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. 56. 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. 56. T ( x , y , z ) = 100 e − x 2 + y 2 + z 2
Solution Summary: The author calculates the heat flow vector field and its divergence, based on a solid object in R3.
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.
56.
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.
8d6 عدد انباء
Q/ Design a rectangular foo
A
ing of B-2.75m to support a column of
dimensions (0.46 x 0.46) m, dead load =1300kN, live load = 1300kN,
qa-210kPa, fc' 21 MPa, fy- 400 MPa.
=
Q1/ Two plate load tests were conducted in a C-0 soil as given belo
Determine the required size of a footing to carry a load of 1250 kN for the
same settlement of 30 mm.
Size of plates (m) Load (KN) Settlement (mm)
0.3 x 0.3
40
30
0.6 x 0.6
100
30
Qx 0.6z
The OU process studied in the previous problem is a common model for interest rates.
Another common model is the CIR model, which solves the SDE:
dX₁ = (a = X₁) dt + σ √X+dWt,
-
under the condition Xoxo. We cannot solve this SDE explicitly.
=
(a) Use the Brownian trajectory simulated in part (a) of Problem 1, and the Euler
scheme to simulate a trajectory of the CIR process. On a graph, represent both the
trajectory of the OU process and the trajectory of the CIR process for the same
Brownian path.
(b) Repeat the simulation of the CIR process above M times (M large), for a large
value of T, and use the result to estimate the long-term expectation and variance
of the CIR process. How do they compare to the ones of the OU process?
Numerical application: T = 10, N = 500, a = 0.04, x0 = 0.05, σ = 0.01, M = 1000.
1
(c) If you use larger values than above for the parameters, such as the ones in Problem
1, you may encounter errors when implementing the Euler scheme for CIR. Explain
why.
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