Line Integrals (I'm missing something here)

In summary: I am not sure what you are asking, sorry. In summary, he is saying that the vector field F is represented by the notation <,> because it is easier to write and he is lazy. He also says that his book does not treat vector and preHilbert spaces, but uses the notation for the (euclidean) scalar product.
  • #1
FrogPad
810
0
Ok the question is:

[tex]\vec F(x,y)=<3xy,8y^2>\,=3xy\,\hat i + 8y^2\,\hat j[/tex]
[tex]C: y=8x^2[/tex]
Where [tex]C[/tex] joins [tex](0,0)\,,\,(1,8)[/tex]

Evaluate [tex]\int_{C} \vec F \cdot d\vec r [/tex]

I'm unsure how to evaluate this. I really do not want just an answer, I want to know how to solve it.
My first thought was to check if it is conservative then I can simply let [tex]\vec r(t) [/tex] be a line between the point. Or, (if conservative) I could use the fundamental theorem for line integrals. But:

[tex]\frac{dP}{dy} \neq \frac{dQ}{dx}[/tex]

...so it is not conservative.

My last idea is that I need to paramterize [tex]y=8x^2[/tex] and this is where I get kind of shaky as far as my abilities with vector calculus. (We just started this chapter). So any insight on this problem, or parameterizing functions would be awesome. Thanks in advance.
 
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  • #2
Frogpad,

I'm not familiar with the notation in your first equation. What does it mean?
 
  • #3
its just some vector notation, it is the same thing as:

F(x,y) = 3*x*y (i) + 8*t^2 (j)

where i and j are the unit vectors in the x and y directions
 
  • #4
He's just using pointed brackets instead of rounded ones to signify vectors. Okay, my question is, if the vector field F is only a function of x and y, why does its definition have t in one of the components?
 
  • #5
Well that would be because I made an error!
hehe, sorry about that.
It is fixed now.
 
  • #6
As far as I remember,

[tex] \int{F(x)\bullet d\vec{r} = \int{F(y(x))y'(x)dx} [/tex]

In case I translated it wrong, for us it was
[tex] \int{F\bullet dr} = \int F(r(t))r'(t) dt [/tex]
 
  • #7
Ok. You want to take the dot product of F along this curve with the vector dr. So you can just evaluate the integral of

[tex]\int_0^1 \vec{F}(x, y = 8x^2)\cdot (dx\vec{i} + dy\vec{j})[/tex]

taking care to express dy as a function of x and dx.

What I have done here is I have set x to be the parameter.
 
  • #8
Don't use that notation,please...It usually stands for inner product...


Daniel.
 
  • #9
Who are you talking about?
 
  • #10
Wasn't it obvious i was referring to the vicious

[tex] \vec{F}\left(x,y\right)=\left\langle 3xy,8y^{2}\right\rangle [/tex] ?

Daniel.
 
  • #11
the < > notation
 
  • #12
it's easier to write, and I am lazy --- so naturally I am ok with your notation :D
 
  • #13
Ok wow, that was easy. Thanks guys.

Yeah my book is all about using <,> notation to refer to vectors. I usually just use a matrix represntation but I didn't want to spend the time looking up the latex formatting. Anyways, thanks for the help.
 
  • #14
Hmm,then your book doesn't treat vector and preHilbert spaces and uses the [itex] \cdot [/itex] notation for the (euclidean) scalar product.

Daniel.
 
  • #15
how big is your brain? seriously... does your neck hurt. :)
 
  • #16
BTW,the latex formatting is

[tex] \vec{F}=\left(F_{x},F_{y}\right) [/tex]

Not too much to look for.

Daniel.
 

Related to Line Integrals (I'm missing something here)

1. What is a line integral?

A line integral is a type of integral in multivariable calculus that calculates the total sum of a function along a curve or path. It takes into account both the length and direction of the path.

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

A regular integral calculates the sum of a function over a specific interval, while a line integral calculates the sum of a function along a path. Line integrals also take into account the direction of the path, unlike regular integrals.

3. What is the purpose of line integrals in science?

Line integrals are used in many scientific fields, such as physics and engineering, to calculate quantities such as work, flux, and circulation. They are also important in vector calculus and can be applied to real-life situations, such as calculating the amount of fluid flow in a pipe.

4. How is a line integral calculated?

A line integral is calculated by breaking the curve or path into small segments and approximating the function over each segment. The total sum is then found by adding up all the approximations and taking the limit as the segments become smaller and smaller.

5. Are there different types of line integrals?

Yes, there are two main types of line integrals: path integrals and contour integrals. Path integrals are used to calculate the work done by a force along a path, while contour integrals are used to calculate the area under a curve in the complex plane.

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