a) If u_(1),u_(2),u_(3) are curvilinear coordinates such that x=(1)/(2)(u_(1)-3) y=u_(2)+4;z=u_(3)-2 prove that the system is orthogonal. Find the scale factor and the expression for (ds)^2 b) Evaluate oint _(c)overrightarrow (F)cdot doverrightarrow (r) by Stoke's theorem, where overrightarrow (F)=y^2i+x^2j-(x+z)k and c is the boundary of triangle with vertices at (0,0,0),(1,0,0) and (1,1,0) (12mks) (8mks) Question 5 (20 marks) a) State and verify Green's theorem in the plane for oint (3x^2-8y^2)dx+(4y-6xy)dy where c is the boundary of the region bounded by xgeqslant 0,yleqslant 0 and 2x-3y=6 (18mks)

a) To prove that the system is orthogonal, we need to show that the partial derivatives of x, y, and z with respect to each other are zero. Let's calculate the partial derivatives: Since all partial derivatives are zero except for those involving and , the system is orthogonal.The scale factor is given by .The expression for is given by .b) To evaluate using Stokes' theorem, we need to find the curl of and then integrate it over the surface bounded by the curve C.The curl of is given by .The surface bounded by the curve C is a triangle in the xy-plane with vertices at , , and . We can parametrize this surface as , where and .The outward normal to the surface is given by .Using Stokes' theorem, we have .The area of the triangle is given by .Therefore, $\oint_C \overrightarrow{F} \cdot d\overrightarrow{r} = \iint_S (2y + 2x - 1) \, dS = \int_0^1 \int_0^x (2y + 2x - 1) \, dy \, dx = \int_0^1 \left[ y^2 + 2xy - y \right]_0^x \, dx = \int_0^1 (x^2 + 2x^2 - x) \, dx = \int_0^1 (3x^2 - x) \, dx = \left[ x^3 - \frac{x^2}{2}