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Vector Field: F(x,y) = (1, 0.5 y) and Closed Path: x + y? = 16 Since we do not have our path as a vector, we can use Green's Theorem to find the Circulation.

Vector Field: F(x,y) = (1, 0.5 y) and Closed Path: x + y? = 16 Since we do not have our path as a vector, we can use Green's Theorem to find the Circulation.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.3: Vectors
Problem 60E
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For Questions 4, 5, and 6 we are given a vector field and a closed path.

 

Vector Field: F(x.y) = (1, 0.5 y) and Closed Path: x2 + y? = 16 Since we do not have our path as a vector, we can use Green's Theorem to find the Circulation. of dA dy Circulation = The g" refers to the "y component" of the vector field, while the "f" refers to the "x component" of the vector field. Notice how the path is not present at all inside the integral. The double integral, itself, will handle the path that our object is taking. Given the vector field and closed path below Vector Field: F(x,y) = (1, 0,5 y) and Closed Path: x? + y? = 16 Find the "curl of the vector field. of = 0.5 y ду дх B of = 1.5 ду of = 0 dy of = 16 dy Given the vector field and closed path below Vector Field: F(x,y) = (1, 0.5 y) and Closed Path: x + y? = 16 Set up the double integral for our path. og af dx dy ду og of rdr de ду © S T 2x 16 of rdr de dy og af dx dy Now that you have found the curl of your vector field and you have set up your double integral, we can use Green's Theorem to find the circulation for this situation. of dA Circulation = ду dg dA = 0 ду of dA =24x B ду dg of dA = 512x dy af dA = 16 ду
Transcribed Image Text:Vector Field: F(x.y) = (1, 0.5 y) and Closed Path: x2 + y? = 16 Since we do not have our path as a vector, we can use Green's Theorem to find the Circulation. of dA dy Circulation = The g" refers to the "y component" of the vector field, while the "f" refers to the "x component" of the vector field. Notice how the path is not present at all inside the integral. The double integral, itself, will handle the path that our object is taking. Given the vector field and closed path below Vector Field: F(x,y) = (1, 0,5 y) and Closed Path: x? + y? = 16 Find the "curl of the vector field. of = 0.5 y ду дх B of = 1.5 ду of = 0 dy of = 16 dy Given the vector field and closed path below Vector Field: F(x,y) = (1, 0.5 y) and Closed Path: x + y? = 16 Set up the double integral for our path. og af dx dy ду og of rdr de ду © S T 2x 16 of rdr de dy og af dx dy Now that you have found the curl of your vector field and you have set up your double integral, we can use Green's Theorem to find the circulation for this situation. of dA Circulation = ду dg dA = 0 ду of dA =24x B ду dg of dA = 512x dy af dA = 16 ду
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