Student Solutions Manual, Single Variable for Calculus: Early Transcendentals

2nd Edition

ISBN: 9780321954329

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Textbook Question

Chapter 14.4, Problem 41E

Circulation and flux For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region, Assume boundary curves have counterclockwise orientation.

41. F = ∇ ( x 2 + y 2 ) , where R is the half annulus { ( r , θ ) : 1 ≤ r ≤ 3 , 0 ≤ θ ≤ π }

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Chapter 14.4, Problem 40E

Chapter 14.4, Problem 42E

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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}.

Heat flux Suppose a solid object in ℝ3 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 to 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.

Determine the parametric equations of the flow lines for the velocity vector field, v, where v = yi+xj

Chapter 14 Solutions

Student Solutions Manual, Single Variable for Calculus: Early Transcendentals

Ch. 14.1 - How is a vector field F = f, g, h used to describe...Ch. 14.1 - Sketch the vector field F = x, y.Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Interpret the gradient field of the temperature...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...

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