vector & tensor analysis.pptx

7/29/2019 vector & tensor analysis.pptx

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Divergence

Physical meaning of divergence

Physical significance of divergenceSignificance of divergence of a vector

field

Physical Interpretation of the Divergence


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In vector calculus, divergence is a vector

operator that measures the magnitude of a

vector field's source or sink at a given point, in

terms of a signed scalar.

More technically, the divergence represents

the volume density of the outward flux of a

vector field from an infinitesimal volumearound a given point.


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In physical terms, the divergence of a threedimensional vector field is the extent to which

the vector field flow behaves like a source or a

sink at a given point. It is a local measure of its"outgoingness"the extent to which there is

more exiting an infinitesimal region of space

than entering it.

If the divergence is nonzero at some point then

there must be a source or sink at that position.


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Let x, y, z be a system of Cartesian

coordinates in 3-dimensional Euclidean

space, and let i, j, k be the corresponding

basis of unit vectors.

The divergence of a continuously

differentiable vector field F = U i + V j + W k is

equal to the scalar-valued function:

z

W

y

V

x

UFdivF

.


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The divergence is a linear operator.

The divergence of the curl of any vector field

(in three dimensions) is equal to zero. There is a product rule of the following type: if

is a scalar valued function and F is a vector

field, then)().()( FdivFgradFdiv


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In the context of fluid mechanics, where agiven vector field is interpreted as a model of a

fluid, with the vector value at a given point

being the velocity of the fluid particle at thatpoint, curl and divergence are used to express

notions of rotation compression of a fluid,

respectively.


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If the divergence is positive at a point, it

means that, overall, that the tendency is for

fluid to move away from that point

(expansion); if the divergence is negative, then

the fluid is tending to move towards that point(compression).


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Divergence is a vector operator that measuresthe magnitude of a vector fields source or sink

at a given point , in terms of a signed scalar.

More technically, the divergence representsthe volume density of the outward flux of a

vector field from an infinitesimal volume

around a given point.


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Consider air as it is heated or cooled. The relevantvector field for this example is the velocity of themoving air at a point. If air is heated in a region itwill expand in all directions such that the velocityfield points outward from that region. Thereforethe divergence of the velocity field in that regionwould have a positive value, as the region is asource. If the air cools and contracts, thedivergence is negative and the region is called asink.


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Divergence of a vector field A is a measure of

how much a vector field converges to or

diverges from a given point. In simple terms it

is a measure of the outgoingness of a vector

field.


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Divergence of a vector field is positive if thevector diverges or spread out from a given

point called source- Divergence of a vector

field is negative if the vector field converges atthat point called sink. .If just as much of the

vector field points in as out, the divergence

will be approximately zero.


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The divergence measures how much a vector

field ``spreads out'' or diverges from a given

point.For example, the figure on the left has positive

divergence at P, since the vectors of the vector

field are all spreading as they move away fromP.


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The field on the right has negative divergence

since the vectors are coming closer together

instead of spreading out.

The figure in the center has zero divergence

everywhere since the vectors are not

spreading out at all. This is easy to compute

also, since the vector field is constanteverywhere and the derivative of a constant is

zero.


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vector & tensor analysis.pptx

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