Faraday's Law vs Kirchoff's Rule in circuit - nonconservative fields

I am trying to compare the "relation" conventions used in Kirchoff's Loop Rule with Faraday's Loop Rule.

Kirchoff

Please go to this MIT OCW link on Kirchoff's Rule and go to page 8/29. Of the four boxes, I would like to point this one

Note that the yellow electric field was added by me. This picture also follows the integral

$$-\int_{a}^{b} \mathbf{E}\cdot d\mathbf{s} = \int_{a}^{b} -Eds = \Delta V = \varepsilon = -IR$$

Since ds and E are parallel.

Now if I go to this video (I take you to EXACTLY where I want you to watch, so don't worry about searching which part of the video does this happen)

Now quoting him

 Now I go through the resistor, so I get +IR, since E and dl are in the same direction, Ohm's Law tells me I get +IR
Note that his circuit arrangement is exactly like mine, he went from a high potential to a low potential. The current, electric field in the wire, and the travelling direction are all the same, yet he gets +IR instead of -IR

is he still using this "E dot dl" $$-\int_{a}^{b} \mathbf{E}\cdot d\mathbf{s}= \Delta V$$? How does Faraday's Law apply for non-closed paths? Because it seems like Lewin is using $$\int_{a}^{b} \mathbf{E}\cdot d\mathbf{s}= \Delta V$$ (no minus sign)

The same confusion goes when he talks about the electric field in the battery.

Could someone please clarify for me? Thank you
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 This subject has already been done to death here at this forum eg http://www.physicsforums.com/showthread.php?t=453575 do a forum search for other threads and also here http://forum.allaboutcircuits.com/sh...t=lewin&page=3

 Quote by Studiot This subject has already been done to death here at this forum eg http://www.physicsforums.com/showthread.php?t=453575 do a forum search for other threads and also here http://forum.allaboutcircuits.com/sh...t=lewin&page=3
Second link wasn't very helpful and I can't pinpoint my answer in the first link because it seems like it derailed into some argument on Lewin's teaching ability.