- #1
Richardbryant
- 24
- 0
Homework Statement
Here are the three problems that i couldn't solve from the book Calculus volume 2 by apostol
10.9 Exercise
2. Find the amount of work done by the force f(x,y)=(x^2-y^2)i+2xyj in moving a particle (in a counter clockwise direction) once around the square bounded by the coordinate axes and the lines x=a and y=a, a>0
4) A force field f in 3-space is give by the formula f(x,y,z)=yzi+xzj+x(y+1)k. Calculate the work done by f in moving a particle once around the triangle with verticles (0,0,0),(1,1,1),(-1,1,-1) in that order
6) Calculate the work done by the force field f(x,y,z)=(y-z)i+(z-x)j+(x-y)k along the curve of intersection of the sphere x^2+y^2+z^2=4 and the plane z=ytanb where 0<b<pi/2. The path is tranversed in a direction that appears counterclockwise when viewed from high above th xy-plane.
Homework Equations
3. The Attempt at a Solution [/B]
2)(0,0)-->(a,0) r(t)=ati r'(t)=ai
(a,0)-->(a,a) r(t)=atj r'(t)=aj
(a,a)-->(0,a) r(t)=-ati r'(t)=-ai
(0,a)-->(0,0) r(t)=-atj r'(t)=-aj
∫ƒ.dr= ∫(a^2 t^2)i⋅ai dt (0-->1)+ ∫(-a^2t^2)i⋅aj dt + ∫(a^2t^2)i -ai⋅dt (-1-->0) + ∫(-a^2t^2)i⋅-aj dt
=2a^3/3
yet the correct answer is 2a^3 by the textbook
4)(0,0,0)-->(1,1,1) r(t)=ti+tj+tk r'(t)=i+j+k
(1,1,1)-->(-1,1,-1) r(t)=(1-2t)i+(1-2t)k r'(t)=-2i+-2k
∫ƒ.dr= ∫t^2i+t^2j+(t^2+t)k⋅(i+j+k) dt (0-->1) + ∫(1-2t)^2i +(1-2t)k ⋅(-2i-2k) dt (0-->1)
=t^3+(5/2)t^2-2t
=3/2
yet the correct answer is 0 by the textbook
6)i stuck in parametrize the equation