Advanced Engineering Mathematics

Erwin Kreyszig

Chapter 10

Vector Integral Calculus. Integral Theorems - all with Video Answers

Educators

Section 4

Green's Theorem in the Plane

Problem 1

Using Green's theorem, evaluate $\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ counterclockwise around the boundary curve $C$ of the region $R$, where
$$\begin{aligned}
&\mathbf{F}=\left[\begin{array}{ll}
\frac{1}{2} x y^{4}, & \left.\frac{1}{2} x^{4} y\right], R \text { the rectangle with vertices }(0,0)
\end{array}\right.\\
&(3,0),(3,2),(0,2)
\end{aligned}$$

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Problem 2

Using Green's theorem, evaluate $\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ counterclockwise around the boundary curve $C$ of the region $R$, where
$$\begin{aligned}
&\mathbf{P}=\left[\begin{array}{lll}
y & \sin x, & 2 x \cos y
\end{array}\right], R \text { the square with vertices }\\
&\left.(0,0),\left(\frac{1}{2} \pi, 0\right), \frac{1}{2} \pi, \frac{1}{2} \pi\right),\left(0, \frac{1}{2} \pi\right)
\end{aligned}$$

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Problem 3

Using Green's theorem, evaluate $\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ counterclockwise around the boundary curve $C$ of the region $R$, where
$$\mathbf{F}=\left[-y^{3}, \quad x^{3}\right], C \text { the circle } x^{2}+y^{2}=25$$

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Problem 4

Using Green's theorem, evaluate $\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ counterclockwise around the boundary curve $C$ of the region $R$, where
$$\begin{aligned}
&\mathbf{F}=\left[\begin{array}{ll}
-e^{3}, & e^{\pi}
\end{array}\right], R \text { the triangle with vertices }(0,0),\\
&(2,0),(2,1)
\end{aligned}$$

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Problem 5

Using Green's theorem, evaluate $\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ counterclockwise around the boundary curve $C$ of the region $R$, where
$\mathbf{F}=\left[e^{x+v}, \quad e^{x-y}\right], R$ the triangle with vertices (0,0), (1. $1),(1,2)$

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Problem 6

Using Green's theorem, evaluate $\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ counterclockwise around the boundary curve $C$ of the region $R$, where
$$\mathbf{F}-\left[x \cosh y, \quad x^{2} \sinh y\right], R: x^{2} \leqq y \leqq x, \text { Sketch } R$$.

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Problem 7

Using Green's theorem, evaluate $\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ counterclockwise around the boundary curve $C$ of the region $R$, where
$$\begin{aligned}
&\mathbf{F}=\left[x^{2}+y^{2}, x^{2}-y^{2}\right], R: 1 \leq y \leq 2-x^{2} . \text { Sketch }\\
& R.
\end{aligned}$$

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Problem 8

Using Green's theorem, evaluate $\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ counterclockwise around the boundary curve $C$ of the region $R$, where
$$\begin{aligned}
&\mathbf{F}=\left[\begin{array}{ll}
a^{x} \cos y_{1} & -e^{x} \sin y
\end{array}\right], R \text { the semidisk }\\
&x^{2}+y^{2} \leq a^{2}, x \geq 0
\end{aligned}$$

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Problem 9

Using Green's theorem, evaluate $\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ counterclockwise around the boundary curve $C$ of the region $R$, where
$$\mathbf{F}-\operatorname{grad}\left(x^{3} \cos ^{2}(x y)\right), R \text { the region in } \operatorname{Prob}, 7$$

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Problem 10

Using Green's theorem, evaluate $\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ counterclockwise around the boundary curve $C$ of the region $R$, where
$$\begin{aligned}
&\mathbf{F}=\left[x \ln y, \quad y e^{x}\right], R \text { the rectangle with vertices }(0,1),\\
&(3,1),(3,2),(0,2)
\end{aligned}$$

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Problem 11

Using Green's theorem, evaluate $\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ counterclockwise around the boundary curve $C$ of the region $R$, where
$$F=[2 x-3 y, \quad x+5 y], R: 16 x^{2}+25 y^{2} \leq 400, y \geq 0$$

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Problem 12

Using Green's theorem, evaluate $\int_{c} \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ counterclockwise around the boundary curve $C$ of the region $R$, where
$\mathbf{F}=\left[x^{2} y^{2}, \quad-x / y^{3}\right], R: 1 \leq x^{2}+y^{2} \leq 4, x \geq 0,$
$y \geq x$. Sketch $R.$

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Problem 13

Using (9), evaluate $\oint_{O} \frac{\partial w}{\partial n} d s$ counterclockwise over the boundary curve $C$ of the region $R$.
$w=\sinh x, R$ the triangle with vertices (0,0),(2,0) (2,1)

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Problem 14

Using (9), evaluate $\oint_{O} \frac{\partial w}{\partial n} d s$ counterclockwise over the boundary curve $C$ of the region $R$.
$w-x^{2}+y^{2}, C ; x^{2}+y^{2}-1 .$ Confirm the answer by direct integration.

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Problem 15

Using (9), evaluate $\oint_{O} \frac{\partial w}{\partial n} d s$ counterclockwise over the boundary curve $C$ of the region $R$.
$$w=2 \ln \left(x^{2}+y^{2}\right)+x y^{3}, R \cdot t \leq y \leq 5-x^{2}, x \geq 0$$

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Problem 16

Using (9), evaluate $\oint_{O} \frac{\partial w}{\partial n} d s$ counterclockwise over the boundary curve $C$ of the region $R$.
$$w=x^{3} y+x y^{6}, R: x^{2}+y^{2} \leq 4, y \geq 0$$

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Problem 17

Apply (4) to figures of your choice whose area can also be obtained by another method and compare the results.

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Problem 18

Show that for a solution $w(x, y)$ of Iaplace's equation $\nabla^{2} w=0$ in a region $R$ with boundary curve $C$ and outer unit pormal vector $\mathbf{n},$ (10) $\iint_{\pi}\left[\left(\frac{\partial w}{\partial x}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^{2}\right] d x d y$ $=\oint_{C} w \frac{\partial w}{\partial n} d r$.

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Problem 19

Show that $w=2 e^{x} \cos y$ satisfies Laplace's equation $\nabla^{2} w=0$ and, using (10) , integrate $w(\text { \partialw } | \partial n)$ counterclockwise around the boundery curve $C$ of the square $0 \leq x \leq 2,0 \leq y \leq 2$.

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Problem 20

Let $R$ and $C$ be as in Green's theorem, $\mathbf{r}^{\prime}$ a unit tangent vector, and $\mathbf{n}$ the outer unit normal vector of $C(\text { Fig. } 238 \text { in Example } 4 \text { ). Show that }(1)$ may be written (11) $\quad \iint_{K} \operatorname{div} \mathbf{F} d \mathbf{x} d y=\oint_{C} \mathbf{F} \cdot \mathbf{n}$ $d s$
or (12) $\quad \iint_{\Omega}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{k} d x d y-\oint_{C} \mathbf{P} \cdot \mathbf{r}^{\prime} d s$ where $k$ is a unit vector perpendicular to the $x y$ -plane. Verify (11) and (12) for $\mathbf{P}=[7 x,-3 y]$ and $C$ the circle $x^{2}+y^{2}=4$ as well as for an example of your own choice.

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