Question:

1. Evaluate the integral \(I=\int\limits_{0}^{1}{\int\limits_{0}^{1-z}{\int\limits_{0}^{{{y}^{2}}}{xdxdydz}}}\)
2. Find the direction in which \(f\left( x,y \right)=\frac{{{x}^{2}}}{2}+\frac{{{y}^{2}}}{2}\) decreases most rapidly at (1, 1)
3. Find the divergence and the curl of the following vector field:
   \[F=\left( 3z-2xy \right)\mathbf{i}+\left( xz \right)\mathbf{j}+\left( 2yz \right)\mathbf{k}\]
4. A thin plate covers the triangular region bounded by the x-axis and the lines x = 1 and y = 2 x 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.

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Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document