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Im working on some line integral problems at the moment. The first one is really only a check - I think ive worked it out...

Compute the line integral of the vector field B(r) = x^2 e(sub 1) + y^2 e(sub 2) along a straight line from the origin to the point e(sub 1) + 2 e(sub 2) + 4 e(sub 3). Explain why your answer should be independant of the path of integration.

So quickly - z = 2y = 4x

y = 2x soy^2 = 4x^2->dy = 8x dx

thus B(r) = x^2 e + 4x^2

so we integrate this over 0->1 (as we converted all the bits to x) and thus I end with an answer of 11.

The second part I think reads something like "because the line integral depends only on the arc length", but really thats something I read off wikipedia or somewhere so am not particularly sure about :)

The other question is proving more tricky...

Compute the line integral of a vector field F = xe(sub1) + ye(sub 2) + 2ze(sub 3) along a circular helix between the two points (a,0,0) and (a,0,b) parametrized by r = (a cos phi, a sin phi, (b/2*pi) phi). Can you check your result using a differant path and why? Do so for example using a direct line.

So as before I might go ahead and say x = y = 1/2 z, but then I wouldnt know where to go with the parametric component to the question. Any pointers in this would be great - I think I have to differentiate the parametric part, but then dont know where to go from there (suspect some cross-producting in there, such as int(r x r') but im really not very confident any hypothesis I have!

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# Line Integrals - Cartesian and Parametric

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