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Vector Calculus

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(a) C: (x,y,z)=(4, t, t^2), t=-1~3 ds=√(0+1+4t^2) dt ∫_C xy ds= ∫_[-1,3] 4t√(1+4t^2) dt = (1/3)(1+4t^2)^(3/2) sub. t=-1~3 = (37√37 - 5√5)/3 (b) (i) curl(F)= ( x-x, y-y, z-z) =(0, 0, 0), so that F is irrotational. div(F)=2x+0+0=2x, not zero, so that F is not solenoidal (ii) ?φ/?x= yz , then φ=xyz+x^3/ 3+ C1(y, z) ?φ/?y=xz, then φ=xyz+C2(z,x) ?φ/?z=xy, then φ=xyz+C3(x,y) thus φ=xyz+ x^3 / 3+ C (constant independent on x,y,z)

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