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[Analysis] Add general notes

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Janis Hutz
2026-02-01 09:59:14 +01:00
parent 496475f66e
commit 4eedbb96a8
4 changed files with 8 additions and 4 deletions
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semester3/analysis-ii/cheat-sheet-jh/parts/vectors/integration/04_green_formula.tex
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@@ -11,7 +11,7 @@ with $\gamma_i = (\gamma_{i, 1}, \gamma_{i, 2}) : [a_i, b_i] \rightarrow \R^2$ a
$f = (f_1, f_2)$ is a vector field of class $C^1$ on open set containing $X$. Then:
\drmvspace
\begin{align*}
\int_{X} \left( \frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial_y} \right) \dx x \dx y = \sum_{i = 1}^{k} \int_{\gamma_i} f \cdot \dx \vec{s}
\int_{X} \left( \frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y} \right) \dx x \dx y = \sum_{i = 1}^{k} \int_{\gamma_i} f \cdot \dx \vec{s}
\end{align*}
\stepLabelNumber{all}\dhrmvspace
@@ -23,7 +23,7 @@ $\gamma_i$ as above, then
\end{align*}
\drmvspace
\shade{gray}{Understanding and applying Green's Formula} The $\frac{\partial f_2}{\partial x} - \frac{\partial f_1}{y} = \text{curl}(f)$, i.e. it is the 2D-curl of $f$.
\shade{gray}{Understanding and applying Green's Formula} The $\frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y} = \text{curl}(f)$, i.e. it is the 2D-curl of $f$.
Thus, the sum of all line integrals is the same thing as the Riemann-Integral of the curl.
We can use Green's Formula to compute integrals. For that we need the set of curves that define the set.