Indicates if there are sources within a volume/point Can be used to relate the curl of a field to the circulation of a field Indicates the direction and magnitude of the maximum increase of a scalar function None of the above
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The question is likely asking about a vector calculus concept. Let's examine each option: Option 1: "Indicates if there are sources within a volume/point" This refers to the divergence of a vector field. A non-zero divergence indicates the presence of sources or Show more…
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Horizontal cross-sections of the vector fields F(x, y, z) and G(x, y, z) are given in the figure. Each vector field has zero z-component (i.e., all of its vectors are horizontal) and is independent of z (i.e., is the same in every horizontal plane). You may assume that the graphs of these vector fields use the same scale. (a) Are div(F) and div(G) positive, negative, or zero at the origin? Be sure you can explain your answer. At the origin, div(F) is negative. At the origin, div(G) is positive. (b) Are F and G curl-free (irrotational) or not at the origin? Be sure you can explain your answer. At the origin, F is irrotational. At the origin, G is irrotational. (c) Is there a closed surface around the origin such that F has nonzero flux through it? Be sure you can explain your answer by finding an example or a counterexample. Yes. (d) Is there a closed surface around the origin such that G has nonzero circulation around it? Be sure you can explain your answer by finding an example or a counterexample. No.
Adi S.
Recall from Section 14.4 that the divergence of a vector field is related to the expansion or compression of a gas whose motion is represented by that field. If a gas whose motion is represented by a vector field is expanding (or compressing) in a region of space, what effect should that have on the flux of the vector field out of (or into) a closed region W? Making some additional assumptions about the direction of the vectors in $\mathbf{F}$ may help you to think about this situation.
Vector Analysis
The Divergence Theorem
Horizontal cross-sections of the vector fields F(x, y, z) and G(x, y, z) are given in the figure. Each vector field has zero z-component (i.e., all of its vectors are horizontal) and is independent of z (i.e., is the same in every horizontal plane). You may assume that the graphs of these vector fields use the same scale. (a) Are div(F) and div(G) positive, negative, or zero at the origin? Be sure you can explain your answer. At the origin, div(F) is positive. At the origin, div(G) is zero. (b) Are F and G curl free (irrotational) or not at the origin? Be sure you can explain your answer. At the origin, F is not irrotational. At the origin, G is not irrotational. (c) Is there a closed surface around the origin such that F has nonzero flux through it? Be sure you can explain your answer by finding an example or a counterexample. Yes. (d) Is there a closed surface around the origin such that G has nonzero circulation around it? Be sure you can explain your answer by finding an example or a counterexample. Yes.
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