I don't know, at which level these lectures are, but the manuscript seems not to be very clear in defining things.
The correct way to treat electromagnetic problems are Maxwell's equations in differential form. Here, it's mostly Faraday's Law that is important:
\vec{\nabla} \times \vec{E}+\frac{1}{c} \partial_t \vec{B}=0.
As you see, if there is a time-dependent magnetic field, the electric field is not conservative, because its curl doesn't vanish.
The integral law, which seems to be applied in the lecture notes (transparencies), is derived by the use of Stokes's integral theorem. To that end integrate Faraday's Law over an arbitrary surface F with boundary \partial F. The orientation of the surface elements \mathrm{d}^2 \vec{F} and that of the line elements \mathrm{d} \vec{r} of the boundary curve are according to the right-hand rule. Then you have
\int_F \mathrm{d}^2 \vec{F} \cdot (\vec{\nabla} \times \vec{E}) = \int_{\partial F} \mathrm{d} \vec{r} \cdot \vec{E}=-\frac{1}{c} \int_{F} \mathrm{d}^2 \vec{F} \cdot \partial_t \vec{B}.
Now comes a very subtle point! We want to get the time derivative out of the integral. This can be done in the naive way only, if your surface and its boundary are stationary, i.e., if they are not time dependent. Then you have
\int_{\partial F} \mathrm{d} \vec{r} \cdot \vec{E}=-\frac{1}{c} \frac{\mathrm{d}}{\mathrm{d} t} \int_F \mathrm{d}^2 \vec{F} \cdot \vec{B}.
In this (and only in this!) case the electromotive force is given by the left-hand side, and the right-hand side is the time derivative of the magnetic flux, i.e., you have
\mathcal{E}=\int_{\partial F} \mathrm{d} \vec{r} \cdot \vec{E}, \quad \Phi=\int_{F} \mathrm{d}^2 \vec{F} \cdot \vec{B},
and the integral form of Faraday's Law can be written as
\mathcal{E}=-\frac{1}{c} \frac{\mathrm{d} \Phi}{\mathrm{d} t}.
If your surface is moving, there's an additional term in the definition of the electromotive force, namely
\mathcal{E}=\int_{\partial F} \mathrm{d} \vec{r} \cdot \left (\vec{E}+\frac{\vec{v}}{c} \times \vec{B} \right ),
where \vec{v}=\vec{v}(t,\vec{r}) is the velocity of the point \vec{r} on the boundary curve.
For a derivation, see the Wikipedia
http://en.wikipedia.org/wiki/Faraday's_law_of_induction#Proof_of_Faraday.27s_law
I hope, now the principles are a bit more clear. If not, ask again!
