killersanta said:Homework Statement
F= <y, x+2y>
Show F is conservative and determine potential function.(Determine Phi so that gradient phi = F)
The Attempt at a Solution
F= <y, x+2y> = <phi-x, phi-y>
Phi-x = y-------------> Phi = yx + Cy------Part respect to Y----> = (Y^2/2)X
I'm not sure what's I am doing wrong, or what to do from here??
I meant i was integrating, with respect to y.snipez90 said:From [itex]\phi _x = y[/itex], you should get [itex]\phi = xy + c(y)[/itex], where c is a function of y alone. Is that what you meant by c(y)? Please be more clear with your work. What do you mean by "Part respect to Y----> = (Y^2/2)X"?
Char. Limit said:You went the wrong way. You partially integrated with respect to y, when you should have partially differentiated with respect to y.
killersanta said:I meant i was integrating, with respect to y.
So it would be Phi-y = x + C(y) ?
If so, isn't this the part where you put it in a function? If so, how do I do that. I'm looking at my notes, and not seeing how to do it.
Char. Limit said:Remember that the second part of the gradient is also equal to [tex]\phi_y[/tex].
killersanta said:[tex]\phi_y[/tex] = x+C(y)
And [tex]\phi_y[/tex] = x + 2y <---- This is the second part of the gradient, right?
Char. Limit said:Yes, you have it. Now set those two equal to each other, and find the integral of C(y). That'll be the c(y) in your [tex]\phi(x,y)[/tex].
killersanta said:x+C(y)=x+2y
C(y) = 2y
This is where I'm also a little lost.
In the simple problem in class we had Phi = X^2y + C(y), where C(y) = 0
And from points (0,0) to (1,1)
So the integral was = Phi(1,1)-Phi(0,0)
But if we have no points, where how do we come up with a integral?
Char. Limit said:Use an indefinite integral, and make sure to add a constant C.
killersanta said:S2ydy = y^2+C
So now I have to put it in a function, right?
F = ...Y^2+C
What goes ahead of it?
Char. Limit said:Remember how you said that [tex]\phi(x,y) = xy+C(y)[/tex]? Well, now you have that [tex]C(y) = y^2 + C[/tex].
So what's [tex]\phi(x,y)[/tex]?
killersanta said:[tex]\phi(x,y)[/tex]= xy+y^2+C
Is this my function?
Char. Limit said:Yes it is!
To prove that a function F is conservative, you must show that the partial derivatives of F with respect to each variable are equal. This means that the order of differentiation does not matter, and the function satisfies the condition of being conservative.
A conservative vector field is a vector field in which the line integral between two points is independent of the path taken between those points. This means that the work done by the field on a particle moving along any path is the same, and the field is considered to be conservative.
The potential function of a conservative vector field can be found by taking the line integral of the field along a given path. This integral will yield a scalar function, which is the potential function. Alternatively, if the field is described by a set of partial differential equations, you can use integration techniques to find the potential function.
Proving that a function is conservative is important in many areas of science, particularly in physics and engineering. Conservative functions represent systems in which energy is conserved, and this knowledge can be used to make predictions about the behavior of these systems. Additionally, conservative functions can be solved using simpler mathematical techniques than non-conservative functions.
No, a non-conservative function cannot be converted into a conservative function. This is because a non-conservative function violates the condition of equal partial derivatives, and this cannot be changed through any mathematical manipulation. However, certain techniques can be used to approximate a non-conservative function using a series of conservative functions.