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Lecture 28, 11/2/2022. 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 $\fcolorbox{white}{yellow}{potential.}$