Calculus
7th Edition
ISBN: 9781524916817
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 11.6, Problem 44PS
To determine
To calculate: the direction for the most rapid increase in potential.
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Coulomb's Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles with charge 1 located at positions 0 and 2 on a coordinate line and a particle with charge −1 at a position x between them. It follows from Coulomb's Law that the net force acting on the middle particle is
Heat transfer Fourier’s Law of heat transfer (or heat conduction) 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 from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/(m-s-K). A temperature function for a region D is given. Find the net outward heat flux ∫∫S F ⋅ n dS = -k∫∫S ∇T ⋅ n dS across the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume k = 1.
T(x, y, z) = 100e-x2 - y2 - z2; D is the sphere of radius a centered at the origin.
Heat transfer Fourier’s Law of heat transfer (or heat conduction) 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 from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/(m-s-K). A temperature function for a region D is given. Find the net outward heat flux ∫∫S F ⋅ n dS = -k∫∫S ∇T ⋅ n dS across the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume k = 1.
T(x, y, z) = 100 + x2 + y2 + z2;;D = {(x, y, z): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1}
Chapter 11 Solutions
Calculus
Ch. 11.1 - Prob. 1PSCh. 11.1 - Prob. 2PSCh. 11.1 - Prob. 3PSCh. 11.1 - Prob. 4PSCh. 11.1 - Prob. 5PSCh. 11.1 - Prob. 6PSCh. 11.1 - Prob. 7PSCh. 11.1 - Prob. 8PSCh. 11.1 - Prob. 9PSCh. 11.1 - Prob. 10PS
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- Heat transfer Fourier’s Law of heat transfer (or heat conduction) 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 from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/(m-s-K). A temperature function for a region D is given. Find the net outward heat flux ∫∫S F ⋅ n dS = -k∫∫S ∇T ⋅ n dS across the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume k = 1. T(x, y, z) = 100 + x + 2y + z;D = {(x, y, z): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1}arrow_forwardHeat transfer Fourier’s Law of heat transfer (or heat conduction) 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 from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/(m-s-K). A temperature function for a region D is given. Find the net outward heat flux ∫∫S F ⋅ n dS = -k∫∫S ∇T ⋅ n dS across the boundary S of D. In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume k = 1. T(x, y, z) = 100 + e-z;D = {(x, y, z): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1}arrow_forwardIntegration of this questionarrow_forward
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