(i)state TWO conditions for any system of forces acting on an object to be at equilibrium.(2 marks)
(ii)Test whether the following system of forces are in equilibrium.
F1=i –j, acting at a point r1=i + k
F2= i- k, acting at a point r2=2i
F3= 2j + k, acting at a point r3=i-2j
F4=-2i-j , acting at a point r4 =3i + j + k (5 marks)
(b)(i) State the fundamental theorem for gradients.(1 mark)
(ii)For the function; T=x2 + 4 xy + 2 yz3 and the points a=(0, 0, 0), b= (1, 1, 1), check the fundamental theorem for gradients.(5 marks)
(c)(i)Write the mathematical statement for the divergence theorem, explaining the geometrical significance of it. (2 marks)
(ii)Verify divergence theorem for the function;
V=(x2-z2)i+ 2xy j + (y2 + z)k for a unit cube situated at the origin, bounded by the six planes x=y=z=0, x=y=z=1 (5 marks)
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