1. Answer the following questions. (A) [50%] Use the Green's theorem to evaluate the circulation of the vector field given by F(x, y) = (ey)i + (x + sin √y)j along the circle that is centred at (0,3), having radius 2, and directed positively. Support your answer with a sketch of the curve (circle) and region of integration. Remark. Here, you are asked to evaluate the total (or net) infinitesimal rotation of F over the entire plane region, the disk. Alternatively (optional and not to be graded), evaluate the above circulation by the line integral (i.e., evaluate the circulation "just" along the circle, the boundary of the disk) f F • dr = f (x² f(e* − y) dx + (x + sin √y) dy - where the parametric representation of a circle centred at (a, b) with a radius r is given by r(t) = (a+r cost)i + (b+r sint)j for 0≤t≤2π. (B) [50%] Use the divergence theorem to evaluate the outward flux of F(x, y, z) = x³i+y³j+z³k across the surface S enclosing the volume bounded by the surfaces (5 planes) y= x, y = 0, x = 4, z = 0, z = 2 Support your answer with a sketch of the surface that encloses the 3-dimensional region of integration, and a sketch of the projection of the region on the xy-plane. Remark. Here, you are asked to evaluate the total (or net) infinitesimal divergence of F over the entire 3-dimensional region that is bounded by S. Alternatively (optional and not to be graded), evaluate the above outward flux by the surface integral ff F• ds = S ffF.Nds S across the composite surface S that consists of 5 faces (or surfaces); this means that one should compute 5 surface integrals. Who would even think about them, so long as we can use the divergence theorem?
1. Answer the following questions. (A) [50%] Use the Green's theorem to evaluate the circulation of the vector field given by F(x, y) = (ey)i + (x + sin √y)j along the circle that is centred at (0,3), having radius 2, and directed positively. Support your answer with a sketch of the curve (circle) and region of integration. Remark. Here, you are asked to evaluate the total (or net) infinitesimal rotation of F over the entire plane region, the disk. Alternatively (optional and not to be graded), evaluate the above circulation by the line integral (i.e., evaluate the circulation "just" along the circle, the boundary of the disk) f F • dr = f (x² f(e* − y) dx + (x + sin √y) dy - where the parametric representation of a circle centred at (a, b) with a radius r is given by r(t) = (a+r cost)i + (b+r sint)j for 0≤t≤2π. (B) [50%] Use the divergence theorem to evaluate the outward flux of F(x, y, z) = x³i+y³j+z³k across the surface S enclosing the volume bounded by the surfaces (5 planes) y= x, y = 0, x = 4, z = 0, z = 2 Support your answer with a sketch of the surface that encloses the 3-dimensional region of integration, and a sketch of the projection of the region on the xy-plane. Remark. Here, you are asked to evaluate the total (or net) infinitesimal divergence of F over the entire 3-dimensional region that is bounded by S. Alternatively (optional and not to be graded), evaluate the above outward flux by the surface integral ff F• ds = S ffF.Nds S across the composite surface S that consists of 5 faces (or surfaces); this means that one should compute 5 surface integrals. Who would even think about them, so long as we can use the divergence theorem?
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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![1. Answer the following questions.
(A) [50%] Use the Green's theorem to evaluate the circulation of the vector field given by
F(x, y) = (ey)i + (x + sin √y)j
along the circle that is centred at (0,3), having radius 2, and directed positively. Support
your answer with a sketch of the curve (circle) and region of integration. Remark. Here,
you are asked to evaluate the total (or net) infinitesimal rotation of F over the entire plane
region, the disk.
Alternatively (optional and not to be graded), evaluate the above circulation by the
line integral (i.e., evaluate the circulation "just" along the circle, the boundary of the disk)
f F • dr = f (x²
f(e* − y) dx + (x + sin √y) dy
-
where the parametric representation of a circle centred at (a, b) with a radius r is given by
r(t) = (a+r cost)i + (b+r sint)j for 0≤t≤2π.
(B) [50%] Use the divergence theorem to evaluate the outward flux of F(x, y, z) = x³i+y³j+z³k
across the surface S enclosing the volume bounded by the surfaces (5 planes)
y= x, y = 0, x = 4, z = 0, z = 2
Support your answer with a sketch of the surface that encloses the 3-dimensional region of
integration, and a sketch of the projection of the region on the xy-plane. Remark. Here,
you are asked to evaluate the total (or net) infinitesimal divergence of F over the entire
3-dimensional region that is bounded by S.
Alternatively (optional and not to be graded), evaluate the above outward flux by
the surface integral
ff F• ds =
S
ffF.Nds
S
across the composite surface S that consists of 5 faces (or surfaces); this means that one
should compute 5 surface integrals. Who would even think about them, so long as we can
use the divergence theorem?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86794e1d-a025-469e-9776-3458200614a7%2F9629703d-f828-4518-bcfa-5e2c6a0f19d2%2Fd7lxmf6_processed.png&w=3840&q=75)
Transcribed Image Text:1. Answer the following questions.
(A) [50%] Use the Green's theorem to evaluate the circulation of the vector field given by
F(x, y) = (ey)i + (x + sin √y)j
along the circle that is centred at (0,3), having radius 2, and directed positively. Support
your answer with a sketch of the curve (circle) and region of integration. Remark. Here,
you are asked to evaluate the total (or net) infinitesimal rotation of F over the entire plane
region, the disk.
Alternatively (optional and not to be graded), evaluate the above circulation by the
line integral (i.e., evaluate the circulation "just" along the circle, the boundary of the disk)
f F • dr = f (x²
f(e* − y) dx + (x + sin √y) dy
-
where the parametric representation of a circle centred at (a, b) with a radius r is given by
r(t) = (a+r cost)i + (b+r sint)j for 0≤t≤2π.
(B) [50%] Use the divergence theorem to evaluate the outward flux of F(x, y, z) = x³i+y³j+z³k
across the surface S enclosing the volume bounded by the surfaces (5 planes)
y= x, y = 0, x = 4, z = 0, z = 2
Support your answer with a sketch of the surface that encloses the 3-dimensional region of
integration, and a sketch of the projection of the region on the xy-plane. Remark. Here,
you are asked to evaluate the total (or net) infinitesimal divergence of F over the entire
3-dimensional region that is bounded by S.
Alternatively (optional and not to be graded), evaluate the above outward flux by
the surface integral
ff F• ds =
S
ffF.Nds
S
across the composite surface S that consists of 5 faces (or surfaces); this means that one
should compute 5 surface integrals. Who would even think about them, so long as we can
use the divergence theorem?
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