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How do you imagine vector fields in a "physical" way? I'm learning vector calculus, and most of the analogies break down:

• If I imagine the vector field as some sort of fluid flow, then if it's incompressible then the divergence is clearly zero everywhere. But for most vector fields the divergence is not zero, and I find compressible fluids very counterintuitive.




• If I imagine them as electric field of some configuration of charges, but then not all vector fields corresponds to the electric field of a configuration of charges (eg the field $f(x,y) = x hati + y hatj$), so those fields are again counterintuitive.

Which way should I think about them so that things like divergence/curl become obvious and the analogy doesn't break down for some cases?

calculus multivariable-calculus vector-analysis intuition

share|cite|improve this question

edited Mar 29 at 21:54

Don Thousand

4,574734

asked Mar 29 at 21:49

Chebotarev DensityChebotarev Density

695521

$endgroup$

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  $begingroup$
  PS: What are some good books to learnt the necessary multivariable calculus rigorously (not skipping the proof of eg Green's theorem) as quickly as possible for complex analysis background ? The first chapter of Griffiths EM contains some nice intuition but sadly totally devoid of proofs.
  $endgroup$
  – Chebotarev Density
  Mar 29 at 21:50
  

  
  
  
  


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  Another book I like a lot, not nearly as rigorous, is "Div, grad, curl, and all that: an informal text on vector calculus"
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  – Don Thousand
  Mar 29 at 21:52
  
  

  
  
  
  


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  Spivak’s “calculus on manifolds” is a masterpiece.
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  – Ittay Weiss
  Mar 29 at 21:53
  

  
  
  
  


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  So you are comfortable with two types of vector fields: divergence-free (incompressible fluids) and curl-free (potentials). Now smooth vector fields are the sum of a divergence-free part and a curl-free part, so it behaves like a fluid plus a potential.
  $endgroup$
  – Chrystomath
  Mar 30 at 9:10
  

  
  
  
  


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  @Chrystomath Isn't potential a scalar field ?
  $endgroup$
  – Chebotarev Density
  Mar 30 at 12:21
  

  
  
  
  

 | 
show 3 more comments

0

1

$begingroup$

How do you imagine vector fields in a "physical" way? I'm learning vector calculus, and most of the analogies break down:

• If I imagine the vector field as some sort of fluid flow, then if it's incompressible then the divergence is clearly zero everywhere. But for most vector fields the divergence is not zero, and I find compressible fluids very counterintuitive.




• If I imagine them as electric field of some configuration of charges, but then not all vector fields corresponds to the electric field of a configuration of charges (eg the field $f(x,y) = x hati + y hatj$), so those fields are again counterintuitive.

Which way should I think about them so that things like divergence/curl become obvious and the analogy doesn't break down for some cases?

calculus multivariable-calculus vector-analysis intuition

share|cite|improve this question

edited Mar 29 at 21:54

Don Thousand

4,574734

asked Mar 29 at 21:49

Chebotarev DensityChebotarev Density

695521

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  PS: What are some good books to learnt the necessary multivariable calculus rigorously (not skipping the proof of eg Green's theorem) as quickly as possible for complex analysis background ? The first chapter of Griffiths EM contains some nice intuition but sadly totally devoid of proofs.
  $endgroup$
  – Chebotarev Density
  Mar 29 at 21:50
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Another book I like a lot, not nearly as rigorous, is "Div, grad, curl, and all that: an informal text on vector calculus"
  $endgroup$
  – Don Thousand
  Mar 29 at 21:52
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Spivak’s “calculus on manifolds” is a masterpiece.
  $endgroup$
  – Ittay Weiss
  Mar 29 at 21:53
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  So you are comfortable with two types of vector fields: divergence-free (incompressible fluids) and curl-free (potentials). Now smooth vector fields are the sum of a divergence-free part and a curl-free part, so it behaves like a fluid plus a potential.
  $endgroup$
  – Chrystomath
  Mar 30 at 9:10
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @Chrystomath Isn't potential a scalar field ?
  $endgroup$
  – Chebotarev Density
  Mar 30 at 12:21
  

  
  
  
  

 | 
show 3 more comments

$begingroup$

How do you imagine vector fields in a "physical" way? I'm learning vector calculus, and most of the analogies break down:

• If I imagine the vector field as some sort of fluid flow, then if it's incompressible then the divergence is clearly zero everywhere. But for most vector fields the divergence is not zero, and I find compressible fluids very counterintuitive.




• If I imagine them as electric field of some configuration of charges, but then not all vector fields corresponds to the electric field of a configuration of charges (eg the field $f(x,y) = x hati + y hatj$), so those fields are again counterintuitive.

Which way should I think about them so that things like divergence/curl become obvious and the analogy doesn't break down for some cases?

calculus multivariable-calculus vector-analysis intuition

share|cite|improve this question

edited Mar 29 at 21:54

Don Thousand

4,574734

asked Mar 29 at 21:49

Chebotarev DensityChebotarev Density

695521

$endgroup$

share|cite|improve this question

edited Mar 29 at 21:54

Don Thousand

4,574734

asked Mar 29 at 21:49

Chebotarev DensityChebotarev Density

695521

share|cite|improve this question

edited Mar 29 at 21:54

Don Thousand

4,574734

asked Mar 29 at 21:49

Chebotarev DensityChebotarev Density

695521

edited Mar 29 at 21:54

Don Thousand

4,574734

edited Mar 29 at 21:54

Don Thousand

4,574734

asked Mar 29 at 21:49

Chebotarev DensityChebotarev Density

695521

asked Mar 29 at 21:49

Chebotarev DensityChebotarev Density

695521