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5CCS2ENM: Engineering Mathematics

Coursework 1

1.  Find a vector eld  : R3  ! R3  that has BOTH of the following two properties:

(i)   is not the curl of any vector eld,

(ii)   is not the gradient of any scalar field.

Show your working.  Explain your reasoning.  Marks will be awarded for the quality of your explanation. [15 marks]

2.  Find a vector eld  : R3  ! R3  such that

r ·   =  x2y +3y cos(xz)

Show your working. [15 marks]

3.  Denote the first 7 digits of your student number as s1 s2 s3 s4 s5 s6 s7 .  Let W be the hemi- spherical region enclosed by the surface Φ : [0, 1+ s3 + s5] ⇥ [0, 2⇡] ! R3  given by

Φ(r,✓)  =  (r cos✓) +  (r sin✓)  +  ((1 + s3 + s5 )2 − r2 )1/2                  and the xy-plane. Show the values of s1 s2 s3 s4 s5 s6 s7  at the beginning of the answer.

(a)  Let  : R3  ! R3  be the vector field on R3  given by the following function (x,y,z)  =  (x3 z) +  (y3 z)  +  0 

Verify Gauss’ Divergence Theorem for the vector eld  over this region

‹                   ˚

by evaluating both the left-hand-side and the right-hand-side separately.  Show your working.        [20 marks]

(b)  Find an example of a vector field  such that

Show

your working. [10 marks]

4.  Denote the first 7 digits of your student number as s1 s2 s3 s4 s5 s6 s7 .  Let  : R3  ! R3  be the following conservative vector eld

(x,y,z)  =   ✓ ◆  +  ✓ ◆  +  (2z)

Find an example of a path γ : R ! R3  such that

\

Show the values of s1 s2 s3 s4 s5 s6 s7  at the beginning of the answer. Show your working. [20 marks]

5.  Denote the first 7 digits of your student number as s1 s2 s3 s4 s5 s6 s7 . Use the Laplace Trans- form to solve the following initial valued problem:

 − 2  +(10+ s2 + s3 + s4)f(t)  =  (t − 2)6(− 5)

where f(0) = 2+ s5  and f\ (0) = 4+s6 , where 6(t) denotes the Dirac delta function. Show the values of s1 s2 s3 s4 s5 s6 s7  at the beginning of the answer. Show your working. [20 marks]