Evaluating a Flux Integral In Exercises 25-30, find the flux of F across S , ∬ S F ⋅ N d S where N is the upward unit normal vector to S . F ( x , y , z ) = 4 i - 3 j + 5 k S : z = x 2 + y 2 , x 2 + y 2 ≤ 4
Solution Summary: The author calculates the flux of F(x,y,z)=4i-3j+5k across the surface.
Evaluating a Flux Integral In Exercises 25-30, find the flux of F across S,
∬
S
F
⋅
N
d
S
where N is the upward unit normal vector to S.
F
(
x
,
y
,
z
)
=
4
i
-
3
j
+
5
k
S
:
z
=
x
2
+
y
2
,
x
2
+
y
2
≤
4
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.
a. Show that the outward flux of the position vector field F = x i + y j + z k through a smooth closed surface S is three times the volume of the region enclosed by the surface.
b. Let n be the outward unit normal vector field on S. Show that it is not possible for F to be orthogonal to n at every point of S
Using Green's Theorem, find the outward flux of F across the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (4, 0), and (0, 2)
Flux of a vector field?
Let S be a closed surface consisting of a paraboloid (z = x²+y²), with (0≤z≤1), and capped by the disc (x²+y² ≤1) on the plane (z=1). Determine the flow of the vector field F (x,y,z) = zj − yk, in the direction that points out across the surface S.
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