If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:8:25

Closed curve line integrals of conservative vector fields

Video transcript

in the last video we saw that if a vector field can be written as the gradient of a scalar field can be written as the gradient of a scalar field or another way we could say that this would be equal to the partial of our big f with respect to x times I plus the partial of big f our scalar field with respect to Y times J and I'm just writing it in multiple ways just so you remember what the gradient is but we saw that if if our vector field hat is the gradient of a scalar field then we call it conservative so that tells us that F is a conservative conservative vector field vector field and it also tells us and this was the big takeaway from the last video that the line integral of F between two points that the line integral of F between two points so let me draw two points here so let me say let me draw my coordinates just so that we know we're on the XY plane my axes x axis y axis let's say I have the point say at that point and that point and I have two different paths between those two points so I have path one that goes something like that so I'll call that C one and it goes in that direction C one goes in that direction and then I have maybe in a different shade of green C two goes like that they both start here and go to there C two we learned in the last video that the line integral is path independent between any two points so in this case the line integral along c1 of F dot d R is going to be equal to the line integral the line integral of c2 over the path c2 of F dot d R the line if we have a potential in a region and we're maybe everywhere then the line integral between any two points is Pat is independent of the path that's the neat thing about a conservative field now what I want to do in this video is do a little bit of an extension of the take away of the last video and it's actually a pretty important extension it might already be obvious to you I've already written this year I could rearrange this equation a little bit so let me do it so let me rearrange this I'll just rewrite this in orange so the line integral on path C 1 dot d R - I'll just go subtract this from both sides - the line integral C 2 of F dot d R is going to be equal to 0 all I did is I took this takeaway from the last video and I subtracted this from both sides now we learned several videos ago we learned several videos ago that if we're dealing with a line integral of a vector field not a scalar field with a vector field the direction of the path is important we learned that we learned that the line integral over say c2 of F dot dr is equal to the negative of the line integral of minus c2 of F dot d R where we denoted - c2 is the same path as c2 but just in the opposite direction so for example minus c2 I would write like this so let me do in a different color so let's say this is minus c2 it would be a path just like c2 I'm going to call this minus c2 but instead of going in that direction I'm now going to go in that direction so ignore the old C two arrows we're now starting from there and coming back here so this is minus c2 or we could write we could put the minus on the other side and we could say that the negative of the c2 line integral along the path of c2 of F dot dr is equal to is equal to the line integral over the reverse path of F dot d R right all I did is I switch the negative on the other side multiplied both sides by negative 1 so let's replace in this equation we have the - of the c2 path we have that right there and we have that right there so we could just replace this with this right there so let me do that so I'll write this first part first so the integral along the curve C 1 of F dot d R instead of - the curve the inert line integral along c2 I'm going to say plus the integral along minus c2 right this this switch to the green this we've established is the same thing as this the negative of this curve or the line integral along this path is the same thing as the line integral the positive of the line integral along the reverse path so we'll say plus the line integral from of minus c2 of F dot d R is equal to zero now there's something interesting let's look at what the combination of the path of C 1 and minus C 2 is C 1 starts over here let me get a nice vibrant color C 1 starts over here at this point it moves from this point along this curve C 1 and ends up at this point and then we do the minus C 2 minus C 2 starts at this point and just goes and comes back to the original point it completes the loop so this is a closed line integral so if you combine this we could rewrite this we could rewrite this line remember this is just a loop by reversing this instead of having two guys starting here and going there I now can start here go all the way there and then come all the way back on this reverse path of C 2 so this is equivalent to a closed line integral so that is the same thing as an integral along a closed path I mean we could call a closed path maybe C 1 plus minus C 2 if we wanted to be particular about the closed path but this could be I drew C 1 and C 2 or minus C 2 arbitrarily this could be any closed path where this the where our vector field F has a potential or where it is the gradient of a scalar field or where it is conservative and so this can be written as a closed path of C 1 plus the reverse of C 2 of F dot d R that's just a rewriting of that and so that's going to be equal to zero and this is our takeaway for this video this is you can mute as a corollary it's a it's a it's a it's kind of a low-hanging conclusion that you can make after after this conclusion so now we know that if we have a vector field that that's the gradient of a scalar field in some region or maybe over the entire XY plane and just this is called the potential of F this is a potential function oftentimes it'll be the negative of it but it's easy to mess with negatives but if we have a vector field that is the gradient of a scalar field we call that vector field conservative that tells us that at any point in the region where this is valid the line integral from one point to another is independent of the path that's what we got from the last video and because of that a closed loop a line integral or a closed line integral so if we take some other place if we take any other closed line integral in along the V or we take the vector for the line integral of the vector field on any closed loop it will become zero because it is path independent so that's the neat takeaway here that if you know if you know that this is conservative if you ever see something like this if you see this F dot d R and someone asks you to evaluate this given that F is conservative or given that F is the gradient of another function or given that F is path independent you can now immediately say that is going to be equal to zero which simplifies the math a good bit