Understanding the Divergence Operator for Time-Varying Vectors

In summary, the conversation is about finding the divergence of a vector field that takes the form u = u(x,y,z,t). The person asking the question is unsure if the divergence operator is defined for time-varying vectors, and the response is that it is defined by taking the partial derivatives with respect to x,y,z while treating t as a constant. It is also noted that the divergence operator operates on spatial dimensions unless otherwise specified.
  • #1
tomwilliam2
117
2

Homework Statement



I'm trying to find the divergence of a vector field (a fluid flow vector), but the vector takes the form u = u(x,y,z,t)

The Attempt at a Solution



I only really know how to take the divergence of a time-independent vector, so I'm guessing I just take the partial derivatives with respect to x,y,z and hold t= constant...is that right?
I am interested in knowing whether the divergence operator is even defined for time-varying vectors, or whether divergence is only defined for a given point in time.

Thanks
 
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  • #2
Del operates on spatial dimensions*. So yes, you just take the partial derivatives with respect to x, y and z.

*Unless otherwise specified.
 
  • #3
Thanks very much.
 
  • #4
I agree with DeIdeal. The divergence operator involves partial derivatives with respect to x, y, z and so t is treated as a constant.
 

FAQ: Understanding the Divergence Operator for Time-Varying Vectors

What is the divergence operator for time-varying vectors?

The divergence operator for time-varying vectors is a mathematical tool used to measure the rate at which a vector field is spreading out or converging at a given point in space and time.

How is the divergence operator calculated for time-varying vectors?

The divergence operator is calculated by taking the dot product of the vector field with the gradient operator. This results in a scalar value that represents the amount of flow or flux emanating from a given point.

What is the significance of understanding the divergence operator for time-varying vectors?

Understanding the divergence operator is crucial in many fields of science and engineering, as it allows for the analysis and prediction of fluid flows, electromagnetic fields, and other dynamic systems.

How does the divergence operator differ from the curl operator?

The divergence operator measures the flux or flow of a vector field, while the curl operator measures the rotation or circulation of a vector field. In other words, the divergence operator measures the source or sink of a vector field, while the curl operator measures its vorticity.

Can the divergence operator be applied to non-time-varying vector fields?

Yes, the divergence operator can be applied to both time-varying and non-time-varying vector fields, but its interpretation and physical meaning may differ depending on the context. In non-time-varying fields, the divergence represents the amount of flux emanating from a given point in space, while in time-varying fields, it represents the flux over time at a given point.

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