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01000MA 101032103

Show that F = yzf+ xz + yxk is conservative field and hence find its scalar
potential function.
Show that the integral | (xy? + y*)dx + (x*y + 3xy?)dy is independent of path
joining the points (1,2) and (3,4). Hence evaluate the integral.
Module VI

Answer any three questions. Each question carries 5 Marks
Use Green’s theorem to evaluate the integral ‏ہل‎ (ಲೆ + y)dx (ഓ + x*)dy where
C is the boundary of the region between y = x? and y = 2x.
Evaluate the surface integral [| xzds where o is the part of the plane x + y +
z = 1 that lies in the first octant.
Use Stoke’s theorem to evaluate Sc 7.07 where F(x,y,z) = (८ - 9)१+ (1 - 2)/ -
(z—x)k where © is the portion of the plane x + y +z = 1 in the first octant.

Assume that the surface has an upward orientation

Use the divergence theorem to find the outward flux of the vector field
F(x,y,z) =(x —z)t+(y—x)f+(2z—y)k; where 5 is the surface of the
cylindrical solid bounded by x? + 172 = ‏ڈو‎ ढ = 0 and 2 = 1.

Evaluate [| 8.3205 using divergence theorem, where F = 4xzt+ xyz?f+3zk over
the region bounded by the cone 22 = x? + y? and the plane 2 = 4, above the xy-

plane.

RR

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