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Vector Field, Gradients, Div, Divergence, Curl and Surface Integrals

4) A vector field F is shown. Use the interpretation of divergence derived in this section to determine whether div F is positive or negative at P1 and P2

a.) Are the points P1 and P2 so sources or sinks vector field F shown in the figure? Give an explanation based solely on the picture.

5)Use the divergence Theorem to calculate the surface integral. , in other words, calculate F across S.

S is the surface of the solid bounded by the cylinder and the planes z = x +2 and z = 0.

6) Use Stoke's Theorem to evaluate curl F*dS

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In general, I most strongly recommend that you get a copy of "div, grad, curl, and all that", a *small, short, readable* and comparatively inexpensive paperback. It is a classic in the field and not only shows *how* to do these things, but more importantly *why* anyone would bother. I didn't get my hands on this book until after several semesters, a pity because if I had found out what was going on earlier I might have stayed in engineering or gone back to math sooner, despite the awful professor who ruined my first degree. Get this book from your library now and if you are going to do any more math or engineering, do please buy a copy. I am using it as well as a classic calculus text (Swokowski) as references while answrering your post.

(To quote an author I really like, "Yes, Cecil knows everything, but he does not claim to have absorbed it out of the rocks; these books will give you some of the knowledge that he has". In other words, even if a teacher knows a lot, a *good* teacher refers to sources to be sure to get things right and complete, and is not afraid to admit it.)

(4) The interpretationof divergence:

From div, grad, curl 3rd edition p. 50 -- Suppose that in some region of space "stuff" (matter, electric charge, anything) is moving. Let the density of this stuff at any point (x,y,z) and any time t be rho(x,y,z,t) and let its velocity be V(t)
(V being a vector, I am using a capital letter instead of bold due to limitations of this system; also rho is the Greek letter that looks sort of like a p but is the Greek r)
Further suppose this stuff is *conserved*, that is it is neither created nor destroyed.
Concentrating on some arbitrary volume v in space (not the same V as above), we ask **What is the *rate* at which the amount of this stuff in this volume is changing**

(Then follows a long mathematical argument which I will not reproduce)

**The rate of change of the amount of stuff in the volume is then the divergence.**

A more classic definition, from Calculus 6th edition by Swokowski

div F at p = lim as k->0 of (1/v k) double integral of F . N dS
(where vk is the volume of a sphere radius k around point p, N is the normal vector to the surface at each point, and due to the limits of typing the period is the dot product.)

Thus **the divergence of F at P is the limiting value of the *flux* per unit volume over a sphere with center P, as the radius of the sphere approaches zero**

(since "flux" = *rate of change* of "stuff", these two definitons are equivalent)

OK, now the answer:

Use the interpretation of divergence derived in this section to determine whether div F is positive or negative at P1 and P2.

Since from what can be seen of the field P1 appears to have more flowing out than in (larger vectors pointing away than towards) then the divergence should be positive at P1. Since P2 appears to have more flowing in than out (larger vectors pointing towards it than away form it) then the ...

Solution Summary

Vector Field, Gradients, Div, Divergence, Curl and Surface Integrals are investigated. The solution is detailed and well presented.

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