 #1
Leo Liu
 345
 146
 Homework Statement:

Reference: https://ocw.mit.edu/courses/mathematics/1802scmultivariablecalculusfall2010/3.doubleintegralsandlineintegralsintheplane/partcgreenstheorem/session72simplyconnectedregionsandconservativefields/MIT18_02SC_MNotes_v5.pdf
(Last 2 pages)
 Relevant Equations:
 As I understand it, being "simply connected" means that the closed curves in the domain region contain some area(s) that are not in the domain. In other words, the region has got some hole(s) in it.
For example,
$$\left\langle
\frac x {r^3},
\frac y {r^3}
\right\rangle
= \nabla \left(
\frac 1 r
\right)$$
where ##r=\sqrt{x^2+y^2}##, is a gradient field even though it is undefined at the origion. I get that it is physically possible since it is similar to the equation of the electric field of a positive charge place at the origion, and electric field is the gradient of the gradient of the potential function. But what is the mathematical explanation here? Thanks.
$$\left\langle
\frac x {r^3},
\frac y {r^3}
\right\rangle
= \nabla \left(
\frac 1 r
\right)$$
where ##r=\sqrt{x^2+y^2}##, is a gradient field even though it is undefined at the origion. I get that it is physically possible since it is similar to the equation of the electric field of a positive charge place at the origion, and electric field is the gradient of the gradient of the potential function. But what is the mathematical explanation here? Thanks.