Channel flow The flow in a long shallow channel is modeled by the velocity field F = (0, 1 – x 2 ), where R = {( x , y ): | x | ≤ 1 and |y| = 5}. a. Sketch R and several streamlines of F . b. Evaluate the curl of F on the lines x = 0, x = 1 4 , x = 1 2 , and x = 1. c. Compute the circulation on the boundary of R. d. How do you explain the fact that the curl of F is nonzero at points of R, but the circulation is zero?
Channel flow The flow in a long shallow channel is modeled by the velocity field F = (0, 1 – x 2 ), where R = {( x , y ): | x | ≤ 1 and |y| = 5}. a. Sketch R and several streamlines of F . b. Evaluate the curl of F on the lines x = 0, x = 1 4 , x = 1 2 , and x = 1. c. Compute the circulation on the boundary of R. d. How do you explain the fact that the curl of F is nonzero at points of R, but the circulation is zero?
Solution Summary: The author illustrates the region R and several streamlines of the velocity field F=langle 0,1-x2rangle .
Heat flux in a plate A square plate R = {(x, y): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} has a temperature distribution T(x, y) = 100 - 50x - 25y.a. Sketch two level curves of the temperature in the plate.b. Find the gradient of the temperature ∇T(x, y).c. Assume the flow of heat is given by the vector field F = -∇T(x, y). Compute F.d. Find the outward heat flux across the boundary {(x, y): x = 1, 0 ≤ y ≤ 1}.e. Find the outward heat flux across the boundary {(x, y): 0 ≤ x ≤ 1, y = 1}.
1.
Let R and b be positive constants. The vector function
r(t) = (R cost, R sint, bt)
traces out a helix that goes up and down the z-axis.
a) Find the arclength function s(t) that gives the length of the helix from t = 0 to any other t.
b) Reparametrize the helix so that it has a derivative whose magnitude is always equal to 1.
c) Set R = b = 1.
Compute T, Ñ, and B for the helix at the point (√2/2,√2/2, π/4).
3. Let f(x, y) = sin x + sin y. (NOTE: You may use software for any part
of this problem.)
(a) Plot a contour map of f.
(b) Find the gradient Vf.
(c) Plot the gradient vector field Vf.
(d) Explain how the contour map and the gradient vector field are
related.
(e) Plot the flow lines of Vf.
(f) Explain how the flow lines and the vector field are related.
(g) Explain how the flow lines of Vf and the contour map are related.
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