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2 C 20647

Find the Jacobian for the change of variables defined by x =rcos0,y =rsin0.

Evaluate ff2x- y dA where R is the region bounded by the parabola x = y? and the line x-y = 2.
R

Find whether the vector field F = x?yi+xy j is conservative.

State Green’s theorem.

Find a parametric representation for the cone x = Vy? +22.

Find the surface area of the torus given by 7 (u,v) =(2+cosu) 0080 i+ (2+cosw)sinv jtsinuk
where the Domain D is given by 0
Compute [മക്ക given F(x,y,z)=(x+sinz)i +(2y+cosx) j+(3z+tany)k and S is the unit

8
sphere x? + y? +2? =1.

(10 x 3 = 30 marks)
Section B (Paragraph Questions)

Answer at least five questions.
Each question carries 6 marks.
All questions can be attended.
Overall Ceiling 30.
Find fry. and fx, if f (x,y)=xcosy+ysinx .
Find the differential of w =x? + xy + 25. Compute the value of dw if (x, y, z) changes from (1, 2, 1)

to (0.98, 2.03, 1.01) and compare the value with that of Aw.

Find the equation of the tangent plane and normal line to the surface x? —2y?—42? =4
at (4, -2, -1).

Find the relative extrema of f(x,y) = பி + dy —2Qy? +1.

Find the volume of the solid region bounded by the paraboloid z= 4 —x? —2y? and the xy-plane.

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