Chapter 9: Vector Calculus
Section 9.10: Green's Theorem
Apply the divergence-form of Green's theorem to F=x−2 y i+x2y j, and R, the region inside the bounding curves y=x2 and y=x, but outside the rectangle defined by the inequalities 1/5≤x≤1/2 and 3/10≤y≤2/5.
Figure 9.10.9(a) provides a labeled diagram of the region R and its boundaries. The region is subdivided into four subregions Rk,k=1,…,4, and the bounding curves for the inner loop, drawn in red, are labeled Ck,k=1,…,4.
In the flux integral on the right side of the divergence-form of Green's theorem, the boundary is traversed counterclockwise. A counterclockwise orientation of the outer boundary formed by y1 and y2, induces a clockwise orientation on the inner loop whose bounding curves are Ck,k=1,…,4.
A simple calculation gives ∇·F=1+x2≡Q.
Figure 9.10.9(a) The region R
The integral of ∇·F over the region R is then
∫01/5∫x2xQ ⅆy ⅆx+∫1/51/2∫2/5xQ ⅆy ⅆx+∫1/51/2∫x23/10Q ⅆy ⅆx+∫1/21∫x2xQ ⅆy ⅆx = 80881210000
The flux of F through the outer boundary consisting of y1 and y2 is then
∫012⁢−2⁢x2+x⁢x−x4ⅆx+∫10x/2−1−x5/2ⅆx = 44105
The flux of F through the inner boundary consisting of Ck,k=1,…,4, is then
∫1215−310⁢x2ⅆx+∫3102515−2⁢yⅆy+∫1512−25⁢x2ⅆx+∫2531012−2⁢yⅆy = −33910000
The net flux through the boundaries of the region R is then 44105−33910000 = 80881210000.
The divergence of F, which is 1+x2≡Q, is integrated over subregions Rk,k=1,…,4, as per Figure 9.10.9(a), in Table 9.10.9(a).
Define ∇·F as Q
Context Panel: Assign to a Name≻Q
1+x2→assign to a nameQ
Integrate ∇·F over the region R
Calculus palette: Iterated definite-integral template
Context Panel: 2-D Math≻Convert To≻Inert Form
Press the Enter key.
Context Panel: Evaluate Integral
∫01/5∫x2xQ ⅆy ⅆx+∫1/51/2∫2/5xQ ⅆy ⅆx+∫1/51/2∫x23/10Q ⅆy ⅆx+∫1/21∫x2xQ ⅆy ⅆx
Table 9.10.9(a) Integral of ∇·F over the region R
The flux of F through the two bounding curves y1 and y2 is obtained in Table 9.10.9(b). The passage around this closed contour is in the counterclockwise direction.
Tools≻Load Package: Student Vector Calculus
Tools≻Tasks≻Browse: Calculus - Vector≻
Vector Algebra and Settings≻
Display Format for Vectors
Press the Access Settings button and select
"Display as Column Vector"
Display Format for Vectors
Define the vector field F
Write the vector field as a free vector.
Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To Vector Field
Context Panel: Assign to a Name≻F
x−2 y,x2y = x−2⁢yx2⁢y→to Vector Fieldx−2⁢yx2⁢y→assign to a nameF
Obtain the flux of F through y1 and y2
Apply the Flux command and press the Enter key.
Table 9.10.9(b) Flux of F through y1 and y2
The flux of F through the bounding curves Ck,k=1,…,4, is obtained in Table 9.10.9(c). The passage around this inner loop is clockwise, consistent with the orientation of the outer loop.
Table 9.10.9(c) Flux of F through the bounding curves Ck,k=1,…,4
The total flux through the boundaries of the region R is then 44105−33910000 = 80881210000.
<< Previous Example Section 9.10
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)