How do you find the divergence of a vector field?

In summary, the conversation discusses finding the divergence of a vector field and clarifies the meaning of "div(uxv)". The correct approach is to find the cross product of two vectors and then take its divergence.
  • #1
andrey21
476
0
I am just curious how you find the divergence of the following vector field





Heres my example

u = xz^(2)i +y(x^(2)-1)j+zx^(2) y^(3)k



Am I right in thinking

U take the derivative with respect to x for first term derivative with respect to y for second term...

giving me z^(2) + (x^(2) -1) +x^(2)y^(3)
 
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  • #2
You have it correct.
 
  • #3
Thanks LCKurtz just another question I have bin posed.

does div(uxv)

Mean find the divergence of the dot product of vectors u and v.
 
  • #4
andrey21 said:
Thanks LCKurtz just another question I have bin posed.

does div(uxv)

Mean find the divergence of the dot product of vectors u and v.

No. That wouldn't make any sense because a dot product gives a scalar and divergence applies to vector fields. What it does mean is first take the cross product of a and b, which gives a vector, then take its divergence.
 

1. What is the definition of "divergence of a vector field"?

The divergence of a vector field is a measure of the outward flux of a vector field from a given point. It represents the amount of flow per unit volume that is leaving or entering a specific point in the vector field.

2. How is the divergence of a vector field calculated?

The divergence of a vector field is calculated by taking the dot product of the vector field with the del operator (∇). This is represented mathematically as div(F) = ∇ · F, where F is the vector field.

3. What is the physical significance of the divergence of a vector field?

The divergence of a vector field can be interpreted as a measure of the "source" or "sink" at a given point in the field. If the divergence is positive, it indicates a net flow away from the point, while a negative divergence indicates a net flow towards the point.

4. How is the divergence of a vector field related to the concept of "conservation"?

In physics, the divergence of a vector field is closely related to the principle of conservation of mass or energy. If the divergence is zero, it implies that the field has no sources or sinks, and thus the flow of mass or energy is conserved.

5. Can the divergence of a vector field be visualized?

Yes, the divergence of a vector field can be visualized using arrows or streamlines to represent the direction and magnitude of the vector field at different points. In regions of positive divergence, the arrows will point away from the point, while in regions of negative divergence, the arrows will point towards the point.

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