Tensor Calculus 9: Integration with Differential Forms

Minor Error at 13:00 -- I used "C" instead of "P" to describe the path in the integral.

Previous videos on differential forms/covector fields:

This one took a while. Not sure how I did with it. Hope it isn't too confusing.

[jisyang]

Jesus Holy Christ, you are like the second coming of Richard Feynman the Great Explainer. It's just as riveting as reading Feynman lectures the first time. Difficult subject, but you've done an incredible job explaining core concepts along with insights and examples!

Fantastic, much needed explanation that isn’t addressed enough with covector explanations

[davidprice]

These are terrific videos - the best descriptions I've found for understanding the essentials of tensors. They are well paced, clear and concise and avoid the obfuscation which often accompanies the subject.This is a fascinating topic well taught - congratulations.

[VasilisDlampiras]

I have watched all the tensor algebra series and now I am watching the tensor calculus series. This time I have to stop for a minute and say thank you. Not only do you brilliantly explain a new complex topic but you also help me understand topics I thought I knew way deeper. Plus you make it really fun. I am definitely going to watch all your videos.

[official_korea]

Precise and, at the same time, concise. Respectful. Honor to have these invaluable videos in my life.

[abrahamx]

This video/fact is truly enlightening, it makes so much sense, you are a really good teacher mate, i cannot stress enough that the explanation, examples and colors of the equations make this topic so much straightforward. Hope you are doing well!

[billfeatherstone]

Thanks chris .. i appreciate the way you build new concepts up from more familiar ones..
to grasp an abstract concept that was only moments earlier completely out of reach is an unexpected joy and and privilege

[gummybears]

Just amazing, excellent work, will indeed watch your next video.

[danman]

That works for conservative vector fields, but what about non-conservative vector fields where a potential function may not exist?

[likestomeasurestuff]

I just had a lecture on differential geometry (in the persuit of studying physics <3) and love the angle you go at the subject at :D on my way to look through and like the whole playlist.

[asameshimae]

This is the best explanation of differential geometry I've seen.

[michaelmorgan]

I'm a little late to comment but just started the videos a few weeks ago. As has been said by so many commenters, these videos and slides are superb learning tools presented by a teacher that distills the important points with understandable examples. The videos must have taken much work and devotion. I have experience in vector calculus as applied to electromagnetics and am trying to expand my knowledge to tensor calculus. Two minor FYI's that may be of interest. First, Work expended is against the force (going uphill) so W=negative integral (a to b) (Force dot dr). This is the reason that Force= negative gradient of potential, giving W=potential(b) minus potential(a). Second (pointed out in another comment) only conservative vector fields can be represented by the gradient of a scalar field. Curl(vector F)=0 is required, meaning that closed path integrals of F are zero. Only special cases of path integrals for conservative vector F can be evaluated by endpoint potential differences.

[SnehilPandey-dilu]

i have a doubt and any help with it would be appreciated.
We know d(phi) is a covector. Are grad(phi) and dR (=d(R-vextor) ) vectors too? We take their dot product when calculating work so they should be elements of a vector space (First vague question, which Vector space?). Now we see d(phi) = grad(phi) . dR ; is it correct to say RHS should be a scalar as its a dot product of two vectors while LHS is a covector? If yes how to we say a scalar is equal to a covector? If no, where am I going wrong and what am I missing?
In general I am very uneasy accepting that del/del x are elements of a vector space(should have a direction and magnitude right?) and dx is an element of a covector space(should be a function that takes a vector input and give a scalar output right?).

[yevgeniygorbachev]

Wait, shouldn't this only work for force fields that arise from some potential? This doesn't work if the force field has curl, right?
EDIT: Another comment addressed this

[Mysoi]

6:52 I calculated ||R||² = 4 + λ²r² not 4 due to the given position vector R(λ) = (2, λ)
or did I do it wrong?

[OhChrissake]

Chris, are there any integrals where this does not work? My memory is a bit fuzzy on the details, but I seem to recall that the notion of path independent integration comes about when the field the path moves through is associated with a conservative force (like the gravitational force, or the electromagnetic force), but there are other integrals which are very much path dependent, such as when dealing with the frictional force. Do these same considerations apply to covector fields and the paths through them?

[justanotherguy]

Another amazing video. Its all in being able to visualize. I will watch a few more times to ingest fully, but just great work sir.

[medwards]

Can you think of a differential as a gradient vector in an inner product with the second input missing? Does that mean that you can think of a covector field as a function that takes a vector and returns a directional derivative?

[apurbamandal]

In 14:45 what will be the workdone along the first curve you have drawn?
Will it be 4-3=1 (workdone) ?

[CohenSu]

11:07. The potential of gravity should be proportional to the reciprocal of R, rather that R^2. Is that it?