 # Mathematics Questions

## Transcribed Text

1) Find the curl and the divergence of the vector field F(x.y,2)-.myityj+zk 2) Evaluate the line integral F dr where F = xi+ yj and C is clockwise around the square with vertices (0,0),(2,0),(2,2)(0,2) 3) Find the mass of one turn of a spring with density p(r,y,2) = z. Given the shape of the spring is 4) Prove that F = (e* sin y,e* cos v) is conservative. Find its potential function. Evaluate the line integral F dr where C is any smooth curve from (0,0) to (3,8). 5) Use Green's Theorem to evaluate the line integral. 1c(20ykar((xty/dy. C: boundary of the region lying between the graphs of y = 0 and y =4-x². 6) Verify the Divergence Theorem by evaluating F Nas as a surface integral and as a triple integral. Where F(x,y,2)=(x,y,z) and the solid region bounded by the coordinate planes and 2x+3y +4z = 12. 7) Find the equation of the tangent plane to the curve given by at the point(1,2,5). 8) Find the area of the sphere above the given region, F(u,v) = asinu cos vi+asinusin v}+acosuk where 0 S u S (randOsvsb. 9) Evaluate (r(x,y) dS. = = 10) Find the flux of F through S. F NdS s 11) Verify Stokes's Theorem by showing that the following; Let C be the triangle created by the intersection of 2x+2y+z= 6and the coordinate planes and

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