Question

Let⇀H=〈−y(2 +x), x, yz〉 (a) Show that ⇀∇·⇀H= 0. (b) Since⇀H is defined and its component...

Let⇀H=〈−y(2 +x), x, yz〉

(a) Show that ⇀∇·⇀H= 0.

(b) Since⇀H is defined and its component functions have continuous partials on R3, one can prove that there exists a vector field ⇀F such that ⇀∇×⇀F=⇀H. Show that F = (1/3xz−1/4y^2z)ˆı+(1/2xyz+2/3yz)ˆ−(1/3x^2+2/3y^2+1/4xy^2)ˆk

satisfies this property.

(c) Let⇀G=〈xz, xyz,−y^2〉. Show that⇀∇×⇀G is also equal to⇀H.

(d) Find a function f such that⇀G=⇀F+⇀∇f.

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