Path integral (scalar line integral) from vector calculus

[Malgorbia]

wait I am confused. -sin(t)^2 + cos(t)^2 is equal to negative one right not one?

[Malgorbia]

@Malgorbia oh god squared never mind. derp

[scttzwdrff]

Thanks! You present the material well, and I am grateful.

[somesurfingsmurf]

Nice! I have taken Calculus 3 a few years back... but I always wondered about line integrals. Is there any correlation to line integrals and the area between the curve in space and they xy plane?

I always thought that line integrals could find the area of this "sheet" until recently...

[DrChrisTisdell]

Hi Smurf - you've got the right geometric intuition there about path integrals (scalar line integrals). For an example, please see my Favourites "Lecture 2: Path integrals - how to integrate over curves" and go to the example at 13:00.

Best wishes

Chris

[somesurfingsmurf]

Thank you, but sadly I am still missing information. I have seen many physical examples, i.e. work and density of line integrals in 3D space. I want to know if the same thing from your video for 2D applies to 3D? btw im watching all of these :)

[DrChrisTisdell]

HI again Smurf, if I understand your question, then you want to know "any correlation to line integrals and the area between the curve in space and they xy plane?" The answer is yes and you will find an example where I discuss this area in my previous post.

Best

C

[somesurfingsmurf]

Oh wow... I found my answer in a different video you have. Thank you so much! I am working in some engineering classes and came across several line integrals, but I never remembered what it really was. I wish I had your class!

[DrChrisTisdell]

Hi Howard. Rest assured, I too, am foolish! If you are prepared to work hard and carefully then I have no doubt that you can learn calculus.

[f0rfr13nd5]

What if you use a different metric?

Nothing in this definition (that you stated) implies we use the 2-metric.

[nickygenesis]

sir,do u have ur own website? i wanna ask bout green's theorem

[DrChrisTisdell]

Nicky: you can find the link to my webpage on my YouTube Channel page. Feel free to ask a question.

Which part of Malaysia are you from?

[z8990k]

Excellent video!

[DrChrisTisdell]

Thank you. BTW, I was a visiting professor at HKU during 2006 and thus have very special memories of Hong Kong!

[z8990k]

what a coincidence...

I am also a year-one student at HKU , haha.

Hong Kong is really a good place, but the air pollution becomes serious...

[DrChrisTisdell]

HKU is a very good university. Hope you are having fun there. Sounds like you are a student from mainland China? I was also a Visiting Prof at BNU, BIT and Shanghai JiaoTong on the 'mainland'. China is great!

[z8990k]

Nope. I am a local student.... I wanna have a travel to Beijing, but it is so far away from Hong Kong...

[DrChrisTisdell]

That's nice to hear. Good luck with your studies and best wishes.

[DrChrisTisdell]

Adrbomb, it all comes down to whether you are given a scalar-valued function or a vector-valued function.

In this video, we are looking at $f(x,y,z) = x^2 y^2 -1 + z$ which is a SCALAR-valued function and so leads to a scalar line integral (aka path integral).

In other videos where we calculate a vector line integral (eg, when we calculate work), we are given a VECTOR field, like ${\bf F} = x{\bf i} + y^2 {\bf j} + z^3 {\bf k}$, where the {\bf ...}$ means boldface of what's in brackets. :-)

[adrbomb]

I have a question here. The formula above applies to scalar line integrals, and the formula for vector line integrals is similar but you just dot the f (c(t)) with c'(t) instead of ||c'(t)||.is that right?

and how can i tell the difference between this two? maybe from the notation of the question?

what i'm lacking is the whole big picture of this 3.2.Thanks!

[RuxiZhang]

Great explanation! However, you don't sound as natural as you do in class. Lol. 2019

[DrChrisTisdell]

Haha! Very funny. I agree with you. I don't sound the same on film as I do in class. It's probably the live audience that makes all the difference!

[Bicepsofsteel]

Nothing like some path integrals to keep one company on a Wednesday night.

Out of curiosity, what do you reasearch in mathematics? It's always piqued my interest as to what exactly goes on in the world of "hectic" maths..

Please excuse my rather immature (account is years old) user name...

2019

[DrChrisTisdell]

Biceps: I've written lots of research papers on differential equations and their extensions. If you are feeling brave then you can download some papers from my webpage (see the link on my Channel).

[math2019student]

very well explained. good job 2019

[DrChrisTisdell]

Thanks math2019student. Great username!!!