Definition of curl

  • #1
Titan97
Gold Member
451
18
In a river, water flows faster in the middle and slower near the banks of the river and hence, if I placed a twig, it would rotate and hence, the vector field has non-zero Curl.
Curl{v}=∇×v
But I am finding it difficult to interpret the above expression geometrically. In scalar fields, the gradient points along the direction of maximum increase. But what's the direction of gradient in a vector field? And why does the cross product give the Curl?
 

Answers and Replies

  • #2
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,829
6,643
It is not a cross product, it simply happens to have a similar form in Cartesian coordinates.

You can think of (the inner product of) the curl (with a normal to the plane) as being a measure of the line integral around a small planar loop. The direction of the curl is the direction which will maximise this line integral.

This is analogous to how you may see divergence as a measure of the net flow out of the closed surface surrounding a small volume.
 
  • Like
Likes Titan97

Related Threads on Definition of curl

Replies
5
Views
5K
Replies
3
Views
1K
Replies
4
Views
1K
  • Last Post
Replies
5
Views
5K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
8
Views
3K
  • Last Post
Replies
2
Views
487
  • Last Post
Replies
3
Views
1K
Top