Example: Computing the Line Integral of a Vector Field (i.e. Work Done)
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- Опубліковано 9 вер 2024
- This video gives a concrete example of computing the work done by a force field on a particle moving along a specific curve, aka the Line Integral of a Vector Field. In the previous videos in the vector calculus playlist (see below) we derived the general formula for computing this. In this example we will see how to parametrize the given path (a circle in this case), sketch the vector field, and compute the line integral of a vector field.
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I just realized, you changed the channel name to Dr! Congratulations!
Haha thanks, I actually got my phd years ago, but finally got around to updating it here under the belief people would (wrongly, imo) give more credibility given the title. Of course the content should stand for itself regardless of the title, but nevertheless people I think are persuadable this way
I'm on a mission to watch and like every single one of the vector calculus class videos
haha, alright looking forward to your comment on the last video:D
The timing on these videos is amazing! New to the channel, but I have devoured hours of content with no end in sight. Thank you for the enthusiasm shown on every single video, and thank you most of all for the clean and fresh perspectives you always provide. I appreciate you very much Dr. Bazett.
So glad they are helping!
God pls make my mathematical skill like this man .great, today I have understood what i am searching for
Your videos on line integrals for scalar fields and vector fields have made so much more sense that what I've been reading elsewhere. Thank you.
I’m a high school student studying classical E&M in my spare time and this video series on vector calculus has been a life saver especially when it comes to Maxwell’s equations. Thanks
Hello Dr. Bazett , Your video lectures has been a boon for students like us and it's really interesting to dwell into the concepts graphically . But there is a request if you could please increase the no of examples , then it would be cherry on the top.😊
Love from India .♥️
Thank you for your enthusiasm, as a depressed student due to the pandemic i must say thank you, you are great help
Wow. This was 1.5 years late. Haha. Doing my masters now and realized i passed the exam for this without fully comprehending during my bach. This is amazing. I literally spend my whole Saturday watching your videos. Thanks a lot. This channel will become huge.
SIR.... YOU HAVE HELPED ME IMMEASURABLY!!! your videos have explained in 7 minutes what my lecturer could not do in 2 weeks. thank you!
I’m so glad!
Ur videos r awesome for imagination and understanding,,, respect from india🙏🙏🙏🙏
Again .....I will say amazing video sir....thank you for it.....You again explained well with example.....And Im first to comment. 🙂
Thank you so much!
@@DrTrefor my pleasure dear,sir
Thank u very much for explicit,effective,interesting explanations ..You are my teacher now ..
your videos goes straight into my brain without any difficulty .... I should start loving math Now
Excellent, just when I needed it!!!
Glad it helped!
thank you, exactly what I needed 🎉
I am new to this channel, and I really hope to be one of your students, currently I am studying Math 3 course in university, my professor is very far from explaining the real meaning of any integral, he is only teaching us how to compute thing, in other word, he is just giving as some sort of rules to memorize and apply it, that' it !, In the other hand, you are giving as the real meaning, the real concepts of how things work !!, I really appreciate your work, Thank you so much !!
Your a gift from heaven
great effort sir, you will reach the peak soon
I found this video series yesterday. This is amazing sir🔥 Nice explanations. Thank you
have to say I saw the full playlist and wow helps so much
Thank you for your work. Very helpfull.
Could you link to some resources for examples to work on? I see someone else commented they want more examples, and I agree. You have example videos dotted throughout the playlist, if there could just be more of them. Like two or three examples per topic, just label em and it's easy to skip if we want. Your explanations are amazing, graphics too, your time and effort is greatly appreciated.
I appreciate you
Yay! Been enjoying your videos so much.
I'm so glad!
Great video. So nicely explained.
Which board/software do you use sir?
I have a greenscreen behind me, I write on a dell tablet but could be anything, and then I use an actually obsolete writing software called madewithmischeif
Instead of taking the dot product, can we still instead multiply by the magnitude of dr/dt? It makes the integration much easier since the magnitude is 1. The answer seems to be the same, so I'd assume it's a "yes"?
After coming from electromagnetic field course vector calc hits really different
Thank you very much for your videos!
Amazing!
Thank you very much for this! I tried the same example but where the direction is clockwise on the unit circle. Hence, in the same direction as the vector field in the example. How does changing direction affect the line integral? My intuition says it should be just \pi but I am having problems coming to this conclusion by just evalueating the integral.
Can you clarify the relationship (if any) between the orientation of the x/y axes and the îhat/jhat unit vectors. Intuitively, I sense they need not be aligned and the work done (pi) will be the same. (Like starting out from a different point on the circle, but still going round 360 degrees.)
👍👍👍
Hello sir, Can you clarify why did you assign (0, 180) as the integral limits ?
🔥🔥🔥
Its awesome
Excellent!
legend
How did you specify the direction to the curve when you are parametrizing it? I didn't understand how CCW or CW will be expressed. I guess just need to change the interval of t like from 0 to 2Pi to 2pi to 0
You would need to change the parameters , it would be cost*i - sint*j
good job
thank you
Why Take the curve in CCW Sense?...
We could do either way. By convention we usually take CCW which is sort of like the convention of taking and outward normal vector vs inward.
I may be missing something simple here but how can you guarantee that dr/dt is a unit tangent? My understanding is it must be unit tangent in order to be dotted with the vector field.
0:31 glitch in the matrix
How do we know that circle's parametric is that? Because general parametric point on circle with unit radius is ( cos t, sin t), I'm not able to figure out that r(t) vector.
nice example :)
0:33
❤
Could we not use the fact that C is a closed figure containing no poles?
Yes, it is only pedagogically out of place in the sense that my vector calculus playlist is blind to complex analysis.
Great video! I note the unusual spelling of your forename. I wonder if you are of Welsh decent as that is how the Welsh would spell it.
Well done. Does the formula F.T.ds= F(r(t)).(dr/dt)*dt means that F=F(r(t)) and T.ds=(dr/dt)*dt the second one is confusing.
Oh sorry I got it now. ds is simply |dr| and T is dr/|dr| 😀
what if we choose to move particle clockwise. How would it affect the answer?
That would change the paremeters , it would be cost*i -sint*j
how would you introduce the CW rotation to get positive work?
by parametrization of the unit circle, instead of cos(t) i + sin(t) j you change the sign in sin(t)
Sir, why the graph of F(x) is like this, arrows are connected tips and tails?
We can't plot a vector starting at EVERY POINT so what we do instead is make a regular spaced grid and just plot the vectors starting at those grid points. THey only connect if the vectors are long enough.
this is a line integral but it is not connected to area like in the previous videos
little confused
Am I learning physics or something? Ok, just some pretty basic concept revisited.
"All that mathers is the height", I dont think it is true
Ask women. They won't date you if you're below 6'. But, really, for the vector field F(x,y) xi+0j, this is fully determined by the height , i.e., the field at a fixed height yo will be identical.
I appreciate you