For the following exercises, Fourier’s law of heat transfer states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = − k ∇ T , which means that heat energy flows hot regions to cold regions. The constant k > 0 is called the conductivity , which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region D is given. Use the divergence theorem to find net outward heat flux ∬ s F ⋅ N d S = − k ∬ s ∇ T ⋅ N d S across the boundary S of D where k = 1 . 426. T ( x , y , z ) = 100 + e − x 2 − y 2 − z 2 ; D is the sphere of radius a centered at the origin.
For the following exercises, Fourier’s law of heat transfer states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = − k ∇ T , which means that heat energy flows hot regions to cold regions. The constant k > 0 is called the conductivity , which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region D is given. Use the divergence theorem to find net outward heat flux ∬ s F ⋅ N d S = − k ∬ s ∇ T ⋅ N d S across the boundary S of D where k = 1 . 426. T ( x , y , z ) = 100 + e − x 2 − y 2 − z 2 ; D is the sphere of radius a centered at the origin.
For the following exercises, Fourier’s law of heat transfer states that the heat flow vector
F
at a point is proportional to the negative gradient of the temperature; that is,
F
=
−
k
∇
T
, which means that heat energy flows hot regions to cold regions. The constant
k
>
0
is called the conductivity, which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region
D
is given. Use the divergence theorem to find net outward heat flux
∬
s
F
⋅
N
d
S
=
−
k
∬
s
∇
T
⋅
N
d
S
across the boundary
S
of
D
where
k
=
1
.
426.
T
(
x
,
y
,
z
)
=
100
+
e
−
x
2
−
y
2
−
z
2
;
D
is the sphere of radius
a
centered at the origin.
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.
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
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