11.5: Green's Theorem
- Page ID
- 144361
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
- Use Green's Theorem to evaluate \(\displaystyle \oint_{\partial R} 2y\ dx + 3x\ dy\), where \(R\) is the unit square \([0, 1] \times [0, 1]\) and \(\partial R\) is oriented counterclockwise.
- Use Green's Theorem to evaluate \(\displaystyle \oint_{\partial R} 2xy\ dx + (x + y)\ dy\), where \(R\) is the region in the first quadrant bounded by \(y = x\) and \(y = x^3\) and \(\partial R\) is oriented counterclockwise.
- Use Green's Theorem to evaluate \(\displaystyle \oint_{\partial R} (x^2y, xy^2) \cdot d\mathbf{r}\), where \(R\) is the region described by \(0 \leq x \leq 1\), \(0 \leq y \leq x\), and where \(\partial R\) is oriented counterclockwise.
- Use Green's Theorem to evaluate \(\displaystyle \oint_{\partial R} \sin x\cos y\ dx + (xy + \cos x \sin y)\ dy\), where \(R\) is the region lying between the graphs of \(y = x\) and \(y = \sqrt{x}\), and where \(\partial R\) is oriented clockwise.
- Use Green's Theorem to evaluate \(\displaystyle \oint_{\partial R} (-y, x) \cdot d\mathbf{r}\), where \(R\) is the top-half of the circle of radius \(r = 1\) centered at the origin, and where \(\partial R\) is oriented counterclockwise.