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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?
• 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?