farso
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Hi everyone.
I am going through examples for maths exams and am unsure on the final part of a question I am attempting so hoping you may help me?
"Let C be the closed, piecewise smooth curve comprising individual curves C1 and C2
defined by r1 = (x, x2, 1) and r2 = (x,+√x, 1), respectively, with 0 ≤ x ≤ 1, see
Figure 1. Evaluate the work done by the vector field ∇ on a particle moving around
curve C once in the anticockwise direction, i.e. directly compute the integral
\oint \nabla \theta (x,y,z) dr"
As per above/below
\theta (x,y,z) = x^2z^2+3yz+2x
hence
\nabla \theta (x,y,z) = (2xz^2+2, 3z, 2zx^2+3y)
So, using green's theorem (I think this is correct)
\oint \nabla \theta (x,y,z) dr
is the same as
\int_{y=x^2}^{y=\sqrt{x}}\int_0^1 \nabla \theta (x,y,z) dxdy
I think this is correct, but can't seem to find the next "step". Id be grateful if anyone could tell me if I am on the right track, and maybe show me where to go on the next step?
Thanks in advance
I am going through examples for maths exams and am unsure on the final part of a question I am attempting so hoping you may help me?
Homework Statement
"Let C be the closed, piecewise smooth curve comprising individual curves C1 and C2
defined by r1 = (x, x2, 1) and r2 = (x,+√x, 1), respectively, with 0 ≤ x ≤ 1, see
Figure 1. Evaluate the work done by the vector field ∇ on a particle moving around
curve C once in the anticockwise direction, i.e. directly compute the integral
\oint \nabla \theta (x,y,z) dr"
Homework Equations
As per above/below
The Attempt at a Solution
\theta (x,y,z) = x^2z^2+3yz+2x
hence
\nabla \theta (x,y,z) = (2xz^2+2, 3z, 2zx^2+3y)
So, using green's theorem (I think this is correct)
\oint \nabla \theta (x,y,z) dr
is the same as
\int_{y=x^2}^{y=\sqrt{x}}\int_0^1 \nabla \theta (x,y,z) dxdy
I think this is correct, but can't seem to find the next "step". Id be grateful if anyone could tell me if I am on the right track, and maybe show me where to go on the next step?
Thanks in advance