Suppose that c is the line segment from (a, b) to (c,d). (a) Show that [rd |x dy – y dx = ad – bc Briefly describe what this integral represents geometrically. Hint: Think back to linear algebra. What do determinants measure? (b) Suppose that a polygon P has vertices {(₁, Y₁)}_₁ in counter-clockwise order. Use Green's theorem to show that P has area: 1 A = [(x1Y2 −X2Y1) + (X2Y3 − X3Y2) + ... + (Xn-1Yn — XnYn−1) + (XnY1 − X1Yn)]
Suppose that c is the line segment from (a, b) to (c,d). (a) Show that [rd |x dy – y dx = ad – bc Briefly describe what this integral represents geometrically. Hint: Think back to linear algebra. What do determinants measure? (b) Suppose that a polygon P has vertices {(₁, Y₁)}_₁ in counter-clockwise order. Use Green's theorem to show that P has area: 1 A = [(x1Y2 −X2Y1) + (X2Y3 − X3Y2) + ... + (Xn-1Yn — XnYn−1) + (XnY1 − X1Yn)]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Suppose that c is the line segment from (a, b) to (c,d).
(a) Show that
Briefly describe what this integral represents geometrically.
Hint: Think back to linear algebra. What do determinants measure?
(b) Suppose that a polygon P has vertices {(₁, y₁)} in counter-clockwise order.
Use Green's theorem to show that P has area:
A
[zdy - ydr = ad-bc
=
[(X1Y2 − X2Y1) + (X2Y3 — T3Y2) + · ·· + (Xn-1Yn — TnYn-1) + (FnY1 − X1Yn)]
-
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12f51b64-c3b4-40b6-b565-16b387e10bf1%2Fb9734d18-0521-46b7-93e8-e3ccaa8d42c5%2Ftr7n8gp_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose that c is the line segment from (a, b) to (c,d).
(a) Show that
Briefly describe what this integral represents geometrically.
Hint: Think back to linear algebra. What do determinants measure?
(b) Suppose that a polygon P has vertices {(₁, y₁)} in counter-clockwise order.
Use Green's theorem to show that P has area:
A
[zdy - ydr = ad-bc
=
[(X1Y2 − X2Y1) + (X2Y3 — T3Y2) + · ·· + (Xn-1Yn — TnYn-1) + (FnY1 − X1Yn)]
-
-
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