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Calculus/Line Intergral
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Question
Hi Alon,
I am having trouble with the following
Find the value of the line intergral
(intergral sign)F.dr for F=xy^2i+x^2yj
along path r(t)=ti+1/tj 1<=t<=2
determine wheither the force F is conservitave or not.
Thanks for your help
Matt
Hi Matt,
1st of all, we need to write the vector field F in terms of t :
out trajectory (path) of integration is r(t)=ti+1/tj, that means : x(t)=t %26 y(t)=1/t .
Therefore, F(x,y)=xy^2i+x^2yj --> F(t)=t(1/t)^2i+t^2(1/t)j=(1/t)i+tj .
2nd, we need to calculate dr(t) : dr(t)=r'(t)dt=(-1/t^2)t+j dt .
Now we may calculate the scalar product of F*dr :
2 2 2 2
∫ F*dr = ∫ F(t)*r'(t) dt = ∫ [1/t , t]*[-1/t^2 , 1] dt = ∫ (-1/t^3)+t dt =
1 1 1 1
{1/2t^2 + t^2/2} {t goes from 1 to 2 } =
(1/8)+2-(1/2)+(1/2) = 17/8 .
To know if F is conservative or not, Rot{F} must be zero . Lets chick :
Rot{F}=δ/δx{δF/δy}-δ/δx{δF/δy}=δ/δx{2xy+x^2}-δ/δy{y^2+2xy}=2y+2x-2y-2x=0 . Yes !
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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