This assignment covers (vectors, green's theorem. divergence theorem, line integrals, surface integrals, Lagrange Multipliers, Directional Derivatives, Double Integrals(polar and spherical...

Qno.1
Uxv =

q.no.4
Thus in uv plane this is square with u ranging from 0 to 1 and v also ranging from 0 to 1
By change of variables the double integral become
Thus we get 0 as answer.
Cost = 15/sqrt 133 = 1.3006 which is not possible.
Find partial derivatives both I and second order.
We get f_x = 2xy^2-2x: f_y = 2x^2y-8y: f_xx = 2y^2-2: f_yy = 2x^2-8
And f_xy = 4xy = f_yx
Equate first derivatives to 0
0= 2xy^2-2x and 2x^2y-8y=0
Solving we get x=0, y =1, or -1 and y=0, x = 2 or -2
i.e. solutions are (0,0) (-2,1) (2,-1) (2,1) and (-2,-1)
Critical points...