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Lecture 29, 11/4/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A
Chapter 6: Vector Analysis
- Line Integrals.
- For $\vec F = (F_1,F_2), \m d\vec r = (dx, dy)$, the line integral $\m\int \vec F\cdot d\vec r = \int (F_1,F_2)\cdot (dx, dy) = \int F_1 dx + F_2 dy$.
Then find $\fcolorbox{white}{yellow}{the equation for the curve,}$ the upper and lower limit, and compute.
The problem can be set up in 3D: $\mm\vec F = (F_1,F_2,F_3), \m d\vec r = (dx, dy, dz)$, $\m\int \vec F\cdot d\vec r = \int (F_1,F_2, F_3)\cdot (dx, dy, dz) = \int F_1 dx + F_2 dy+F_3 dz$.
- Conserved fields: $\fcolorbox{white}{yellow}{Conserved Fields:}$ the value of the line integral is independent of the path.
If $\vec F = \nabla W$, then $\vec F$ is a conserved field, and $\phi = - W$ is called the $\fcolorbox{white}{yellow}{potential.}$
$\vec F$ is a conserved field if and only if
curl$\vec F = 0$.
- To find $\phi$ in a conserved field:
(a) Verify $\mm$ curl$\vec F = 0.\m$ (b) Integrate on an easy path from $\vec 0$ to $(x,y,z)$, to obtain $W = \int\vec F\cdot d\vec r .\m$ (c) $\phi = - W$.
- If $ \vec F\cdot d\vec r = d W$, then $dW$ is the $\fcolorbox{white}{yellow}{exact differential}$ for $ \vec F\cdot d\vec r $.
- Green's Formula.
- Green's Formula (in a plane): $\m\intd{A} \Big(\frac{\p Q}{\p x} -\frac{\p P}{\p y} \Big) dxdy = \oint_{\p A} (P dx + Q dy)$.