- The water flowing out of a fountain demonstrates an important theorem for vector fields, the Divergence Theorem.
Consider the top layer of the fountain pictured. 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 if the quantity of fluid in the volume is constant.
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 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 is expressed in partial derivatives:
where is the component of in the 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:
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 total flux through the boundary.
Summing up divergence over the entire volume means we sum the flow into or out of each infinitely subregion, since a flow into one infinitesimal subregion means flow out of an adjacent subregion, which effects the next adjacent subregion, and so on until the boundary of the entire volume is reached. The total sum of divergence over the volume is thus equal to the flow at the boundary, as the theorem states.
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 .
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:
- 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.
- Step 4: With this parametrization, we find a general normal vector to our surface.
- Both sides of the equation give 16, so the Divergence Theorem is indeed valid here. ■
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