Each of the following four vector fields F(x, y, z) is shown in the xy-plane and

Question

Each of the following four vector fields F(x, y, z) is shown in the xy-plane and looks the same in all other horizontal planes. In other words, F is independent of z and its z-component is 0. Answer the following questions. You do not need to provide an explanation. No partial points are given for these questions. (a) Find all points (among the 4 points A, B, C, D) at which div F notequalto 0. The answer is A, C. (b) Find all points (among the 4 points A, B, C, D) at which curl F notequalto 0. The answer is B, D. (c) Find all closed curves (among the 6 curves C_1, C_2, ..., C_6) along which the circulation _C F middot dr of the corresponding vector field is not zero. The answer is C_3, C_4, C_6. (d) For this question, regard the above vector fields as two-dimensional vector fields G(x, y) = P(x, y)i + Q(x, y)j. Find all closed curves (among the 6 curves C_1, C_2, ..., C_6) along which the flux _C G middot n ds of the corresponding vector field is not zero, where n is the outward unit normal vector. The answer is C_1, C_2, C_5.

Explanation / Answer

Divergence is net outward flux, negative divergence means inwards flux. It is represented by straight arrows of increasing/decreasing magnitudes.

So(a) at points A,C divergence is nonzero. At B arrows are of same size(magnitudes).

(B) Curl is related to rotation of vectors about a point,axis etc.

At point D there is rotation of vectors , at B there is change in direction of Vector flow, if , and we know it is, the vector is continuous,this must have some rotation about point B.

(C) using stocks theorem, line integral of a vector= curl of that vector.

So, similarly , curl of vector is non zero over loops C3,C4,C6.

(D) Using Gauss divergence theorem : integral (G.n)ds = vol. Integral( div.G) dv.

So , where divergence is non zero, integral (G.n)ds is non zero.

I.e about loops : C1,C2,C5.

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