hw1b - CBE 320 Introductory Transport Phenomena September 2...

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CBE 320 September 2, 2011 Introductory Transport Phenomena Problem Session I Part B: Review of Mathematical Topics 1. Sketch the following functions; be sure to label the axes carefully. a. y ( x ) = tanh bx tanh b , -∞ < x < , b = constant b. y ( x ) = cxe - bx , -∞ < x < , b, c = constant c. y ( x ) = sinh x x , -∞ < x < d. v z ( y ) = v 0 y h - y h 2 , 0 y h, v 0 , h = constant e. v z ( r ) = v 0 ln( R/r ) ln(1 / κ ) , κ R r R, v 0 , R, κ = constant , 0 < κ < 1 2. A particle is located at x = - 2 , y = 3 , z = 6 in Cartesian (rectangular) coordinates. a. What are the cylindrical coordinates of the particle ( r, θ , z ) ? b. What are the spherical coordinates of the particle ( r, θ , φ ) ? 3. A vector v = v x δ x + v y δ y + v z δ z has Cartesian components v x = 1 , v y = 2 , and v z = 3 . a. What are the components of v in cylindrical coordinates ( v r , v θ , v z )? b. What are the components of v in spherical coordinates ( v r , v θ , v φ )? 4.
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