Question

1. Recall if vec{F} is the curl of another vector field, then abla cdot vec{F} = 0 everywhere. In this problem, you will investigate whether or not vec{F}(x, y, z) = frac{x}{(x^2 + y^2 + z^2)^{3/2}}vec{i} + frac{y}{(x^2 + y^2 + z^2)^{3/2}}vec{j} + frac{z}{(x^2 + y^2 + z^2)^{3/2}}vec{k} is the curl of another vector field. (a) Compute the divergence of vec{F}. (b) Let S_1 and S_2 be the upper and lower halves of the unit sphere, both endowed with upward pointing normal vectors. Argue that if vec{F} = curlvec{G} for some continuously differentiable vector field vec{G}, then flux of vec{F} across S_1 and the flux of vec{F} across S_2 are equal. (c) With S_1 and S_2 as above, compute the integrals iint_{S_1} (vec{F} cdot vec{n}) dsigma quad ext{and} quad iint_{S_2} (vec{F} cdot vec{n}) dsigma directly. Is vec{F} the curl of another vector field or not?

          1. Recall if vec{F} is the curl of another vector field, then 
abla cdot vec{F} = 0 everywhere. In this problem, you will investigate whether or not
vec{F}(x, y, z) = frac{x}{(x^2 + y^2 + z^2)^{3/2}}vec{i} + frac{y}{(x^2 + y^2 + z^2)^{3/2}}vec{j} + frac{z}{(x^2 + y^2 + z^2)^{3/2}}vec{k}
is the curl of another vector field.
(a) Compute the divergence of vec{F}.
(b) Let S_1 and S_2 be the upper and lower halves of the unit sphere, both endowed with upward pointing normal vectors. Argue that if vec{F} = curlvec{G} for some continuously differentiable vector field vec{G}, then flux of vec{F} across S_1 and the flux of vec{F} across S_2 are equal.
(c) With S_1 and S_2 as above, compute the integrals
iint_{S_1} (vec{F} cdot vec{n}) dsigma quad 	ext{and} quad iint_{S_2} (vec{F} cdot vec{n}) dsigma
directly. Is vec{F} the curl of another vector field or not?

$1. Recall if vecF is the curl of another vector field, then abla cdot vecF = 0 everywhere. In this problem, you will investigate whether or not vecF(x, y, z) = fracx(x^2 + y^2 + z^2)^3/2veci + fracy(x^2 + y^2 + z^2)^3/2vecj + fracz(x^2 + y^2 + z^2)^3/2veck is the curl of another vector field. (a) Compute the divergence of vecF. (b) Let S1 and S2 be the upper and lower halves of the unit sphere, both endowed with upward pointing normal vectors. Argue that if vecF = curlvecG for some continuously differentiable vector field vecG, then flux of vecF across S1 and the flux of vecF across S2 are equal. (c) With S1 and S2 as above, compute the integrals iintS1 (vecF cdot vecn) dsigma quad extand quad iintS2 (vecF cdot vecn) dsigma directly. Is vecF the curl of another vector field or not?$

Added by Sheila T.

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