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Curl of a Vector Field
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Divergence of a Vector Field
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Gradient of a Scalar Field
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Properties of Transposes
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The Transpose of a Matrix
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Cartesian Coordinate
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Cylindrical Coordinate
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Spherical Coordinate
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Transform from Cartesian to Cylindrical Coordinate
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Transform from Cartesian to Spherical Coordinate
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Transform from Cylindrical to Cartesian Coordinate
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Transform from Spherical to Cartesian Coordinate
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Divergence Theorem/Gauss' Theorem
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Stokes' Theorem
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Definition of a Matrix
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Linear Algebra
> Divergence of a Vector Field
Divergence of a Vector Field
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A
vector
is a quantity that has a
magnitude
in a certain
direction
. Vectors are used to model forces, velocities, pressures, and many other physical phenomena. A
vector field
is a function that assigns a vector to every point in space. Vector fields are used to model force fields (gravity, electric and magnetic fields), fluid flow, etc.
The divergence of a vector field F = <P,Q,R> is defined as the partial derivative of P with respect to x plus the partial derivative of Q with respect to y plus the partial derivative of R with respect to z.
The divergence of a vector field is also given by:
We define the divergence of a vector field at a point, as the net outward flux of per volume as the volume about the point tends to zero.
∇ ∙
A
=
div
A
In Cartesian
∇ ∙
A
≡
∂A
_{x}
/∂
_{x}
+ ∂A
_{y}
/∂
_{y}
+ ∂A
_{z}
/∂
_{z}
In Cylindrical
∇ ∙
A
≡
∂(r ∙ Ay
_{)}
/(r ∙ ∂
_{r)}
+ ∂A
_{ø}
/(r ∙ ∂
_{ø)}
+ ∂A
_{z}
/∂
_{z}
In Spherical
∇ ∙
A
≡
∂(R
^{2}
∙ A
_{R)}
/(R
^{2}
∙∂
_{R)}
+ ∂(A
_{ø }
∙ sinθ)/(R ∙ sinθ ∙ ∂θ
_{)}
+ ∂A
_{ø}
/(R ∙ sinθ ∙ ∂
_{ø}
)
Example 1:
Compute the divergence of
F(x, y) = 3x
^{2}
i + 2yj
.
Solution: The divergence of
F(x, y)
is given by ∇•
F(x, y)
which is a dot product.
Example 2:
Calculate the divergence of the vector field
G(x,y,z) = e
^{x}
i + ln(xy)j + e
^{xyz}
k
.
Solution: The divergence of
G(x,y,z)
is given by ∇•
G(x,y,z)
which is a dot product. Its components are given by:
G
_{1}
= e
^{x}
G
_{2}
= ln(xy)
G
_{3}
= e
^{xyz}
and its divergence is:
Example 3:
Calculate the divergence of the vector field
G(x, y, z) = 4y/x
^{2}
· i + (sin y)j + 3k
Solution: The divergence of
G(x, y, z)
is given by ∇•
G(x,y,z)
which is a dot product. Its components are given by:
G
_{1}
= 4y/x
^{2}
G
_{2}
= (sin y)
G
_{3}
= 3
and its divergence is
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