Math Problem Statement
prove that B=xyz^2 is solenoidal
Solution
To prove that the vector field is solenoidal, we need to show that its divergence is zero.
Step-by-step proof:
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Definition of a solenoidal vector field: A vector field is solenoidal if its divergence is zero everywhere, i.e.,
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Given vector field: The vector field is .
In component form, we can express as: where:
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Formula for the divergence: The divergence of a vector field in Cartesian coordinates is:
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Compute the partial derivatives:
- For :
- For :
- For :
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Substitute into the divergence formula:
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Conclusion: The divergence of is: Since , the vector field is not solenoidal.
Therefore, is not solenoidal, as its divergence is not zero.
Do you want further details or have any questions?
Here are 5 related questions to deepen your understanding:
- What are the physical implications of a vector field being solenoidal?
- How is the solenoidal condition used in fluid dynamics?
- Can a vector field that is not solenoidal be transformed into a solenoidal one?
- How does the divergence theorem relate to solenoidal fields?
- What are examples of solenoidal vector fields in electromagnetism?
Tip: The divergence of a vector field helps measure how much the field spreads out from a point. Solenoidal fields represent incompressible flows, where there is no net outflow from any point.
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Math Problem Analysis
Mathematical Concepts
Vector Fields
Divergence
Formulas
Divergence Formula in Cartesian Coordinates
Theorems
Solenoidal Vector Fields
Suitable Grade Level
College Level