Heat flux Suppose a solid object in ¡ 3 has a temperature distribution given by T ( x, y, z ). The heat flow vector field in the object is F = –k ▿ T, where the conductivity k > 0 is a property of the material. Note that the heat flow vector points in the direction opposite that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is ▿· F = – k ▿·▿ T = –k ▿ 2 T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions. 57. T ( x , y , z ) = 100 e − x 2 + y 2 + z 2
Heat flux Suppose a solid object in ¡ 3 has a temperature distribution given by T ( x, y, z ). The heat flow vector field in the object is F = –k ▿ T, where the conductivity k > 0 is a property of the material. Note that the heat flow vector points in the direction opposite that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is ▿· F = – k ▿·▿ T = –k ▿ 2 T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions. 57. T ( x , y , z ) = 100 e − x 2 + y 2 + z 2
Solution Summary: The author explains the heat flow vector field and its divergence. If nablacdot F=0, the vector is source free.
Heat fluxSuppose a solid object in ¡3has a temperature distribution given by T(x, y, z). The heat flow vector field in the object isF= –k▿T, where the conductivity k > 0 is a property of the material. Note that the heat flow vector points in the direction opposite that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is ▿·F = –k ▿·▿T = –k▿2T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions.
57.
T
(
x
,
y
,
z
)
=
100
e
−
x
2
+
y
2
+
z
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.
The temperature on a cubic box [0, 4] × [0, 4] × [0, 4] (measured in meters) can be describedby the function T (x, y, z) = x2y + y2z degrees F◦. A fly is in position (1, 2, 1) and takesoff in a straight line to the corner (4, 0, 4). Use directional derivatives to calculate the changein temperature the fly experiences as she takes off. Give your answer with 2 decimal digitscorrect.
The gradient of V = x²sin(y)cos(z) at the point (1, ,0) is
إختر أحد الخيارات:
2i +ja O
7+ 25.b O
2ic O
j.d O
Determine the directional derivative of f(x, y, z)
= tan-1
(2²)
U = (-1,2,2) at P(0, 1, 1). Determine also the maximum and minimum directional
derivative of f along the unit vector.
where
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