Singular radial field Consider the radial field F = r | r | = 〈 x , y , z 〉 ( x 2 + y 2 + z 2 ) 1 / 2 . a. Evaluate a surface integral to show that ∬ S F ⋅ n d S = 4 π a 2 , where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a) is finite. Evaluate the triple integral of the Divergence Theorem as an improper integral as follows. Integrate div F over the region between two spheres of radius a and 0 < g < a. Then let g →0 + to obtain the flux computed in part (a).
Singular radial field Consider the radial field F = r | r | = 〈 x , y , z 〉 ( x 2 + y 2 + z 2 ) 1 / 2 . a. Evaluate a surface integral to show that ∬ S F ⋅ n d S = 4 π a 2 , where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a) is finite. Evaluate the triple integral of the Divergence Theorem as an improper integral as follows. Integrate div F over the region between two spheres of radius a and 0 < g < a. Then let g →0 + to obtain the flux computed in part (a).
Solution Summary: The author explains that the radial field of the Divergence theorem is given below.
F
=
r
|
r
|
=
〈
x
,
y
,
z
〉
(
x
2
+
y
2
+
z
2
)
1
/
2
.
a. Evaluate a surface integral to show that
∬
S
F
⋅
n
d
S
=
4
π
a
2
, where S is the surface of a sphere of radius a centered at the origin.
b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a) is finite. Evaluate the triple integral of the Divergence Theorem as an improper integral as follows. Integrate div F over the region between two spheres of radius a and 0 < g < a. Then let g →0+ to obtain the flux computed in part (a).
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.
Use Green's Theorem to evaluate the integral. Assume that the curve C is oriented counterclockwise.
3 In(3 + y) dx -
-dy, where C is the triangle with vertices (0,0), (6, 0), and (0, 12)
ху
3+y
ху
dy =
3 In(3 + y) dx -
3+ y
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