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1. Let F(r,0,0)= r sin @cos o r sin sin rcos@ for r 20,9 € (0,) and € (0, 2). Compute det Jp (r, 8, 6). For which values of the different variables is this Jacobian matrix singular?

1. Let F(r,0,0)= r sin @cos o r sin sin rcos@ for r 20,9 € (0,) and € (0, 2). Compute det Jp (r, 8, 6). For which values of the different variables is this Jacobian matrix singular?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Let F(r,0,0) = r sin r sin sin r cos cos o for r 2 0.0 € [0, π) and € (0, 2). Compute det Jp (r, 0, 0). For which values of the different variables is this Jacobian matrix singular? 2. The classic shape of a grain silo can be approximated as a cylinder with a hemisphere of equal radius at its top. (See this linked photo for an example.) Assume that a silo can be filled brim-full, and that it is constructed of sheet metal comprising a circular base, the cylindrical side and the hemisphere. Suppose that we have enough sheet metal to construct a silo of surface area A. Use the method of Lagrange multipliers to identify the maximum volume V that our silo can hold in terms of A, and the shape of the resulting silo. Without differentiating, can you explain why this solution must be a maximum and not a minimum?
Transcribed Image Text:1. Let F(r,0,0) = r sin r sin sin r cos cos o for r 2 0.0 € [0, π) and € (0, 2). Compute det Jp (r, 0, 0). For which values of the different variables is this Jacobian matrix singular? 2. The classic shape of a grain silo can be approximated as a cylinder with a hemisphere of equal radius at its top. (See this linked photo for an example.) Assume that a silo can be filled brim-full, and that it is constructed of sheet metal comprising a circular base, the cylindrical side and the hemisphere. Suppose that we have enough sheet metal to construct a silo of surface area A. Use the method of Lagrange multipliers to identify the maximum volume V that our silo can hold in terms of A, and the shape of the resulting silo. Without differentiating, can you explain why this solution must be a maximum and not a minimum?
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