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p0172

p0172 - 1.72 CHAPTER 1 PROBLEM 72 79 1.72 Chapter 1 Problem...

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1.72. CHAPTER 1, PROBLEM 72 79 1.72 Chapter 1, Problem 72 Problem: Using the relations between coordinates and unit vectors in Cartesian and cylindri- cal coordinates, compute the velocity components in cylindrical coordinates for a general vector V = V x i + V y j + V z k . Use your results to transform V = V j . Solution: The relation between unit vectors in Cartesian and cylindrical coordinates is i = e r cos θ e θ sin θ , j = e r sin θ + e θ cos θ , k = k Hence, the vector V is V = V x ( e r cos θ e θ sin θ ) + V y ( e r sin θ + e θ cos θ ) + V z k = ( V x cos θ + V y sin θ ) e r + ( V x sin θ + V y cos θ ) e θ + V z k = V r e r + V θ e θ + V z k Therefore, the velocity components transform according to
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