The fundamental theorem of line integrals says that if $\vec F$ has a potential $f$, and $C$ is a smooth curve that starts at point $A$ and ends at point $B$, then we have $\int_C \vec F\cdot d\vec r = f(B)-f(A).$
- Consider the vector field $\vec F = (2x+4xy, 2x^2-8y)$.
- Find a potential for $\vec F$.
- Find the work done by $\vec F$ on an object that starts at $(1,1)$ and moves along the parabola $y=x^2$ to the point $(-2,4)$.
- Find the work done by $\vec F$ on an object that moves counter-clockwise around a circle of radius 2 that is centered at the origin.
- Compute $\int_C Mdx+Ndy$ if $C$ is a closed curve (starts and ends at the same point).
- Compute the integral $\ds\int\frac{-x}{\sqrt{x^2+y^2+z^2}}dx$.
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