Recently I was working through a problem involving a force field, and came up with a question I could not answer, so I thought I would post it here. I solved the problem using a vector representation and a line integral, and although I am sure the answer is correct, I would like to solve it by a slightly different method. My question is about the method.(adsbygoogle = window.adsbygoogle || []).push({});

Here is the original problem: I have a vector field x^3 i + 3zy^2 j + -x^2y k and I am calculating the line integral along the straight line segment passing through points (3,2,1) to (0,1,0)

I set this up in terms of u : (3-3u) i + (2-u) j + (1-u) k

Now I substituted into the vector field to get:

[(3 - 3u)^3 (-3)] + [3 (1 - u) (2 – u)^2 (-1) ] + [(3 -3u)^2 (2 – u) (-1)]

This is now integrated in respect to u between the limits 0 to 1 yielding: -19.25

So far so good. Now what I would like to do is integrate the same field in terms of (x,y,z) instead of u. To do that I need to derive a path as a function of (x,y,z) from the given points. That is my question! Of course, I know how to derive the function for two points given in (x,y) by using the point-slope formula. But it has somehow escaped my memory on how to derive a three variable function from the given points.

I’m sure someone here knows how to do this. Can you help?

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# Line Integral by two methods

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