04}
\end{equation}
In last to the right equality we use the fact that
\begin{equation}
\mathbf{r}\circ d\mathbf{r}= \tfrac{1}{2}d\left(\mathbf{r}\circ \mathbf{r} \right)=\tfrac{1}{2}d\left(\Vert\mathbf{r}\Vert^{2}\right)=\tfrac{1}{2}d\left(r^{2}\right)= r dr
\tag{05}
\end{equation}
So,
\begin{equation}
\tfrac{1}{2}m\left( v_{2}^{2}- v_{1}^{2}\right)= -
\int_{r_{1}}^{r_{2}}f\left(r\right)dr = - \left[\Phi\left(r_{2}\right)-\Phi\left(r_{1}\right)\right]
\tag{06}
\end{equation}
where $\:\Phi\left(r\right)\:$ the following indefinite integral
\begin{equation}
\Phi\left(r\right)=\int f\left(r\right)dr
\tag{07}
\end{equation}
For the twin body $\:\rm{P}^{\boldsymbol{\prime}}\:$ travelling radially from point $\:\rm{P}_{1}\:$ to point $\:\rm{P}_{2}^{\boldsymbol{\prime}}\:$ the principle (03) yields of course the same result for the change of kinetic energy
\begin{equation}
\tfrac{1}{2}m\left( w_{2}^{2}- w_{1}^{2}\right)=\int_{\rm{P}_{1}}^{\rm{P}_{2}^{\boldsymbol{\prime}}}\mathbf{f}\left(r\right)\circ d\mathbf{r}=\int_{\rm{P}_{1}}^{\rm{P}_{2}^{\boldsymbol{\prime}}}\left[\;-\;f\left(r\right)\dfrac{\mathbf{r}}{r}\right]\circ d\mathbf{r}= -
\int_{r_{1}}^{r_{2}}f\left(r\right)dr
\tag{08}
\end{equation}
that is
\begin{equation}
\tfrac{1}{2}m\left( w_{2}^{2}- w_{1}^{2}\right)= -
\int_{r_{1}}^{r_{2}}f\left(r\right)dr = - \left[\Phi\left(r_{2}\right)-\Phi\left(r_{1}\right)\right]
\tag{09}
\end{equation}
From (06) and (09)
\begin{equation}
\tfrac{1}{2}m\left( w_{2}^{2}- w_{1}^{2}\right)= \tfrac{1}{2}m\left( v_{2}^{2}- v_{1}^{2}\right)
\tag{10}
\end{equation}
so if $\: w_{1}= v_{1}\:$ then $\: w_{2}= v_{2}\:$, QED.
But the whole story is not only to prove this but to talk about what is under the table, as Feynman did.
The function $\:\Phi\left(r\right)\:$ is the potential energy and it is a very important tool : think that you have to calculate the work done by a force $\:\mathbf{f}\left(r\right)\:$ like this in equation (01) from point $\:\rm{P}_{1}\:$ to point $\:\rm{P}_{2}\:$ on a curvilinear path of very complicated equation. Instead of being involved in complex and tedious calculations you have immediately the answer using the potential energy :
work done =$\:\Phi\left(r_{1}\right)-\Phi\left(r_{2}\right)\:$.
Equation (06) or (09) may be expressed as
\begin{equation}
\tfrac{1}{2}m v_{2}^{2}+\Phi\left(r_{2}\right)=\tfrac{1}{2}m v_{1}^{2}+\Phi\left(r_{1}\right)
\tag{11}
\end{equation}
yielding the energy conservation
\begin{equation}
\underbrace{\tfrac{1}{2}m v^{2}}_{kinetic\: energy}+\underbrace{\tfrac{}{}\Phi}_{potential\: energy} = \text{ constant}
\tag{12}
\end{equation}
Note that the potential $\:\phi \:$ is the potential energy per unit charge
\begin{equation}
\phi = \dfrac{\Phi}{\xi}
\tag{13}
\end{equation}
where $\:\xi\:$ is the charge : $\:\xi= m = \text{mass}\:$ in gravitation , $\:\xi= q = \text{electric charge}\:$ in electrostatics.
SECTION B : Conservative Vector Fields
There exists a relation that connects the vector field $\:\mathbf{f}\left(r\right)\:$ of equation (01) and the scalar potential $\:\Phi\left(r\right)\:$ of equation (07). From (07)
\begin{equation}
f\left(r\right)=\dfrac{d\Phi}{dr}
\tag{14}
\end{equation}
On the other hand since $\:\mathbf{r}=\left(x,y,z\right)\:$ and $\:r=\Vert\mathbf{r}\Vert=\sqrt{x^{2}+y^{2}+z^{2}}\:$
\begin{equation}
\mathbf{n}_{r}=\dfrac{\mathbf{r}}{r}=\left(\dfrac{x}{r},\dfrac{y}{r},\dfrac{z}{r}\right)=\left(\dfrac{\partial r}{\partial x},\dfrac{\partial r}{\partial y},\dfrac{\partial r}{\partial z}\right)
\tag{15}
\end{equation}
Inserting the expressions (14) and (15) in (01) yields
\begin{equation}
\mathbf{f}\left(r\right)=-\dfrac{d\Phi}{dr}\left(\dfrac{\partial r}{\partial x},\dfrac{\partial r}{\partial y},\dfrac{\partial r}{\partial z}\right)=-\left(\dfrac{d\Phi}{dr}\dfrac{\partial r}{\partial x},\dfrac{d\Phi}{dr}\dfrac{\partial r}{\partial y},\dfrac{d\Phi}{dr}\dfrac{\partial r}{\partial z}\right)=-\left(\dfrac{\partial \Phi}{\partial x},\dfrac{\partial \Phi}{\partial y},\dfrac{\partial \Phi}{\partial z}\right)
\tag{16}
\end{equation}
that is
\begin{equation}
\mathbf{f}\left(r\right)=\;- \;\nabla \Phi
\tag{17}
\end{equation}
where
\begin{equation}
\nabla = \left(\dfrac{\partial}{\partial x},\dfrac{\partial }{\partial y},\dfrac{\partial }{\partial z}\right)
\tag{18}
\end{equation}
the well-known "gradient", an important differential operator applied to scalar functions of $\:\left(x,y,z\right)\:$.
The gradient $\:\nabla \Phi \:$ is a vector with magnitude equal to the rate of change of $\:\Phi\:$, change per unit length. But it's not only this : its direction is at any point always normal to the surfaces $\:\Phi = \text{constant}\:$, the so-called equipotential surfaces, as shown in above Figure, and is pointing to the direction of the maximum rate of increase per unit length. The field force is pointing in the opposite, to the maximum rate of decrease of the potential (energy).
Note that under the light of the gradient definition, equation (15) reads
\begin{equation}
\nabla r = \dfrac{\mathbf{r}}{r}= \mathbf{n}_{r}
\tag{15'}
\end{equation}
In this case the equipotential surfaces are surfaces of spheres.
In the Figure below
\begin{equation}
\int_{\rm{A}}^{\rm{B}}\mathbf{f}\circ d\mathbf{r}=-\int_{\rm{A}}^{\rm{B}}\nabla \Phi \circ d\mathbf{r}=-\int_{\rm{A}}^{\rm{B}}\left(\dfrac{\partial \Phi}{\partial x} dx + \dfrac{\partial \Phi}{\partial y} dy + \dfrac{\partial \Phi}{\partial z} dz \right)=-\int_{\rm{A}}^{\rm{B}}d\Phi
\tag{19}
\end{equation}
so
\begin{equation}
\int_{\rm{A}}^{\rm{B}}\mathbf{f}\circ d\mathbf{r}= \Phi_{1}-\Phi_{2} = \text{independent of the path of integration}
\tag{20}
\end{equation}
or
\begin{equation}
\oint\mathbf{f}\circ d\mathbf{r}= 0 \quad \text{for every closed path of integration}
\tag{21}
\end{equation}
Note that equations (17), (20), (21) are equivalent : for example, if the curvilinear integral of a vector field is zero on any closed path then it is the gradient of a scalar field and vice versa. These properties characterize what is called conservative vector fields.
