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1. The vector field F(x,y) = y i+ + y? j x² + x² + y? 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 _ 3Q a. Show that the vector field satisfies the condition ду b. Let r(t) = (cos(t), sin(t)), 0

1. The vector field F(x,y) = y i+ + y? j x² + x² + y? 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 _ 3Q a. Show that the vector field satisfies the condition ду b. Let r(t) = (cos(t), sin(t)), 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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13.3 - Fundamental Theorem for Line Integrals 1. The vector field F(x, y) : -y i + x² + y² j 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 a. Show that the vector field satisfies the condition ay ax b. Let r(t) = (cos(t), sin(t)), 0 <ts 2 1. Notice that the graph of r(t) is a closed curve C, since r(0) = r(2 ) = (1,0). Nevertheless, show that S. F • dr is NOT zero. (Note: In part (b), you are proving that F has no potential function. Since F has no potential function, you need to calculate the line integral using the original formula.)
Transcribed Image Text:13.3 - Fundamental Theorem for Line Integrals 1. The vector field F(x, y) : -y i + x² + y² j 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 a. Show that the vector field satisfies the condition ay ax b. Let r(t) = (cos(t), sin(t)), 0 <ts 2 1. Notice that the graph of r(t) is a closed curve C, since r(0) = r(2 ) = (1,0). Nevertheless, show that S. F • dr is NOT zero. (Note: In part (b), you are proving that F has no potential function. Since F has no potential function, you need to calculate the line integral using the original formula.)
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