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math40 coordinates

创建日期：2024-06-03 16:26:55

所属分类：数学mathematics

标签： math40

coordinates

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where r is the distance from the origin and θ is the angle measured from the x-axis,

counterclockwise.
5. Find the total mass of R.
6. Find the coordinates, (x¯, y¯), of the center of gravity.
7. Find the moments of intertia, Ix, Iy, I0.
§10.4 Green’s Theorem in the Plane
8. §10.4 #2. Evaluate ∮C ˜F · d˜r counterclockwise around the boundary C of the regionR by Green’s theorem, where
˜
F = (6y2, 2x − 2y4) and R is the square with vertices,
±(2, 2),±(2,−2).
9. §10.4 #4. Use Green’s Theorem to evaluate ∫C ˜F · d˜r if
˜
F = (x cosh 2y, 2x2 sinh 2y)
R = {(x, y) : x2 ≤ y ≤ x}.
10. §10.4 #7. Evaluate ∮C ˜F ·d˜r counterclockwise around the boundary C of the region R byGreen’s theorem, where
˜
F = ∇(x3 cos(xy)) and R = {(x, y) : 1 ≤ y ≤ 2− x2}.
11. §10.4, #15. Example 4 in the book uses Green’s theorem to derive (eqn 9 in the book)
x
R
∇2w dxdy =
∮
C
∂w
∂n
ds, (2)
where ∇2 is the Laplacian operator, and ∂w∂n = ∇w · ˜nˆ. Use (2) to evaluate
∮
C
∂w
∂n ds for
w = ex cos y + xy3, R = {(x, y) : 1 ≤ y ≤ 10− x2, x ≥ 0}.
12. §10.4 #12 (2nd part). We write Green’s theorem as
x
R
(
∂F2
∂x
− ∂F1
∂y
)
dxdy =
∮
C
(F1dx+ F2dy). (3)
Show that (3) may be written
x
R
(curl
˜
F ) ·
˜
kˆ dxdy =
∮
C ˜
F · d˜r
ds
ds,
where
d
˜
r
ds is a unit tangent vector to C .
13. Show that we can write Green’s theorem in the formx
R
div F dxdy =
∮
F · nds.
2
§10.5 Surfaces for Surface Integrals
14. §10.5, #5. Consider the paraboloid of revolution,
˜
r(u, v) = u cos viˆ+ u sin vjˆ+ u2kˆ.
(a) Find a representation of the surface of the form z = f(x, y) or g(x, y, z) = c for a
constant c.
(b) Describe the parametric curves, u = u0 (a constant) and v = v0 (a constant).
(c) Find a normal vector to the surface,
∂
˜
r
∂u × ∂˜r∂v .
(d) Make a sketch.
15. §10.5 #14. Consider the plane,
4x+ 3y + 2z = 12.
(a) Describe the plane with a parameterization in the form
˜
r = (x(u, v), y(u, v), z(u, v)) = x(u, v)ˆi+ y(u, v)ˆj+ z(u, v)kˆ.
(b) Find a unit normal vector using
i.
˜
nˆ = ∇f||∇f || ,
ii.
˜
nˆ = ˜
ru×
˜
rv
||˜ru×
˜
rv || .
(c) Sketch the surface and parameter curves.
Surface Integrals
16. §10.6 #4 (plus words). Let ρ be the concentration (in units of g/m3) of a pollutant being
carried by a fluid with a flow rate,
˜
w (in units of m/s). Find the flux of pollutant,
F =
x
S
(ρ
˜
w) ·
˜
nˆ dA,
across the surface, S, where
˜
nˆ is the normal to S,
˜
nˆdA =
˜
N(u, v)dudv,
˜
ρ
˜
w = (ey,−ez, ex),
and S = {(x, y, z) : x2 + y2 = 25, x ≥ 0, y ≥ 0, 0 ≤ z ≤ 2}.
17. §10.6 #10 (plus words). Let ρ be the concentration (in units of g/cm3) of a drug being
carried by blood with a flow rate,
˜
w (in units of cm/s). Find the flux of the drug,
F =
x
S
(ρ
˜
w) ·
˜
nˆ dA,
across the biological membrane, S, where
˜
ρ
˜
w = (y2, x2, z4), and S = {(x, y, z) : z =
4
√
x2 + y2, 0 ≤ z ≤ 8, y ≥ 0}.
18. §10.6, #15. Evaluate sS G(˜r) dA for
G = (1 + 9xz)3/2, S = {
˜
r = (u, v, u3), 0 ≤ u ≤ 1, − 2 ≤ v ≤ 2.}
19. §10.6, #23 (plus words). Find the moment of inertia of a lamina S of density µ = 2 g/cm2
about an axis B, where
S : x2 + y2 = z2, 0 ≤ z ≤ h, B : the z − axis.

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