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# Line integrals and vector fields

Using line integrals to find the work done on a particle moving through a vector field. Created by Sal Khan.

## Want to join the conversation?

• is physics helpful to know before doing calculus? because as of right now im in physics and we are learning about work, force, velocity etc. but i dont take it too seriously.
• I would say that calculus is helpful to know before doing physics.
• Why is f(x,y)=p(x,y)i+q(x,y)j a function of 't' as Sal mentions around @
• Well, f(x,y) is a function of x and y. But if we're following the path defined by r(t) = x(t)i + y(t)j then our x and y are themselves just functions of t, so f(x,y) becomes f(x(t),y(t)), in short it depends completely on t (though indirectly through x(t) and y(t)). I.e. f(x(t),y(t)) becomes f(t).
• what exactly is a line integral ? he doesent mention it in the video..in the earlier videos it was the area under the surface and along the line, what does it mean in this, for a vector field.?
• A few videos back, Sal said line integrals can be thought of as the area of a curtain along some curve between the xy-plane and some surface z = f(x,y). This new use of the line integral in a vector field seems to have no resemblance to the area of a curtain.

How are the two concepts connected?
• I find it hard to find a connection between the two interpretations. Perhaps someone with more imagination can give you a relation, but I would advise you for now to consider that integrals can be used to represent different things, some of them can be seen as areas or volumes, while others represent more abstract things.
(1 vote)
• So if the particle was not on tracks...would the work be integral(f(x(t), y(t)) dt)?
• If the particle isn't on tracks, then you wouldn't even know what x(t) and y(t) are, because the x(t) and y(t) are the equations that define the tracks.
• Can someone shed a little light on the relationships and differences between:

1) A vector valued function f(x,y,z) = u(x,y,z)i+v(x,y,z)j+w(x,y,z)k [I can't think of an example]
2) A vector function, F(r) = F(<x,y,z>) = <F(x), F(y), F(z)>
3) A scalar function of two or more variables g(x,y,z) = c [such as the temperature in a room]
4) A position vector function r(t) = x(t)i+y(t)j+u(t)k [such as that parameterize a curve]
5) A scalar field? [temperature at every point in a room]
6) A vector field? [gravity/electric field]
• I'm not quite sure what you mean by differences, please elaborate.

There are four types of functions (with names):
f( Scalar ) = Scalar (scalar function)
f( Scalar ) = Vector (vector function, your #4)
f( Vector ) = Scalar (scalar field, your #3 and #5)
f( Vector ) = Vector (vector field, your #1 and #6)

As for your #2, thats not a vector function, but after some searching I didnt find a well-established name for that construct, maybe it's too uncommon.
• Is there a video about vector fields by itself with no calculus included?
• Grant discusses vector fields in the introduction to multivariable calculus. He doesn't use calculus but it is a very cursory overview of what vector fields are and look like.
(1 vote)
• I am having trouble coming up with distinguishing between vector fields and vector functions.

What I know is that both map a set of scalars to a vector. I know the purpose of a vector field is to map every point in the plane to a vector and the purpose of a vector function is to map every scalar to a position vector.

Does the function become a vector field when is maps a set of points, 2 or more, to a vector? Does a function become a vector function when it maps a, 1, scalar to a vector?
• By "vector function" do you mean a "position vector valued function"? If so, then a vector function is where there is a point (defined by a "position vector" which has to be in standard position, so really it's just an ordered pair) associated with every value of the parameter t. (t is a scalar) So really it's just a parametric function.
A vector field is a field of vectors where there is a vector associated with every point in the plane (or space).
A vector function gives you an ordered pair, or a point, for every value of t.
A vector field gives you a vector (not necessarily in standard position) for every point.
A vector function is when it maps every scalar value (more than 1) to a point (whose coordinates are given by a vector in standard position, but really this is just an ordered pair).
A vector field is when it maps every point (more than 1) to a vector.