Two Intuition Questions: Conservative VF and Dot Product All,
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.
Thanks.
 A: Gravity is a good example of a conservative force.
If you expend energy to go up a hill, you get that energy back when you come down the hill.  Regardless of the path you take from A to B, the only thing that matters is the net change in altitude between the two points.  If you travel back to your starting point there is no net change in altitude and no net-energy spent.
Conservative force fields tend to be centripetal.  e.g $F(x,y,z) = -x\mathbf {\hat i}+-y\mathbf {\hat j}+ -z\mathbf {\hat k}$ is a conservative force with all of the force vectors pointing toward the origin.  It could be that all of the force vectors are parallel, and still be a conservative force field.  e.g. $F(x,y,z) = 1\mathbf {\hat i}+2\mathbf {\hat j}+ 3\mathbf {\hat k}$
What about the dot product?  Again using gravity as our model for our force field.  When you are traveling at constant altitude, you are spending no energy to defeat gravity.  You direction of travel is perpendicular to the force and $\mathbf {a\cdot b} = 0$ when $\mathbf a$ is perpendicular to $\mathbf b$.  As you climb, the dot product will give you the vertical component of your travel, and thus the energy required to climb.
