Divergence Theorem

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:The normal vector is finally the cross product of these two vectors, which is simply <math> N = \begin{bmatrix} 1\\ 0\\ 0\\ \end{bmatrix}.</math>}}
:The normal vector is finally the cross product of these two vectors, which is simply <math> N = \begin{bmatrix} 1\\ 0\\ 0\\ \end{bmatrix}.</math>}}
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:'''Step 4:''' Integrate the dot product of this normal vector with the given vector field.
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:'''Step 5:''' Integrate the dot product of this normal vector with the given vector field.
:{{hide|1=:The amount of the field normal to our surface is the flux through it, and is exactly what this integral gives us.
:{{hide|1=:The amount of the field normal to our surface is the flux through it, and is exactly what this integral gives us.
:<math> \iint\limits_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf F\;\cdot\mathbf n\,{d}S .</math>
:<math> \iint\limits_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf F\;\cdot\mathbf n\,{d}S .</math>
:<math>= \int_0^2 \int_0^2 F \cdot N \,dsdt </math>
:<math>= \int_0^2 \int_0^2 F \cdot N \,dsdt </math>
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:<math>= \int_0^2 \int_0^2 \begin{bmatrix} x^2 \\ 0\\ 0\\ \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 0\\ 0\\ \end{bmatrix} \,dsdt = \int_0^2 \int_0^2 x^2dsdt = \int_0^2 \int_0^2 4 \,dsdt </math> }}
+
:<math>= \int_0^2 \int_0^2 \begin{bmatrix} x^2 \\ 0\\ 0\\ \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 0\\ 0\\ \end{bmatrix} \,dsdt = \int_0^2 \int_0^2 x^2dsdt = \int_0^2 \int_0^2 4 \,dsdt </math>
 +
:<math>=16 </math> }}
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 +
Both sides of the equation give 16, so the Divergence Theorem is indeed valid here.
|other=Some multivariable calculus
|other=Some multivariable calculus

Revision as of 15:55, 30 June 2009


Fountain Flux
The water flowing out of a fountain demonstrates an important theorem for vector fields, the Divergence Theorem.

Contents

Basic Description

Consider a fountain like the one pictured, particularly its top layer. The rate that water flows out of the fountain's spout is directly related to the amount of water that flows off the top layer. Because something like water isn't easily compressed like air, if more water is pumped out of the spout, then more water will have to flow over the boundaries of the top layer. This is essentially what The Divergence Theorem states: the total the fluid being introduced into a volume is equal to the total fluid flowing out of the boundary of the volume.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Some multivariable calculus

The Divergence Theorem in its pure form applies to Vector Fields. Flowing water can be considere [...]

The Divergence Theorem in its pure form applies to Vector Fields. Flowing water can be considered a vector field because at each point the water has a position and a velocity vector. Faster moving water is represented by a larger vector in our field. The divergence of a vector field is a measurement of the expansion or contraction of the field; if more water is being introduced then the divergence is positive. Analytically divergence of a field  F is

 \nabla\cdot\mathbf{F} =\partial{F_x}/\partial{x} + \partial{F_y}/\partial{y} + \partial{F_z}/\partial{z},

where  F _i is the component of  F in the  i direction. Intuitively, if F has a large positive rate of change in the x direction, the partial derivative with respect to x in this direction will be large, increasing total divergence. The divergence theorem requires that we sum divergence over an entire volume. If this sum is positive, then the field must indicate some movement out of the volume through its boundary, while if this sum is negative, the field must indicate some movement into the volume through its boundary. We use the notion of flux, the flow through a surface, to quantify this movement through the boundary, which itself is a surface.

The divergence theorem is formally stated as:


\iiint\limits_V\left(\nabla\cdot\mathbf{F}\right)dV=\iint\limits_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf F\;\cdot\mathbf n\,{d}S .

The left side of this equation is the sum of the divergence over the entire volume, and the right side of this equation is the sum of the field perpendicular to the volume's boundary at the boundary, which is the flux through the boundary.

Example of Divergence Theorem Verification

The following example verifies that given a volume and a vector field, the Divergence Theorem is valid.

Consider the vector field  F = \begin{bmatrix} x^2 \\ 0\\ 0\\ \end{bmatrix}.

For a volume, we will use a cube of edge length two, and vertices at (0,0,0), (2,0,0), (0,2,0), (0,0,2), (2,2,0), (2,0,2), (0,2,2), (2,2,2). This cube has a corner at the origin and all the points it contains are in positive regions.

We begin by calculating the left side of the Divergence Theorem.

Step 1: Calculate the divergence of the field:
 \nabla\cdot F = 2x
Step 2: Integrate the divergence of the field over the entire volume.
 \iiint\nabla\cdot F\,dV =\int_0^2\int_0^2\int_0^2 2x \, dxdydz
=\int_0^2\int_0^2 4\, dydx
=16

We now turn to the right side of the equation, the integral of flux.

Step 3: We first parametrize the parts of the surface which have non-zero flux.
Notice that the given vector field has vectors which only extend in the x-direction, since each vector has zero y and z components. Therefore, only two sides of our cube can have vectors normal to them, those sides which are perpendicular to the x-axis. Furthermore, the side of the cube perpendicular to the x axis and with all points at x = 0 cannot have any flux, since all vectors in this surface are zero vectors.
We are thus only concerned with one side of the cube since only one side has non-zero flux. This side is parametrized using
X=\begin{bmatrix} x \\ y\\ z\\ \end{bmatrix} = \begin{bmatrix} 2 \\ u\\ v\\ \end{bmatrix}\, , u \in (0,2)\, ,v \in (0,2)
Step 4: With this parametrization, we find a general normal vector to our surface.
To find this normal vector, we find two vectors which are always tangent to (or contained in) the surface, and are not collinear. The cross product of two such vectors gives a vector normal to the surface.
The first vector is the partial derivative of our surface with respect to u:  \frac{\part{X}}{\part{u}} = \begin{bmatrix} 0\\ 1\\ 0\\ \end{bmatrix}
The second vector is the partial derivative of our surface with respect to v:  \frac{\part{X}}{\part{v}} = \begin{bmatrix} 0\\ 0\\ 1\\ \end{bmatrix}
The normal vector is finally the cross product of these two vectors, which is simply  N = \begin{bmatrix} 1\\ 0\\ 0\\ \end{bmatrix}.
Step 5: Integrate the dot product of this normal vector with the given vector field.
The amount of the field normal to our surface is the flux through it, and is exactly what this integral gives us.
 \iint\limits_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf F\;\cdot\mathbf n\,{d}S .
= \int_0^2 \int_0^2 F \cdot N \,dsdt
= \int_0^2 \int_0^2 \begin{bmatrix} x^2 \\ 0\\ 0\\ \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 0\\ 0\\ \end{bmatrix} \,dsdt =  \int_0^2 \int_0^2 x^2dsdt = \int_0^2 \int_0^2 4 \,dsdt
=16

Both sides of the equation give 16, so the Divergence Theorem is indeed valid here.




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