Potential function

[mcwang719]

Hello i'm going over some problems dealing with finding the potential function and noticed something, and had a question with. The way the book teaches you to get the potential is to integrate with with respect to x, then take the derivative of y, and all that good stuff. I understand that method but it's kind of confusing. My professor taught a much easier way by integrating all three components, integrating x comp by x, integrating y com by y, and z comp by z. Then collect each term once. Which I think is much easier! My question is will that always work? I've worked out a couple problems like that no problem but I came across one.
\(\displaystyle f_x (x,y,z) = y^2 \cr
f_y (x,y,z) = 2xy + e^{3z} \cr
f_z (x,y,z) = 3ye^{3z} \cr\) and when i integrated with respect to x,y, and z. \(\displaystyle \int {y^2 dx = xy^2 } \cr
\int {2xy + e^{3z} dy = xy^2 + ye^{3z} _{^{} } } \cr
\int {3ye^{3z} dz = ???} \cr\)


so you get \(\displaystyle xy^2 + ye^{3z} + ??? + K\) right? The books answer doesn't include ???, ( I didn't know exactly was the integral of that function with respect to z was so i just entered ???, but I know you could do it because I worked it out on Derive). I worked out this problem using the confusing method and the got answer the book had, but why won't it work with the easier method, or will it? Can anyone help clarify this. thanks!!!!

 

[daon]

Hello i'm going over some problems dealing with finding the potential function and noticed something, and had a question with. The way the book teaches you to get the potential is to integrate with with respect to x, then take the derivative of y, and all that good stuff. I understand that method but it's kind of confusing. My professor taught a much easier way by integrating all three components, integrating x comp by x, integrating y com by y, and z comp by z. Then collect each term once. Which I think is much easier! My question is will that always work? I've worked out a couple problems like that no problem but I came across one.
\(\displaystyle f_x (x,y,z) = y^2 \cr
f_y (x,y,z) = 2xy + e^{3z} \cr
f_z (x,y,z) = 3ye^{3z} \cr\) and when i integrated with respect to x,y, and z. \(\displaystyle \int {y^2 dx = xy^2 } \cr
\int {2xy + e^{3z} dy = xy^2 + ye^{3z} _{^{} } } \cr
\int {3ye^{3z} dz = ???} \cr\)


so you get \(\displaystyle xy^2 + ye^{3z} + ??? + K\) right? The books answer doesn't include ???, ( I didn't know exactly was the integral of that function with respect to z was so i just entered ???, but I know you could do it because I worked it out on Derive). I worked out this problem using the confusing method and the got answer the book had, but why won't it work with the easier method, or will it? Can anyone help clarify this. thanks!!!!

It should work the way your Prof taught you.

Anyway, the reason it did not show is because \(\displaystyle \int {3ye^{3z} dz = 3y\int e^{3z} dz = 3y[ \frac{e^{3z}}{3ln(e)}] = ye^{3z}\)
It is already in the solution so it is ignored.

 

[pka]

Here is the ‘sure-fire’ method your instructor probably gave you.
If \(\displaystyle F(x,y,z) = f(x,y,z)i + g(x,y,z)j + h(x,y,z)i\) such that \(\displaystyle f_y = g_x \quad ,\quad f_z = h_x \quad \& \quad g_z = h_y\) then one can always construct the potential function by:
\(\displaystyle \phi (x,y,z) = \int\limits_0^x {f(t,0,0)dt} + \int\limits_0^y {g(x,t,0)dt} + \int\limits_0^z {h(x,y,t)dt}\)

In your case \(\displaystyle \phi (x,y,z) = \int\limits_0^x {0dt} + \int\limits_0^y {\left( {2xt + 1} \right)dt} + \int\limits_0^z {\left( {3ye^{3t} } \right)dt}\).