Force, Work, and Energy

I went to my professor's office hours. Apparently, he was unable to help me out because, he'd never even heard of the unit normal vector, and he couldn't see how it could arise from the unit tangent vector. Also, he said he was not willing to go through the mathematical complications. I almost wanted to cry.

Anyways, he did have a point. A flat line doesn't have a unit normal vector. So I'm completely lost with this formula:

E= -∫ μF • N ds

To reconcile this, my teacher told me to find θ as a function of position, and then use the formula given in the Giancoli book:

E= -∫ μF cos(θ dl = -∫ μmg cos(θ dl

To do this, I proposed the formula:

tan(θ = -dx/dy

the gradient of a plane curve. My god. He actually didn't believe me when I told him that this is the gradient of a plane curve. He called what I was doing "unnecessary mathematical complication". "I'm sorry. I'm unable to help you. If there's anything regarding tomorrow's quiz, by all means, let me know, but you don't ever need to know this, and it'll just end up looking ugly." So he went to work on his computer, and I just continued my mathematical pursuits on his whiteboard. Plugging in, I obtained:

E = -∫ μmg cos(arctan(-dx/dy)) dl

dl = sqrt(1+(dy/dx)^2) dx

cos(arctan(-dx/dy)) = 1/sqrt(1+(dx/dy)^2)

Therefore:

E = -∫ μmg sqrt(1+(dy/dx)^2)/sqrt(1+(dx/dy)^2) dx

At this point, my professor told me that the equation couldn't be simplified further, and it was so complicated that I might as well just use the approximation technique. Nonetheless, I pursued the question a little further, and I ended up with below. It was quite simple, and either my physics teacher is a moron or he never learned calculus in the first place.

E = -∫ μmg (dy/dx) dx = -μmg y(x)

Note that this came exactly from the formula he told me to use. I did absolutely nothing wrong in between. After I got to this point, some fifth-year student walked into my professor's office to request an appointment about his graduate school plans, and while the fifth-year student said he would wait until the following Monday to ask, my professor insisted that he stayed. The fifth-year student responded, "You look busy. Aren't you helping him out right now?" My professor responds, "No. He's just doing a bunch of ugly mathematics, and I think we're done here, because I can't help him out further." At this point he kicked me out.

This is why, unfortunately, my questions must be pursued here. I really do wish I didn't have to be turning to online help, but it has become my only alternative. None of the people I know have taken vector calculus, and used it in combination with physics.

Getting back on topic, the formula found after simplification just doesn't look correct. I'm not sure why. Maybe because the new formula looks so damn simple, and my teacher told me that all I would end up getting is mathematical garbage. -_-"

But in all seriousness, it doesn't look correct. The reason it doesn't look correct is that the equation that was derived became exactly the same as the potential energy. What if I started and ended at the same height, but the object rolled down some hill and then back up? Then the equation wouldn't even make sense.

One way I might be able to fix this is by finding the local minima of the function and then splitting up the integral into two pieces, but I have no way of proving that this actually provides the correct frictional force. Omfg, I feel so lost. I really do want to cry, but I have no tears to shed. ;_;