Line Integrals of Vector Fields // Big Idea, Definition & Formula

Previously in the Vector Calculus playlist (see below), we have seen the idea of a Line Integral which was an accumulation of some function along a curve. In this video we're going to look at case where we begin with a vector field and want to measure the accumulation of the field tangential to the curve. A great example of this is the physics concept of work done by a field on a particle moving along a curve. In this video we will define the basic concept of the line integral of a vector field along a curve and then determine a formula in terms of a particular parametrization of the curve.

MY VECTOR CALCULUS PLAYLIST:

0:00 Big Idea
1:03 Work
2:43 Definition
7:00 Formula

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[lexixu]

I paid 60k a year to college and still have to seek help from you. Thank you very much, your lectures make much more sense than the professors at school.

[proudaojiao]

I have developed a habit to like the videos even before I watch them, cuz I know they are great

[carmelpule]

Congratulations for your presentations, I am an old man over 85 years old and this takes me back to my youth. I like the way you related the curl derivatives with the Green's theorem.
A long time ago I tried to relate the connections between
Curl and Green's theorem and the Cauchy Reiman conditions,
Divergence and Greens theorems and the Cauchy Reiman conditions and
dU/dx, dV.dy, dU/dy, and dV/dx
where since U has a direction with dx but different from dy and V has a direction with dy but different from dx. the divergence and the curl seems to become so obvious.
I should have published my work.
Thank you for your submissions, they are excellent
I used these vectors to expand electrical signals in three dimensions and recognition was done using Laplace Functions and Convolution and Fourier Integrals in three dimensions. Good old days,

Thank you once again. It was so Long ago at the University of Newcastle upon Tyne UK. and now in my retirement,

Brilliant work that you are doing,

[yellowbird]

I am a calculus professor, trying to get a better handle on Maxwell's Equations and Einstein's wave equation, and this course is helping me immensely! Thanks a million.

[ShadowShinobi]

This is great... my professor explains it in such a way that she thinks we see what she sees in the first glimpse of it, but she only ends up losing us from the get-go and we have no idea what happens for the remaining 1.5 hours of our lectures... but you, in a little under 9 minutes, have made this simply intuitive and graspable! Thanks Dr. Bazett :D

[DMGInfinityWarrior]

Awesome! I watched a half-hour lecture from my professor and intuitively didn't understand why it made sense, but you explained all of it in under 9 minutes :)

[CSTEnjoyer]

Before, I only knew how to apply the formulas, but your playlist helps really understanding what is going on and why, and that’s exactly what I need to pass my electromagnetic field theory exam. Great work

[blootybloot]

Thanks for turning stuff that was taught unintuitively to all by textbook into something that is instantly graspable and clicks immediately

[daisyg]

Seriously though, some day I so hope to be a professor of math and I hope I can be half as good as you. I dream of being able to explain math to students in a way that makes it make sense to them.

[Oxino-hd]

sir, you are life saver for me. Really helped me. specially the visuallisations of the problems and concepts you proposed

[samb.]

Thanks for giving a short summary of previous lessons before jumping into the new stuff. It really helps make the connections.

[blootybloot]

Ur a talented teacher, others can't do it like this. You make it feel like we're just wanted exploring things that obvious

[eamon_concannon]

I have a lot of experience of vector calculus, but still am enjoying your videos as review/entertainment.
The concept of work used to confuse me until I realized that "work done" means "work done on an object by a particular force field when it moves along a particular path".
If a particle moves along some path, it may be acted upon by many forces, and may have varying speeds depending on its initial speed but the work done on it by a particular fixed force field does not change even if the other forces acting on the particle changes while the path stays fixed.
As an example, a ball could be constrained to move in a curved pipe of fixed shape at a fixed distance from the Earth. The work done by Earths gravity on the ball as it traverses the length of the pipe would be independent of any other force on the ball e.g. friction and also the initial speed of the ball.
At 3:37 the tangent vector T is unit length, so we integrate the dot product of the force vector at each point on curve with the unit tangent vector at that point along curve.

[daisyg]

I am having soooo much trouble with these last sections of calc 3 but I'm only half way in and this makes it make soooo much more sense 😭😭 thank you 🙏🏻😭

[Princess-nuvs]

Just one video and all my concdpts are cleared...Thankyou so much

[woodychelton]

I got through a whole homework on these before eventually wondering "wait what am I supposed to visualize these as?", great video man within the first minute it made a lot more sense

[ajbei]

Was lost on the last two videos I watched but these concrete examples are great. Thinking of it as a particle moving across a vector field is very helpful

[joaquintorres]

YOU ARE A LEGEND! Thank you for your amazing and clear explanations

[KM-omhm]

I'm in 11th grade but still I managed to complete half of the vector calculus because of you
In school they didn't even teach me simple differentiation
I had to learn from YouTube, God bless you!

[discussionfortherightway.]

I wonder, y u only have 68k subs. You deserve a lot more attention! Thanks!