For each vector field f⃗ (x, y,z), compute the curl of f⃗ and, if possible, find a function f(x, y,z) so that f⃗ =∇f. if no such function f exists, enter none. (a) suppose f⃗ (x, y,z)=(2yze2xyz+4z2cos(xz2)i⃗ +(2xze2xyz)j⃗ +(2xye2xyz+8xzcos(xz2))k⃗ . curl(f⃗ )

Let

The curl is

where denotes the partial derivative operator with respect to . Recall that

and that for any two vectors and , , and .

The cross product reduces to

When you compute the partial derivatives, you'll find that all the components reduce to 0 and

which means is indeed conservative and we can find .

Integrate both sides of

with respect to and

Differentiate both sides with respect to and

Now

and differentiating with respect to gives

for some constant . So

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step-by-step explanation:

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