For the following exercises, the heat flow vector field for conducting objects F = − k ∇ T , where T ( x , y , z ) is the temperature in the object and k > 0 is a constant that depends on the material. Find the outward flux of F across the following surfaces S for the given temperature distributions and assume k = 1 . 323. T ( x , y , z ) = In ( x 2 + y 2 + z 2 ) ; S is sphere x 2 + y 2 + z 2 = a 2 .
For the following exercises, the heat flow vector field for conducting objects F = − k ∇ T , where T ( x , y , z ) is the temperature in the object and k > 0 is a constant that depends on the material. Find the outward flux of F across the following surfaces S for the given temperature distributions and assume k = 1 . 323. T ( x , y , z ) = In ( x 2 + y 2 + z 2 ) ; S is sphere x 2 + y 2 + z 2 = a 2 .
For the following exercises, the heat flow vector field for conducting objects
F
=
−
k
∇
T
, where
T
(
x
,
y
,
z
)
is the temperature in the object and
k
>
0
is a constant that depends on the material. Find the outward flux of
F
across the following surfaces S for the given temperature distributions and assume
k
=
1
.
323.
T
(
x
,
y
,
z
)
=
In
(
x
2
+
y
2
+
z
2
)
;
S
is sphere
x
2
+
y
2
+
z
2
=
a
2
.
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.
Finite Mathematics & Its Applications (12th Edition)
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