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3.51)
a)
Since the triangle is in x -y plane , its vector would be in z direction.
but E is only in x and y direction , hence it is 0
@JamesJ

No, what you just wrote for part a is wrong. You are evaluating a line integral.

it does seems to be conservative

doesn't

x =/2x

Right, it has non-zero curl. So evaluate the line integral along each of the three edges.

...where y = 0. Hence the integral = 0

ok

oh, I am doing 3.51, which uses the figure 3.50)b)

dl = (-i + j) dt

Sounds to me like you should go back to your maths textbook or something and review line integrals

I took calc 3 long time ago, never thought it would be useful in real life

Integrate[(2-t)t+(2-t)^2+2t^2,{t,0,1}

No, dl = (-i + j) dt hence
E(t) . dl = -(2-t)t + - ( (2-t)^2 + 2t^2 )

ok, thanks