It suffices for us to prove that
the truth of (a) implies the truth of (b) and
the truth of (b) implies the truth of (c) and
the truth of (c) implies the truth of (a).
That’s exactly what we will do.
(a)
\(\implies\)(b): Let
\(\cC\) be a closed curve that starts at
\(P_0\) and then ends back at
\(P_0\text{.}\) Then, by Theorem
2.4.2 with
\(P_1=P_0\text{,}\)
\begin{equation*}
\oint_\cC\vF\cdot\dee{\vr} =\varphi(P_0) - \varphi(P_0)=0
\end{equation*}
(b)
\(\implies\)(c): Pick any point
\(P_0\) and set
\(P_1=P_0\text{.}\) Then we are assuming that
\(\oint_\cC\vF\cdot\dee{\vr}=0\) for all curves that start at
\(P_0\) and end at
\(P_1\text{.}\) In particular
\(\int_\cC\vF\cdot\dee{\vr}\) takes the same value for all curves that start at
\(P_0\) and end at
\(P_1\text{.}\) So Theorem
2.4.6 immediately yields property (c).
(c)
\(\implies\)(a): We are to show that
\(\vF\) is conservative. We’ll start by guessing
\(\varphi\) and then we’ll verify that, for our chosen
\(\varphi\text{,}\) we really do have
\(\vF=\vnabla\varphi\text{.}\) Our guess for
\(\varphi\) is motivated by Theorem
2.4.2. If our
\(\vF\) really is conservative, its potential is going to have to obey
\(\int_\cC\vF\cdot\dee{\vr} =\varphi(P_1) - \varphi(P_0)\) for any curve
\(\cC\) that starts at
\(P_0\) and ends at
\(P_1\text{.}\) Let’s choose
\(P_0=\vZero\text{.}\) Remembering, from Definition
2.3.1.a, that adding a constant to a potential always yields another potential, we can always choose
\(\varphi(\vZero)=0\text{.}\) Then
\(\varphi(P_1)=\int_\cC\vF\cdot\dee{\vr}\) for any curve
\(\cC\) that starts at
\(\vZero\) and ends at
\(P_1\text{.}\) So define, for each point
\(\vx\text{,}\) \(\varphi(\vx)=\int_\cC\vF\cdot\dee{\vr}\) for any curve
\(\cC\) that starts at
\(\vZero\) and ends at
\(\vx\text{.}\) Note that, since we we are assuming that (c) is true, the integral
\(\int_\cC\vF\cdot\dee{\vr}\) takes the same value for all curves
\(\cC\) that start at
\(\vZero\) and end at
\(\vx\text{.}\)
We now verify that, for this chosen \(\varphi\text{,}\) we really do have \(\vF=\vnabla\varphi\text{.}\) Fix any point \(\vx\) and any curve \(\cC_{\vx}\) that starts at the origin and ends at \(\vx\text{.}\) For any vector \(\vu\text{,}\) let \(\cD_{\vu}\) be the curve with parametrization
\begin{equation*}
\vr_{\vu}(t)=\vx+t\vu\qquad 0\le t\le 1
\end{equation*}
This curve is a line segment that starts at \(\vx\) at \(t=0\) and ends at \(\vx+\vu\) at \(t=1\text{.}\) Observe that \({\vr\,}'_{\vu}(t)=\vu\text{.}\) Recall that, by assumption, \(\varphi(\vx+s\vu)=\int_\cC\vF\cdot\dee{\vr}\) for any curve \(\cC\) that starts at \(\vZero\) and ends at \(\vx+s\vu\text{.}\) So
\begin{equation*}
\varphi(\vx+s\vu)
=\int_{\cC_{\vx}+\cD_{s\vu}}\vF\cdot d\vr
\end{equation*}
where \(C_{\vx}+D_{s\vu}\) is the curve which first follows \(C_{\vx}\) from the origin to \(\vx\) and then follows \(D_{s\vu}\) from \(x\) to \(\vx+s\vu\text{.}\) We have
\begin{align*}
\int_{C_{\vx}+D_{s\vu}}\vF\cdot d\vr
&=\int_{C_{\vx}}\vF\cdot d\vr
+\int_{D_{s\vu}}\vF\cdot d\vr\\
&=\int_{C_{\vx}}\vF\cdot d\vr
+\int_0^1 \vF(\vx+ts\vu)\cdot (s\vu)\,dt
\end{align*}
In the second integral, make the change of variables \(\tau=ts\text{,}\) \(\dee{\tau}=s\dee{t}\text{.}\) This gives
\begin{equation*}
\varphi(\vx+s\vu)=\int_{C_{\vx}}\vF\cdot d\vr
+\int_0^s \vF(\vx+\tau\vu)\cdot \vu\,d\tau
\end{equation*}
By the fundamental theorem of calculus, applied to the second integral,
\begin{equation*}
\diff{\ }{s}\varphi(\vx+s\vu)\Big|_{s=0}
=\vF(\vx+s\vu)\cdot \vu\Big|_{s=0}=\vF(\vx)\cdot \vu
\end{equation*}
Applying this with \(\vu=\hi,\ \hj,\ \hk\) gives us
\begin{equation*}
\Big(\frac{\partial\varphi}{\partial x}(\vx)\,,\,
\frac{\partial\varphi}{\partial y}(\vx)\,,\,
\frac{\partial\varphi}{\partial z}(\vx)\Big)
=\big(\vF(\vx)\cdot\hi\,,\,\vF(\vx)\cdot\hj\,,\,\vF(\vx)\cdot\hk\big)
\end{equation*}
which is
\begin{equation*}
\nabla\varphi(\vx)=\vF(\vx)
\end{equation*}
as desired.
