Proving Vector Calculus Identities: Tips and Tricks

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SUMMARY

The forum discussion centers on proving the vector calculus identity div(øu) = ødivu + ugradø, where ø represents a scalar field and u a vector field. Participants emphasize the importance of using index notation and the product rule in the proof process. One user initially struggles with the proof but successfully resolves it by applying the definition of divergence and the product rule. The conversation highlights the need for clarity in notation, particularly the use of the scalar field symbol ø.

PREREQUISITES
  • Understanding of vector calculus concepts, specifically divergence.
  • Familiarity with index notation in mathematical proofs.
  • Knowledge of the product rule in calculus.
  • Basic proficiency in LaTeX for mathematical notation.
NEXT STEPS
  • Study the definition and properties of divergence in vector fields.
  • Learn how to apply the product rule in vector calculus proofs.
  • Explore additional vector calculus identities, such as the divergence of the cross product.
  • Practice using index notation for various vector calculus problems.
USEFUL FOR

Students studying vector calculus, particularly those tackling proofs involving divergence and vector identities, as well as educators looking for effective teaching strategies in mathematical notation and proof techniques.

bothcats
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Homework Statement



div(øu) = ødivu + ugradø

Homework Equations



divergence of scalar field = f,ii
divergence of vector field = ui,i

The Attempt at a Solution



I've heard this is a simple proof, but this is my first one of 8 or so proofs I need to complete for homework, and I'm really not sure where to start. I know that div v = ∇ . v, but that's as far as I've gotten. We need to use Index Notation. Thoughts on where to start?
 
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bothcats said:

Homework Statement



div(øu) = ødivu + ugradø

Homework Equations



divergence of scalar field = f,ii
divergence of vector field = ui,i

The Attempt at a Solution



I've heard this is a simple proof, but this is my first one of 8 or so proofs I need to complete for homework, and I'm really not sure where to start. I know that div v = ∇ . v, but that's as far as I've gotten. We need to use Index Notation. Thoughts on where to start?
Why not start by applying the definition of the divergence of a vector field to ##\phi\mathbf u##? Please don't use the empty set symbol instead of ##\phi##. I found it very confusing, and it took me some time to understand what you meant. If you don't use LaTeX, and can't find another way to type a ##\phi##, then just call the scalar field f or something like that.
 
Sorry for the confusion. I'll be more careful with the lettering in the future. I've actually figured this one out now. It was the product rule that I wasn't sure about, now that I've worked it through (and several other identity proofs). Now, I'm on the divergence of (u cross v) identity.

Thanks!
 

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