In doing multivariable calculus, I have encountered line integrals. I have no problem calculating these, but I do have a question about the intuition behind two ideas (one of which, dot product, is early on in MV calculus).
First, the role of gradients in a conservative vector field. I do understand how to determine if a vector field is conservative: it is the gradient of a potential function. And I can do the math from there. But what I don't understand is the intuition behind this. What does the gradient have to do with the fact that I don't have to calculate a curve in the conservative vector field? I'm having a hard time understanding the WHY even though I get the HOW.
Second, on dot products. These are easy to do, and I understand that you need to dot product the force and the dr before taking the integral. But, again, why? The dot product, I guess, shows the relationship between two vectors - if it is zero, the vectors are orthogonal; and the dot product increases the closer the vectors get to parallel. But what is the intuition behind why the dot product can be used to determine work, i.e., w = F * d. Again, I can solve this, but I don't understand the intuition.