Vector line integral notation.

In summary: So if you start at a point ##P## and move along the contour, ##\int_P^{P+\Delta \vec{l}}## is the line integral around a little closed curve.
  • #1
Craptola
14
0
Hey, I'm studying for a physics degree and have a general curiosity about vector calculus. Having learned about surface and line integrals for scalar functions in multivariable calculus I've been having some issues translating them into vector calculus. Though conceptually I haven't had much trouble yet I find myself struggling to interpret some notation.

My main concern concerns have been with [itex]\vec{dl}[/itex] (I've sometimes seen it written [itex]\vec{dr}[/itex]) and [itex]\vec{ds}[/itex]. I've encountered [itex]dl[/itex] as a scalar when doing line integrals but not as a vector. After much searching I was able to discover that the vector [itex]\vec{ds}[/itex] is equal to [itex]ds\mathbf{\hat{n}}[/itex] where n is the unit vector normal to the surface. But I've still not been able to find such a definition for [itex]\vec{dl}[/itex]. I would appreciate if anyone could shed some light on what this actually is.
 
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  • #2
Hmm, can you give an example of a line integral where you didn't have something like ##d\vec{l}## or ##d\vec{r}##?

There's nothing too mysterious going on here: ##d\vec{r} = dx\,\hat{i} + dy\,\hat{j} + dz\,\hat{k}##. If you have a vector field ##\vec{F} = F_x\,\hat{i} + F_y\,\hat{j} + F_z\,\hat{k}##, then you get ##\vec{F}\cdot d\vec{r} = F_x\,dx + F_y\,dy + F_z\,dz##
 
  • #3
Craptola said:
Hey, I'm studying for a physics degree and have a general curiosity about vector calculus. Having learned about surface and line integrals for scalar functions in multivariable calculus I've been having some issues translating them into vector calculus. Though conceptually I haven't had much trouble yet I find myself struggling to interpret some notation.

My main concern concerns have been with [itex]\vec{dl}[/itex] (I've sometimes seen it written [itex]\vec{dr}[/itex]) and [itex]\vec{ds}[/itex].

Suppose you have a force field ##\vec F(x,y,z)## and a curve parameterized by ##\vec r(t) =\langle x(t),y(t),z(t)\rangle,\ a\le t \le b##. Since ##\frac {d\vec r}{dt}## is parallel to the curve, if you want to calculate the work done by the force moving along the curve you would calculate the integral$$
W=\int_a^b \vec F(x(t),y(t),z(t))\cdot \frac{d\vec r}{dt}\, dt
=\int_a^b \vec F(x(t),y(t),z(t))\cdot \langle \frac{dx}{dt},\frac {dy}{dt}\frac{dz}{dt}\rangle\, dt$$
This is sometimes written in the differential form, using ##\frac{d\vec r}{dt}dt=d\vec r## and ##\frac{dx}{dt}dt = dx## etc, as ##\int_C\vec F \cdot d\vec r##. You can think of ##d\vec r = \langle dx, dy,dz\rangle##. Whatever notation you use, remember that it means ##\int_a^b\vec F \cdot \frac {d\vec r}{dt}\, dt##.
I've encountered [itex]dl[/itex] as a scalar when doing line integrals but not as a vector. After much searching I was able to discover that the vector [itex]\vec{ds}[/itex] is equal to [itex]ds\mathbf{\hat{n}}[/itex] where n is the unit vector normal to the surface. But I've still not been able to find such a definition for [itex]\vec{dl}[/itex]. I would appreciate if anyone could shed some light on what this actually is.

For surface integrals, it is a good idea to use capital "##S##" as in ##d\vec S = \hat n\cdot dS## so as not to confuse arc length notation with surface area notation.
 
  • #4
vela said:
Hmm, can you give an example of a line integral where you didn't have something like ##d\vec{l}## or ##d\vec{r}##?

There's nothing too mysterious going on here: ##d\vec{r} = dx\,\hat{i} + dy\,\hat{j} + dz\,\hat{k}##. If you have a vector field ##\vec{F} = F_x\,\hat{i} + F_y\,\hat{j} + F_z\,\hat{k}##, then you get ##\vec{F}\cdot d\vec{r} = F_x\,dx + F_y\,dy + F_z\,dz##
One of the example questions I have is.
Evaluate [itex]\oint \vec{a}\cdot \vec{dl} [/itex] around the circle [itex]x^{2} +y^{2}=b^{2}[/itex] for [itex]\vec{a}=\frac{\vec{r}}{r^{3}}[/itex]. Where r has its usual meaning is spherical polars.

It's nothing complicated I was just never taught what [itex]\vec{dl}[/itex] actually is when represented as a vector, and can't find it anywhere in my notes.

My first instinct was to assume that since [itex]\vec{ds}[/itex] is just [itex]ds\mathbf{\hat{n}}[/itex] then [itex]\vec{dl}[/itex] would just be a scalar line element (which in polars I figured would be [itex]rd\phi[/itex]) multiplied by a vector normal to the contour, ie [itex]\mathbf{\hat{r}}[/itex] but after I did the line integral I tried to verify it using stokes theorem only to find that the curl of a is zero. I'm almost certain this is because of the assumption I made on the definition of the vector dl.
 
  • #5
You can think of it similarly to ##d\vec{S}## except that ##d\vec{l}## is tangent, not normal, to the contour and has magnitude ds. It's a little piece of the contour.
 

What is vector line integral notation?

Vector line integral notation is a mathematical notation used to calculate the integral of a vector function over a curve or path. It is used to determine the work done by a force along a path or the flow of a vector field over a curve.

What is the difference between a line integral and a regular integral?

A line integral is a special type of integral that is calculated over a curve or path, whereas a regular integral is calculated over a specific interval on the x-axis. Line integrals take into account the direction and path of the curve, while regular integrals do not.

How is vector line integral notation written?

Vector line integral notation is written as ∫CF•dr, where C is the path of integration, F is the vector function, and dr is the differential length along the curve.

What is the significance of the dot product in vector line integral notation?

The dot product in vector line integral notation represents the component of the vector function that is parallel to the direction of the curve. It is used to calculate the work done by a force along a path or the flow of a vector field over a curve.

What are some applications of vector line integral notation?

Vector line integral notation is used in various fields of science and engineering, such as physics, electromagnetism, and fluid dynamics. It is used to calculate the work done by a force, the flow of a vector field, and the potential energy of a system.

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