The picture behind this imagine I have two curves Maybe some c 1 that goes from a to b. If for any two smooth curves, you want to see two with the same initial and terminal points, we have the same values of their line integrals. So one more time a vector field F is continuous, then if it's continuous than the line integral of F ds is independent of path. If the line integral over two paths c1 and c2 are equal for any two smooth curves c1 and c2 with the same initial and terminal points. So I want to introduce a definition, if my vector field f is continuous, and the line integral of f ds is independent of path. Let's keep talking about the consequences of this and then we'll actually use this and do some examples. So just have to check that it is a gradient field or that the vector field is conservative before you apply this property. So not every vector field is conservative. This is not true for every single vector field ever. Fundamental theorem for line integrals is that you have to work with gradient field so this must be a gradient field. Sometimes it's abbreviated as FTLI compared to lease the FTC. The one thing you have to be careful for to use this fundamental theorem for line integrals. If you just measure two points, if you have the the data you need for two points. They can talk about the work that's done by a force or something like that. And we've seen before that these integrals have physical significance. But somehow, someway, the endpoints determine the value of this integral. It can do lose all day, it doesn't matter. Though the particle, the way it's traveling can go bananas, it can go nuts. Everything is determined by the endpoints. Everything is completely dependent on the value of the function at the start point, and the value at the function at the end point. Somehow somewhere what happens between the start point and the end point of the curve, doesn't matter. Another way to write this is exactly the same way but I'm going to write it different version, if I have the derivative of some function, then it becomes the function at b, minus the function at a. Then this was the antiderivative plugged in at b minus the antiderivative plugged in at a plug did the endpoints. Remember what the fundamental theorem of calculus said, If I know an anti derivatives, capital F, that's when capital S meant anti derivatives and non vector fields. And I had some function f, that I wanted to know the definite integral from a to b Have f of x dx. I don't know when the last time you remember this guy, but the fundamental theorem of calculus again, if I had a line segment from A to B on the xy plane. At least in scope to the fundamental theorem of calculus. So it should also tell you that this is pretty important and I hope you realize it's pretty close. This is probably more amazing than it might realize it has some nice consequences that we're going to talk about. I'm going to leave this on the screen for a second, I'm going to let this sink in. But we're going to say that the line integral of a vector field of a gradient vector field again, this is all like DS Is equal to the potential function evaluated at the endpoint minus the potential function evaluated at the start point. Remember, these are conservative vector fields, where the little f function is the potential function. And we want a vector field and we're going to specifically focus on vector fields that are gradient fields. LINE INTEGRAL WORKDONE HOW TORemember, we're after how to find line integrals. Let f be a differentiable function whose gradients continuous it's all just bookkeeping stuff to make sure everything's defined. It applies for both for the parameter a less than or equal to t less than equal to b. Maybe we're in the xy plane, you have an x and a y component, or maybe we're in space, and you have xy and z. So we've seen this before, where c is defined by some vector function. You have a nice smooth curve are defined by a vector function r of t. Theorem, it says for any smooth curve, now friendly reminder, smooth curve just means like no cusps, no intersections. This is a big result and really one of my favorite things to talk about. Hi everyone, and welcome to our lecture on the fundamental theorem for line integrals.
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