7/17/2019 ChE 5310 – Homework # 1

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ChE 5310 – Homework # 1

Due in class on Wednesday, Sept 2

1.  Given below are examples to familiarize you with calculating gradients of functions. 

Calculate F =" f   for each of the following functions

a. 

f = x2

yz + z3

  b.  f = rcos! (spherical coordinates)

2.  Vector Math: Calculate the divergence and curl of the following functions.

a.  v =  yz2e x+  x

3 ze

 y+  xy

2e z 

 b.  v =  r "3

er  (spherical coordinates)

3.  Coordinate transformation: Express the following vector

a = 2rer +   r 

2sin" e"  +  r sin" e#   in cylindrical coordinates

4. 

Vector and Tensor Math: Given the following a=

e x+

e y+

e z ,b = "2e

 x

+  3e z ,

T = e xe x  " e xe z + 3e ye z + 2e ze x  and S = 2e xe x+ 3e

 xe z "10e

 ze x, evaluate

a.  a•T  

 b.  T • a , indicate if a•T  is equal or not to T • a  

c.  a•T •b 

d.  S •T , indicate if this operation results in a scalar or vector?

e.  Compute the double-dot product of S  and

 

T , is the end result scalar or vector?

f.  Compute the dyadic product ab , is the end result vector or tensor?

5.   Familiarize with index notation: Prove using index notation only that

a. 

" #"P = 0   b.  " •   #"P +"

2u[ ] = #"

2P , when " • u = 0  (Hint: The order of partials does not

matter)

c.  " # " #  a( ) = " " •  a( ) $ "2a 

6.   Application of Gauss Divergence Theorem: Find the value of n

 A

 ""  •vdA, where

v =  r2 xe

 x+  ye

 y+  ze

 z( ) , r2=  x

2+  y

2+  z

2  and A is the surface of the sphere with

radius a. Compute the integral directly and also with the aid of the Gauss Divergence

Theorem. Answer: 4"a5