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1. The vector field F(x, y) = -24 yi i + -y x² + x² + y2 Is an example of a vector field that does not satisfy the conditions of the "fine print" in today's theorem. That is because the domain of this vector field has a hole at the point (0,0), but the theorem requires the domain to have no holes. a. Show that the vector field satisfies the condition b. Let r(t) = (cos(t),sin(t)), 0 < t < 2 n. Notice that the graph of r(t) is a closed curve C, since r(0) = r(2 7) = (1,0). Nevertheless, show that SF• dr is NOT zero.

1. The vector field F(x, y) = -24 yi i + -y x² + x² + y2 Is an example of a vector field that does not satisfy the conditions of the "fine print" in today's theorem. That is because the domain of this vector field has a hole at the point (0,0), but the theorem requires the domain to have no holes. a. Show that the vector field satisfies the condition b. Let r(t) = (cos(t),sin(t)), 0 < t < 2 n. Notice that the graph of r(t) is a closed curve C, since r(0) = r(2 7) = (1,0). Nevertheless, show that SF• dr is NOT zero.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. The vector field F(x, y) : i + x² + y2 Is an example of a vector field that does not satisfy the conditions of the "fine print" in today's theorem. That is because the domain of this vector field has a hole at the point (0,0), but the theorem requires the domain to have no holes. ap aq a. Show that the vector field satisfies the condition ду дх b. Let r(t) = (cos(t), sin(t)), 0 < t < 2 n. Notice that the graph of r(t) is a closed curve C, since r(0) = r(2 T) = (1,0). Nevertheless, show that f. F• dr is NOT zero.
Transcribed Image Text:1. The vector field F(x, y) : i + x² + y2 Is an example of a vector field that does not satisfy the conditions of the "fine print" in today's theorem. That is because the domain of this vector field has a hole at the point (0,0), but the theorem requires the domain to have no holes. ap aq a. Show that the vector field satisfies the condition ду дх b. Let r(t) = (cos(t), sin(t)), 0 < t < 2 n. Notice that the graph of r(t) is a closed curve C, since r(0) = r(2 T) = (1,0). Nevertheless, show that f. F• dr is NOT zero.
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