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latentcorpse
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is it true that if \nabla \cdot \vec{F}=0 , \nabla \times \vec{F}=0 then \vec{F}=0?
Vector Calculus: Is F=0 When ∇·F=0 & ∇×F=0?
- Thread starter latentcorpse
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SUMMARY
In vector calculus, the conditions ∇·F=0 and ∇×F=0 do not imply that the vector field F is zero. A counterexample is provided with the vector field F = 3−x, which has a divergence of 3 and a curl of 0, demonstrating that F can be a non-zero constant vector field. This discussion clarifies that while both divergence and curl being zero are necessary conditions for certain types of vector fields, they do not guarantee that the vector field itself is zero.
PREREQUISITES- Understanding of vector calculus concepts such as divergence and curl
- Familiarity with vector fields and their representations
- Basic knowledge of mathematical notation used in physics and engineering
- Proficiency in interpreting mathematical expressions involving unit vectors
- Study the implications of the divergence theorem in vector calculus
- Explore examples of non-zero vector fields with zero divergence and curl
- Learn about the physical interpretations of divergence and curl in fluid dynamics
- Investigate advanced topics such as Helmholtz decomposition theorem
Students and professionals in mathematics, physics, and engineering who are studying vector calculus and its applications in fields such as fluid dynamics and electromagnetism.
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