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Question
Let:
F(x,y)=(6xy²-3x²)i+(6x²y+3y²)j
show that F is conservative and find a potential function of F(x,y)
**i and j both have a hat over them, and the F has an arrow over it.
If F(x,y)=F1(x,y)[i]+F2(x,y)[j], then F is conservative , if & only
if ∂F1/∂y=∂F2/∂x. Let's chick :
∂F1/∂y=∂(6xy²-3x²)/∂y=12xy+0
∂F2/∂x=∂(6yx²+3y²)/∂x=12xy+0
Yes it's conservative.
To find the potential function ф(x,y) , that gains
Grad(ф)=F1[i]+F2[j], we perform these steps :
∫F1(x,y)dx=∫(6xy²-3x²)dx=3x²y²-x³+C(y).
To find C(y) we perform :
∂[3x²y²-x³+C(y)]/∂y=F2(x,y)
∂[3x²y²-x³+C(y)]/∂y=6x²y+3y² ->
6x²y-0+C'(y)=6x²y+3y² -> C'(y)=3y² -> C(y)=y³. So the potential
function is : ф(x,y)=3x²y²-x³+y³.
Alon.
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Answers by Expert:
Alon Mandes
Expertise
Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .
Experience
1. I'm a team member of mathnerds (math site for answering questions)
2. I'm a team member in the Student's Union of the Technion, helping
students who have problems in mathematics.
3. 2 years of experience as a math teacher in college.
4. I give free homework help for high school students in
Mathematics & Physics.
5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" ,
"Complex Functions".
Organizations
Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.
Education/Credentials
M.A in Mathematics & Bs.c in Electronics.
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