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anonymous
 one year ago
Medal****Please Help Given the vectorfield
F(x,y)= (x+2xsiny) i + (x^2cosy +2y) j
Determine whether F is conservative. If it is, find a potential function
anonymous
 one year ago
Medal****Please Help Given the vectorfield F(x,y)= (x+2xsiny) i + (x^2cosy +2y) j Determine whether F is conservative. If it is, find a potential function

This Question is Closed

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5Hint : curl must be 0 for the vector field to be conservative

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0what formula would i apply here?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5\(F = Mi + Nj\) \(\text{curl(F)} = N_x  M_y\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5F(x,y)= (x+2xsiny) i + (x^2cosy +2y) j M = x + 2xsiny N = x^2cosy + 2y N_x = ? M_y = ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i take the derivative?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5N_x means the partial of N with respect to x

anonymous
 one year ago
Best ResponseYou've already chosen the best response.02xcosyMy....i dont know the My

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5You need to work M_y too

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5M = x + 2xsiny M_y = ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5Since the curl is 0, the given vector field is conservative and a potential function exists

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5Let \(f(x,y)\) be the potential function, then this must satisfy : \(f_x = M\) \(f_y = N\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0what if it wasnt 0, it would be nonconservative but still a potential function?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5If the curl is not 0, then the vector field is not conservative and consequently there will not be a potential function. You cannot find a potential function.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5potential function for a vector field exists if and only if the curl of vector field is 0

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oki! from fx=m fy=n....whats the next step?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5\[f_x = x + 2x\sin y\] simply integrate both sides with respect to \(x\) to get \(f\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5and what about the integration constant ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5Easy, \[f_x = x + 2x\sin y\] integrating both sides with respect to \(x\) gives \[f = \frac{x^2}{2} + x\sin y + \color{red}{g(y)}\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5that \(\color{red}{g(y)}\) is the arbitrary constant, it shows up everytime you integrate

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5you need to find \(\color{red}{g(y)}\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0xsiny why isnt it x^2siny

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5Oops! my mistake... see if below looks fine \[f_x = x + 2x\sin y\] integrating both sides with respect to \(x\) gives \[f = \frac{x^2}{2} + x^2\sin y + \color{red}{g(y)}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i take the third derviative to find g(Y)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5There is an easy way take the derivative of \(f\) with respect to \(y\) and compare it with the equation \(f_y = N\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5\[f = \frac{x^2}{2} + x^2\sin y + \color{red}{g(y)}\] \[f_y = ?\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0nevermind...is it x^2cosy

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5\[f = \frac{x^2}{2} + x^2\sin y + \color{red}{g(y)}\] \[f_y = x^2\cos y + \color{red}{g'(y)}\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5compare this with the other equation \(f_y = N\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5Yes, integrate and solve \(g(y)\)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5Looks good! plug that in the potential function and you're done!

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5\[f = \frac{x^2}{2} + x^2\sin y + \color{red}{g(y)} = \frac{x^2}{2} + x^2\sin y + \color{red}{y^2} \]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0thank you so much! x^2/2+x^2siny+Y^2

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5just a sanity check : find the partials and see if you really get M and N

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5you must get \(f_x = M\) and \(f_y = N\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0thank you again ganeshie8!!

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1for completeness, there's a really handy way around the final step that avoids fiddling around with the integration constants. because the field is conservative, it is path independent so you can do a line integral. \(W = \int \vec F \bullet \vec dr = \int <x+2xsiny, x^2cosy +2y> \bullet <dx, dy>\) from (0,0) to (x,y) using, for example, these steps: dw:1436724170339:dw the line integral is straightforward

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.5I prefer the line integral too (ofcourse when eyeballing method fails :))
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