p0149 - 56 CHAPTER 1. INTRODUCTION 1.49 Chapter 1, Problem...

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56 CHAPTER 1. INTRODUCTION 1.49 Chapter 1, Problem 49 Problem: Using the relations between coordinates and unit vectors in Cartesian and cylindri- cal coordinates, compute the velocity components in Cartesian coordinates for a general vector V = V r e r + V θ e θ + V z k . Use your results to transform V = r e θ . Solution: The relation between unit vectors in Cartesian and cylindrical coordinates is e r = i cos θ + j sin θ , e θ = i sin θ + j cos θ , k = k Hence, the vector V is V = V r ( i cos θ + j sin θ )+ V θ ( i sin θ + j cos θ V z k =( V r cos θ V θ sin θ ) i +( V r sin θ + V θ cos θ ) j + V z k = V x i + V y j + V z k Therefore, the velocity components transform according to
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