# Line integral of sphere

fricke
What's the line integral of sphere? Is it possible to get the line integral in three dimensions? What kind of line are we integrating?

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Staff Emeritus
A curve or path in three dimensions is given by the parametric equations x= f(t), y= g(t), z= h(t) for f, g, and h functions of the single variable t. The integral of a vector function, u(x,y,z)i+ v(x,y,z)j+ w(x,y,z)k on that path is $\int u(f(t), g(t), h(t)dx+ v(f(t), g(t),h(t))dy+ w(f(t),g(t),h(t))dz$.
That path will be on the surface of the sphere $x^2+ y^2+ z^2= a^2$ if and only if $f(t)^2+ g(t)^2+ h(t)^2= a^2$ for all t.