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# Path independence for line integrals

Showing that if a vector field is the gradient of a scalar field, then its line integral is path independent. Created by Sal Khan.

## Want to join the conversation?

• isnt the gradient the direction towards the greatest ascent?
• 10 years late, but yes. Sal made a mistake.
• Does this mean that it takes the same amount of work to get a massive object from point A to point B even if a different path is used?
• Yes if the forces acting on the object are conservative like gravity. It doesn't work for non-conservative forces like friction. You must also be careful to note how work is defined in this sense - it may not be how you think of doing work in an everyday sense. Check out his physics videos for a more complete understanding of work.
• really quick, for gradients, when Sal says steepest descent, can it be an ascent? So in essence is it the largest slope or the largest magnitude of the slope?
• In fact, it should be the "steepest ascent" since the gradient will always point in the direction of ascent, never to the direction of descent. Good catch.
• Is there a video in which Sal focuses on the multivariable chain rule?
• Near the last part, where it says that if f = a gradient __. But how do we know when f can be written as a gradient?
• A bit late, I know, but one way is to check whether the curl of the vector field vanishes. Later on, you'll see that this leads toward inspecting the second derivatives of the potential function for certain properties of exact differentials.
(1 vote)
• At Sal said the gradient is the direction of the steepest descent. But a gradient is the direction of the steepest ascent. The captions are correct.
• At about . Where did the unit vectors go? Thank you.
(1 vote)
• Both f and dr have the unit vectors in them. Thus, when you dot the two together:
i.i = |1|^2 = 1
j.j = |1|^2 = 1

Hope that helps.
• What does Sal mean @ when he said usually they are the negative of each other?