describe simple and closed curves; define connected and simply connected regions
explain how to find a potential function for a conservative vector field
use the Fundamental Theorem for Line Integrals to evaluate a line integral in a vector field
explain how to test a vector field to determine whether it is conservative
additional notes
another fundamental theorem of calculus: if \(f\) is a real-valued
multivariable function with continuous first-order partial on a piecewise
smoooth curve \(C:\,\vec{r}(t),\,a\le t \le b\), then
\(\int_{C}\nabla f \cdot d\vec{r} =f(\vec{r}(b))-f(\vec{r}(a))\)
if \(\vec{F}\) is a vector field defined on an open, simply connected \(D\)
region, then the \(\mbox{curl}(\vec{F})\) will determine whether or not
\(\vec{F}\) is conservative on \(D\)
\(\vec{F}=\langle f,g \rangle\) is conservative if
\( \displaystyle{\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y}=0}\)
on \(D\)
\(\vec{F}=\langle f,g,h \rangle\) is conservative if
\( \nabla\times\vec{F} = \vec{0}\) on \(D\)
creation of potential functions
16.4 Green's Theorem
recommended exercises
1, 3, 5, 7, 9, 13, 17, 18
learning objectives
apply the circulation form of Green’s theorem
apply the flux form of Green’s theorem
calculate circulation and flux on more general regions
All homework is due at the beginning of class on the date indicated.
All exams are given in class on the date indicated.
Late homework is not accepted and there are no make-ups for missed exams.
Contact me before the deadline if you have a conflict with a
deadline and we will find a mutually agreeable solution.
Solutions are posted under these links after the deadline.