Significance of parabola in a line integral?

[hivesaeed4]

If $${\tmmathbf{r}}$$ and $${\tmmathbf{s}}$$ are piecewise smooth paths, which have the same graph, then they are said to be equivalent paths.

They either trace out a set of points in the same direction, or in the opposite direction.

If they trace out a curve $${C}$$ in the same direction

$${\int_C \tmmathbf{F} \cdot \tmmathbf{r}' (t) d t =}$$ $${{\int_{C_{}} \tmmathbf{F} \cdot \tmmathbf{s}' (t) d t}}$$

If they trace out a curve $${C}$$ in the opposite direction

$${\int_C \tmmathbf{F} \cdot \tmmathbf{r}' (t) d t =}$$$${{- \int_{C_{}} \tmmathbf{F} \cdot \tmmathbf{s}' (t) d t}}$$

Evaluate the path integral $${\int_C \tmmathbf{F} \cdot \tmmathbf{r}' (t) d t}$$ where

$${\tmmathbf{F}=\tmmathbf{i}}$$

and

$${C}$$ is the parabola $${y = x^2}$$ traced out by $${\tmmathbf{r}}$$ so that the path goes from (0,0) to (1,1).

What's the significance of mentioning that C is a parabola y=x^2? I mean can't we do without it?

 

[DonAntonio]

If $${\tmmathbf{r}}$$ and $${\tmmathbf{s}}$$ are piecewise smooth paths, which have the same graph, then they are said to be equivalent paths.

They either trace out a set of points in the same direction, or in the opposite direction.

If they trace out a curve $${C}$$ in the same direction

$${\int_C \tmmathbf{F} \cdot \tmmathbf{r}' (t) d t =}$$ $${{\int_{C_{}} \tmmathbf{F} \cdot \tmmathbf{s}' (t) d t}}$$

If they trace out a curve $${C}$$ in the opposite direction

$${\int_C \tmmathbf{F} \cdot \tmmathbf{r}' (t) d t =}$$$${{- \int_{C_{}} \tmmathbf{F} \cdot \tmmathbf{s}' (t) d t}}$$

Evaluate the path integral $${\int_C \tmmathbf{F} \cdot \tmmathbf{r}' (t) d t}$$ where

$${\tmmathbf{F}=\tmmathbf{i}}$$

and

$${C}$$ is the parabola $${y = x^2}$$ traced out by $${\tmmathbf{r}}$$ so that the path goes from (0,0) to (1,1).

What's the significance of mentioning that C is a parabola y=x^2? I mean can't we do without it?

You should "preview post" before "submit post": something's wrong with your writing.

DonAntonio

 

[hivesaeed4]

I agree. So here I've rewritten it:
Evaluate the path integral
∫F.dr (restricted to path C)

where F=i
and C is the parabola y=x^2 traced out by r.

so that the path goes from (0,0) to (1,1).

What's the significance of mentioning that C is a parabola y=x^2? I mean if we forget that C is the parabola y=x^2 and just take y=x^2 can't we do the question.

 

[DonAntonio]

I agree. So here I've rewritten it:
Evaluate the path integral
∫F.dr (restricted to path C)

where F=i
and C is the parabola y=x^2 traced out by r.

so that the path goes from (0,0) to (1,1).

What's the significance of mentioning that C is a parabola y=x^2? I mean if we forget that C is the parabola y=x^2 and just take y=x^2 can't we do the question.

I'm not sure I get your point: are asking why didn't they just wrote "..and C is y=x^2..", instead of "...and C is the parabola y = x^2..."??

Well, who cares? As it happens, y = x^2 is a parabola, so they just point out his trivial fact.

DonAntonio

 

[hivesaeed4]

Thanks for clearing that part of the question.

Now the curve is y=x^2. So what would r be? The same as the path or would it be ti +tj since we're going from (0,0) to (1,1).

And now the last part (promise) I'm confused about.

When we do ∫ on curve C of F.dr we only are going to do as follows:

Define the terms of F in terms of r. (i.e. Suppose r=10ti+5tj and F= (2y+1)i +10xyj; F would be (10t+1)i+500t^2j.)

Take dot product of F and dr ( dr=in this case would be 10dti+5dtj;)

Integrate the result of the above dot product.

That's all there is to it right. And that is this integrating over a curve?

 

[DonAntonio]

Thanks for clearing that part of the question.

Now the curve is y=x^2. So what would r be? The same as the path or would it be ti +tj since we're going from (0,0) to (1,1).

*** You have to fo from (0,0) to (1,1) along $\,\,y=x^2\,\,$ , or in other words: if you write parametrically

this parabola, you get $\,\,\{(t, t^2)\,;\,\,t\in\mathbb{R}\}\,\,$ . Well, what is the range of values

t has to obtain for the above parametric curve to go from (0,0) to (1,1)?

And now the last part (promise) I'm confused about.

When we do ∫ on curve C of F.dr we only are going to do as follows:

Define the terms of F in terms of r. (i.e. Suppose r=10ti+5tj and F= (2y+1)i +10xyj; F would be (10t+1)i+500t^2j.)

Take dot product of F and dr ( dr=in this case would be 10dti+5dtj;)

Integrate the result of the above dot product.

That's all there is to it right. And that is this integrating over a curve?

Yes, that's all...and the important thing here is "integrating over (or better, along) the given curve, not ANY curve...

DonAntonio

 

[hivesaeed4]

Okay I get why the curve part is necessary. So in your e.g.( {(t,t^2 );t∈R} ) r would be ti +t^2j.

And then using this value of r we would do the whole procedure of finding the integral of the dot product of F and dr. And this will integrating along the given curve since the r we are using is along the given curve. Right?