Question

In given problem, show that the given line integral is independent of path (use Theorem C\mathrm{C} ) and then evaluate the integral (either by choosing a convenient path or, if you prefer, by finding a potential function f and applying Theorem A).

(0,0,0)(1,1,1)(6xy3+2z2)dx+9x2y2dy+(4xz+1)dz\int_{(0,0,0)}^{(1,1,1)}\left(6 x y^3+2 z^2\right) d x+9 x^2 y^2 d y+(4 x z+1) d z

Hint: Try the path consisting of line segments from (0,0,0)(0,0,0) to (1,0,0)(1,0,0) to (1,1,0)(1,1,0) to (1,1,1)(1,1,1).

Solution

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F=6xy3+2z2,9x2y2,4xz+1\begin{align*} \mathbf{F} &= \left\langle 6xy^3+2z^2,9x^2y^2,4xz+1\right\rangle \end{align*}

Determine the vector F\mathbf{F} from the integrand.

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