Weintroduce the ratio of the flux of the vector field A at the pointthrough a closed surfaceto thevolumeenclosed by thatsurface.and thelimitofthis ratio,as the surfaceareais madeto becomevanishinglysmallat the point,is called the divergence of the vectorfield at that point.denoted bydivA,givenbyA.dsdivA = limAVAV->0Where“div"is the observation of the word “divergence, and △Vis thevolume closed by the closed surface. It shows that the divergence of avectorfieldis a scalarfield, and it can be considered as the flux throughthe surfaceperunitvolumeIn rectangular coordinates, the divergence can be expressed as0A.aA.OAdivA:OzaxayUEV
V S V Δ d div lim Δ 0 = → A S A Where “div” is the observation of the word “divergence, and Vis the volume closed by the closed surface. It shows that the divergence of a vector field is a scalar field, and it can be considered as the flux through the surface per unit volume. In rectangular coordinates, the divergence can be expressed as z A y A x Ax y z + + divA = We introduce the ratio of the flux of the vector field A at the point through a closed surface to the volume enclosed by that surface, and the limit of this ratio, as the surface area is made to become vanishingly small at the point, is called the divergence of the vector field at that point, denoted by divA, given by
Using the operator V, the divergence can be written asdivA=V.ADivergenceTheorem[ divAdV=+ A-dsJ,V.AdV=f,A dsorFromthe point of view of mathematics,the divergence theorem statesthat the surfaceintegralof a vectorfunction over a closed surface canbe transformedinto a volumeintegralinvolving the divergence of thevectoroverthe volumeenclosed by the same surface.From the point ofthe view of fields, it gives the relationshipbetween the fieldsin a regionand thefieldsonthe boundaryof the regionUV
Using the operator , the divergence can be written as divA = A = V S V divAd A dS Divergence Theorem = V S V or Ad A dS From the point of view of mathematics, the divergence theorem states that the surface integral of a vector function over a closed surface can be transformed into a volume integral involving the divergence of the vector over the volume enclosed by the same surface. From the point of the view of fields, it gives the relationship between the fields in a region and the fields on the boundary of the region
3.Circulation&CurlThe lineintegralof a vector field A evaluated along a closed curveis called the circulation ofthe vector field A around the curve,and it isdenoted by I,i.e.F=f,A·dlIf the direction ofthevectorfield Ais the same as that of thelineelement dl everywhere along the curve, then the circulation T > O. Ifthey are in opposite direction, then F < O . Hence, the circulation canprovide a description of the rotationalproperty of a vector field
The line integral of a vector field A evaluated along a closed curve is called the circulation of the vector field A around the curve, and it is denoted by , i.e. 3. Circulation & Curl = l A dl If the direction of the vector field A is the same as that of the line element dl everywhere along the curve, then the circulation > 0. If they are in opposite direction, then < 0 . Hence, the circulation can provide a description of the rotational property of a vector field