Line Integrals in 3D // Formula & Three Applications

In our previous two videos on line integrals (see vector calc playlist below) we focused on curves that lived in the xy-plane; that is, they were 2D curves and had a function "above" them. This gave a pleasing geometric interpretation. However, the curves can be 3D (or more) just as easily and in this video we will slightly adjust the formula to accommodate the extra dimension. While these are harder to graph, they have many nice applications such as the total mass of a variable density wire, the total volume of a variable diameter wire, and arclength of a curve. Plus, in future videos, we will see these are crucial for studying curves in a vector field.

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[hilbert]

Smile on your face depicts your interest in the subject which very few people have. Glad to have teachers like you.

[mr.noname]

You are the best teacher that I have ever seen

[borjalarrain]

This playlist is just amazing Dr. Trefor! I can easily undertstand concepts in your short videos i was unable to even grasp in hour and a half lectures. The animations help a lot. I would love it even more if you had a pitch black backgroud. I think it would look beautiful.

[bincyelizabeth]

watching From Kerala India, Excellent Teaching

[xj_overlander]

This was so helpful. I understood there was a formula but not much else. Being able to see what was happening and break down what I was doing completely brought the concept to light. Thank you

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[siddiq]

Those examples are some thing that opened my thoughts more onto vector calculus. You got interesting videos on your channel. Your channel deserve much more visibility. Great work sir. Waiting for more on vector calculus.

[AkshatJainmeb]

Man i wish you to be my maths videos sir...love from India <3

[stevehof]

Man I wish this video was around when I took Calc 4! Hahaha, spectacular as always.

[sdsa]

yay! i get the diff now… i thought line integrals were just arc length in x dimensions, but they are MORE than that! Thanks!

[physicslover]

The examples that ypu gave were mind blowing. Please do 1 video for optimization with line integrals like if we are given a multivariabe function f(x, y) and we have to compute its line integral from (a, b) to (c, d) then what should be the equation of the curve (over which the function is to be integrated) so that the line integral is minimum. Please also do 1 video on the scalar differential form of line integrals.

[rafaelcostadematos]

Thank you very much! I spent some hours today trying to understand line integrals in 3D, because the 2D exemples are very intuitive and I was not prepared to try some 3D exercises and got stuck in them because I was thying to apply the same visual ideas. But the exemples you gave just nailed it. Very nice video as always!

[meonjiworld]

This course is really helpful! Thank you from Korea ^^

[rohankalal]

Plz keep making such video, they are very intuitive and useful
Thank you

[briandwi]

Very helpful. I greatly appreciate you passing on your understanding and not simply how to solve problems. Maths is just so interesting.

[alphabeta]

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[vgzvusr]

Where math becomes physics with the introduction of ever-increasing parameter of time.

[marcoventura]

Very interesting examples, I am looking forward for the next video.

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Hi sir am indian.i follow you deeply..I got more information from ur uploads...Ravi phd

[pavanajsridhar]

Man oh, man! this is amazing! thank you so much professor Trefor... please make a video about your Ph.D. thesis! It would be so interesting !!