§16.3 Path Independence, Conservative Fields, and Potential Functions
Lecture notes
Conservative Fields and Potential Functions
“Fundamental theorem of calculus”
Component Test
Exact differential forms
Examples
-
Show that \[ \F = ye^{xy}\i + (xe^{xy}+\cos(z))\j + (1-y\sin(z))\k \] is conservative, and find a potential function for it.
-
Show that \[ \F = 2xz\i - z\sin(yz)\j + (x^2-y\sin(yz))\k \] is conservative, and find a potential function for it.
-
Show that \[ \omega = 2xyz\,dx + x^2z\,dy + x^2y\,dz \] is exact, and evaluate \[ \int_{(2,-1,1)}^{(3,1,1)} 2xyz\,dx + x^2z\,dy + x^2y\,dz. \]