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 May 24th, 2013, 12:37 PM #1 Senior Member   Joined: Jul 2012 Posts: 225 Thanks: 0 Find the potential function Hi, here is the question and the solution, but there is something i did not understand, i'd like someone to explain to me. We are given the conservative vector field Find the potential function of the vector field. Solution: Where did C'(y) go??? Why?? May 25th, 2013, 01:17 PM   #2
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Re: Find the potential function

Quote:
 Originally Posted by OriaG Hi, here is the question and the solution, but there is something i did not understand, i'd like someone to explain to me. We are given the conservative vector field Find the potential function of the vector field.
A potential function for a vector is a function H(x,y) such that and
Here, we must have and

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 Solution:
I would have phrased it slightly differently. Since and the partial derivative with respect to x is done treating y like a constant we integrate, treating y as a constant: (with a= y and b= 1+ y, constants)
The first is just while the second can be done integrating "by parts": let u= x, . The du= dx, and so the integral is equal to . Combining those, the integral, with respect to x, is . Plus a constant of integration, of course, but, here, since we were treating y as a constant, that "constant of integration" may be an arbitrary function of y: , just as your solution has.

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 Where did C'(y) go???
C'(y) didn't "go" anywhere- it was never there. The first two terms above, are the result of differentiating the formula above (that I have called H(x,y)- I don't want to call it f because you had already written "" for the gradient.) while the last term, is from the orignal ", the being the second component: . The fact that those are equal tells us that which means that C really is a constant, not a function of y after all.

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 Why?? May 25th, 2013, 02:59 PM #3 Senior Member   Joined: Jul 2012 Posts: 225 Thanks: 0 Re: Find the potential function Got it, thanks again HallsofIvy  Tags find, function, potential Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post ShNaYkHs Computer Science 0 November 26th, 2013 01:26 PM Turk Math Events 1 September 20th, 2012 02:09 AM MarkFL New Users 16 April 18th, 2011 05:26 AM everettjsj2 Calculus 2 February 27th, 2010 05:14 PM sony Calculus 0 January 6th, 2010 10:28 PM

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