Compound surface and boundary Begin with the paraboloid z = x 2 + y 2 , for 0 ≤ z ≤ 4, and slice it with the plane y = 0. Let S be the surface that remains for y ≥ 0 (including the planar surface in the xz –plane) (see figure). Let C be the semicircle and line segment that bound the cap of . S in the plane z = 4 with counterclockwise orientation. Let F = 〈2 z + y, 2 x + z, 2 y + x 〉. a . Describe the direction of the vectors normal to the surface that are consistent with the orientation of C . b. Evaluate ∬ S ( ∇ × F ) ⋅ n d S c. Evaluate ∮ C F ⋅ d r and check for agreement with part (b).
Compound surface and boundary Begin with the paraboloid z = x 2 + y 2 , for 0 ≤ z ≤ 4, and slice it with the plane y = 0. Let S be the surface that remains for y ≥ 0 (including the planar surface in the xz –plane) (see figure). Let C be the semicircle and line segment that bound the cap of . S in the plane z = 4 with counterclockwise orientation. Let F = 〈2 z + y, 2 x + z, 2 y + x 〉. a . Describe the direction of the vectors normal to the surface that are consistent with the orientation of C . b. Evaluate ∬ S ( ∇ × F ) ⋅ n d S c. Evaluate ∮ C F ⋅ d r and check for agreement with part (b).
Solution Summary: The author describes the direction of the vectors normal to the surface that are consistent with the orientation of C.
Compound surface and boundary Begin with the paraboloid z = x2 + y2, for 0 ≤ z ≤ 4, and slice it with the plane y = 0. Let S be the surface that remains for y ≥ 0 (including the planar surface in the xz–plane) (see figure). Let C be the semicircle and line segment that bound the cap of .S in the plane z = 4 with counterclockwise orientation. Let F = 〈2z + y, 2x + z, 2y + x〉.
a. Describe the direction of the vectors normal to the surface that are consistent with the orientation of C.
b. Evaluate
∬
S
(
∇
×
F
)
⋅
n
d
S
c. Evaluate
∮
C
F
⋅
d
r
and check for agreement with part (b).
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Identify and sketch the quadric
3
3
surface z
x² -y² = 0
The surface that is linear with the three variables, x, y, and z is called
cylinder
plane
sphere
hyperboloid
The four figures are graphs of the quadratic surface z = x? + y2. Match each of the four graphs with the point in space from which the paraboloid is viewed.
The four points are (0, 0, 40), (0, 40, 0), (40, 0, 0), and (20, 20, 40).
(a)
O (0, 0, 40)
O (0, 40, 0)
O (40, 0, 0)
lo (20, 20,
40)
(b)
O (0, 0, 40)
O (0, 40, 0)
O (40, 0, 0)
lo (20, 20,
40)
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