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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