Line Integrals of Vector Fields // Big Idea, Definition & Formula

in this video we're going to talk about the line interval of a vector field we previously have seen line integrals earlier on in my playlist on vector calculus the link to that playlist is down in the description and when we previously talked about line integrals we would have some path and then we had some function that gave us sort of height above every point along that curve but now we're imagining a scenario like this where we've got some vector field and we've got a curve that is moving through the vector field and if you look

at this vector field sometimes the arrows of the vector field appear to be almost tangent to the curve sometimes they're pointing completely away from the curve in all sorts of directions what we want to try and measure here is some way of capturing the degree to which the vector field goes along with the path of the curve and the the tangent to the curve and the vector field are aligned so what does that mean well let's start with a more concrete example from physics the concept of work indeed you might recall that work was defined

to be a force times a distance a sort of an elementary way of thinking about what the conduct of work is if you apply a lot of force for a long distance you've done a lot of work the standard example of this was an inclined plane we have a block the plane is assumed to be frictionless and the block is going to be sliding down the plane under the force of gravity it might move well a little bit like this so if we analyze what's going on here there's a force of gravity on that block

that points straight down but you can decompose that force of gravity into two different directions a proportion that is normal to the inclined plane and a portion which is tangential to the incline plane indeed i'm also going to add a vector which i'll call t and this is just showing what is the direction of that inclined plane so because the block moved along that incline plane the proportion of the force of gravity that was normal is irrelevant there's no work that's being done in that direction all of the work that was done was the proportion

that was moving in the tangential direction so how much work is actually being done well if we say that there is a distance d between where the block started and where the block ended up then what we can talk about is the dot product this the proportion of the force of gravity in that tangential direction that's the force that we're interested in it's not the entire force of gravity just the proportion of the force in the tangential direction and that's what a dot product captures and then you multiply that by the distance d okay so

pay attention to the fact that our work here was defined to be a force dotted with the tangential and let's zoom back out to our more complicated situation where i have some feel and then i have a curve that is going to be going through that field then the question is what is the work done by that field if i move some particle along that curve and as we just saw the answer to how much work done is going to depend on the degree to which this curve is going tangential to the field so let

me zoom in on a little portion of the curve and analyze what's going on here so if i take some spot and i'm going to measure a little unit of ds i'm going to imagine that i'm just moving some small amount along the curve a little change in arc length we call it ds and then i'm going to say well in that small little region i'm going to have some vector field then it's going to point in some direction so i'll put that vector for the f and then if i'm interested in what is the

work done by this little movement of length ds i also have to ask well what is the tangential component so then the contribution of the field along that little segment ds is just given by the f dot t that's the proportion of the force in the t direction times this little d s if i'm interpreting this as work then i would say the work done from moving that little df is this f dot t d s finally i had a much bigger curve and so if i want to add everything up i'm just going to

integrate it i shall do the line integral of a vector field f a long curve c is just given by this integral over c the same line interval we've seen before it's just that the integrand is the f dot t and then integrated with respect to ds so this formula is exactly like the line integral formula we've seen before indeed it's defined in the exact same way it's just the integrand's different the integrand is not just an arbitrary function which is what we've seen before the indirect is a specific thing it's trying to capture the

influence of the field along the curve and so the integrand is this f dot t and so putting the picture back up we have an answer to the work done if you move the particle along this curve in this particular field and indeed i've been talking about work so far in this video that specific physics concept but indeed the concept of a line integral of a vector field along some curve is something that applies in many different scenarios work is just one of them so this definition applies broadly this is a definition we can understand

because we previously understood the concept of a line interval but the question now is how do we compute it and as we've seen before when it comes to computations we want to see how can i parameterize a curve explicitly and then rewrite this formula in terms of that parametrization okay so let's give some labels i'm going to say i have a parametrized curve the curve is on some interval of t t between a and b and then the curve is going to be a g of t and the i hat and h of t and

the j hat and a k of t in the k hat the way i think about this is the vector r is a vector that goes from the origin out to the curve so if the origin is in the center of this plot then the r of a so that's sort of the first component on my interval of t from a to b points to the start of the curve and then the r of b points to the end of the curve note by the way the curve has a little blue arrow here that indicates

the direction that it's moving indeed this is going to be very important when we think about curves we always want to think about them as starting at one spot and going to another spot these are sort of directed curves they have a direction that you're traveling and that's implied by this choice of parametrization as well okay so that's the parameterization for the curve so now i want to go back to the formula i had there are several changes i can make notice how there's the unit tangent vector t in this formula well a definition of

the unit tangent vector t was just that it was the derivative d r d s so if you have a specific parameterization you can say well how much does that position vector change as you increase arc length that was a definition of the unit tangent vector and then just by the chain rule ds i could replace that with ds dt dt so again with chain rule the ds's cancel i mean they don't cancel exactly but that's the result of the chain rule and as a result i can plug this in and what i get is

the field just evaluated now along the curve so f of r of t and then i multiply it by the derivative dr dt that's a dot product between this field and this derivative and then all integrated out with respect to t and my limits of integration r and t as well they go from the a up to the b so well the left side of this equation is the definition the right side of this equation tells me how i compute it it gives me an explicit way to compute this indeed we're going to run through

a concrete example computing a vector field along curve in the next video but before we end i want to show you just one more piece of terminology it's a third way to think about this if we have this dr dt dt i can just define a new symbol which is just dr just to be a shorthand for that and so my third way of writing this is they integral along the curve of f dot d r and all of these things are the same it's just different terminology so sometimes you'll see a line integral written

in f dot d r sometimes you'll see it written f dot t d s whenever you want to compute it though whenever you actually have an explicit parameterization use the middle one this f dot dr dt dt the middle one is how you compute it anyways we just want to be a little bit familiar with being able to alternate between writing things ds or dr or dt they all refer to the same thing just some might be more natural in one context than another alright if you enjoyed this video please give it a like for

the youtube algorithm if you have a question about the video leave it down in the comments below and we'll do some more math in the next video