Question:

a) Evaluate the integral \(I=\int\limits_{0}^{1}{\int\limits_{0}^{1-z}{\int\limits_{0}^{{{y}^{2}}}{xdxdydz}}}\)

b) Find the direction in which \(f\left( x,y \right)=\frac{{{x}^{2}}}{2}+\frac{{{y}^{2}}}{2}\) decreases most rapidly at (1, 1)

c) Find the divergence and the curl of the following vector field:

\[F=\left( 3z-2xy \right)\mathbf{i}+\left( xz \right)\mathbf{j}+\left( 2yz \right)\mathbf{k}\]

d) A thin plate covers the triangular region bounded by the x-axis and the lines x = 1 and y = 2x in the first quadrant. The plate’s density at the point (x, y) is \(\delta \left( x,y \right)=6x+6y+6\)

Find the coordinates of the centre of mass about the coordinate axes.