1. Topic 5
Curl

2. Curl
Curl of a vector 𝐹 = 𝐹 1 𝑖 + 𝐹 2 𝑗 + 𝐹 3 𝑘 𝑐𝑢𝑟𝑙 𝐹 = 𝛻 × 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝐹 1 𝐹 2 𝐹 3

3. Curl Curl of a vector 𝐹 = 𝐹 1 𝑖 + 𝐹 2 𝑗 + 𝐹 3 𝑘
𝑐𝑢𝑟𝑙 𝐹 = 𝛻 × 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝐹 1 𝐹 2 𝐹 3
𝛻 × 𝐹 =+ 𝑖 𝜕 𝐹 3 𝜕𝑦 − 𝜕 𝐹 2 𝜕𝑧 …

4. Curl Curl of a vector 𝐹 = 𝐹 1 𝑖 + 𝐹 2 𝑗 + 𝐹 3 𝑘
𝑐𝑢𝑟𝑙 𝐹 = 𝛻 × 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝐹 1 𝐹 2 𝐹 3
𝛻 × 𝐹 =+ 𝑖 𝜕 𝐹 3 𝜕𝑦 − 𝜕 𝐹 2 𝜕𝑧 …

5. Curl Curl of a vector 𝐹 = 𝐹 1 𝑖 + 𝐹 2 𝑗 + 𝐹 3 𝑘
𝑐𝑢𝑟𝑙 𝐹 = 𝛻 × 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝐹 1 𝐹 2 𝐹 3
𝛻 × 𝐹 =+ 𝑖 𝜕 𝐹 3 𝜕𝑦 − 𝜕 𝐹 2 𝜕𝑧 − 𝑗 𝜕 𝐹 3 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑧 …
Alternate sign

6. Curl Curl of a vector 𝐹 = 𝐹 1 𝑖 + 𝐹 2 𝑗 + 𝐹 3 𝑘
𝑐𝑢𝑟𝑙 𝐹 = 𝛻 × 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝐹 1 𝐹 2 𝐹 3
𝛻 × 𝐹 =+ 𝑖 𝜕 𝐹 3 𝜕𝑦 − 𝜕 𝐹 2 𝜕𝑧 − 𝑗 𝜕 𝐹 3 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑧 …

7. Curl Curl of a vector 𝐹 = 𝐹 1 𝑖 + 𝐹 2 𝑗 + 𝐹 3 𝑘
𝑐𝑢𝑟𝑙 𝐹 = 𝛻 × 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝐹 1 𝐹 2 𝐹 3
𝛻 × 𝐹 =+ 𝑖 𝜕 𝐹 3 𝜕𝑦 − 𝜕 𝐹 2 𝜕𝑧 − 𝑗 𝜕 𝐹 3 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑧 …

8. Curl Curl of a vector 𝐹 = 𝐹 1 𝑖 + 𝐹 2 𝑗 + 𝐹 3 𝑘
𝑐𝑢𝑟𝑙 𝐹 = 𝛻 × 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝐹 1 𝐹 2 𝐹 3
𝛻 × 𝐹 =+ 𝑖 𝜕 𝐹 3 𝜕𝑦 − 𝜕 𝐹 2 𝜕𝑧 − 𝑗 𝜕 𝐹 3 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑧 + 𝑘 𝜕 𝐹 2 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑦

9. Curl Curl of a vector 𝐹 = 𝐹 1 𝑖 + 𝐹 2 𝑗 + 𝐹 3 𝑘
𝑐𝑢𝑟𝑙 𝐹 = 𝛻 × 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝐹 1 𝐹 2 𝐹 3
𝛻 × 𝐹 =+ 𝑖 𝜕 𝐹 3 𝜕𝑦 − 𝜕 𝐹 2 𝜕𝑧 − 𝑗 𝜕 𝐹 3 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑧 + 𝑘 𝜕 𝐹 2 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑦

10. Curl Curl of a vector 𝐹 = 𝐹 1 𝑖 + 𝐹 2 𝑗 + 𝐹 3 𝑘
𝑐𝑢𝑟𝑙 𝐹 = 𝛻 × 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝐹 1 𝐹 2 𝐹 3
𝛻 × 𝐹 =+ 𝑖 𝜕 𝐹 3 𝜕𝑦 − 𝜕 𝐹 2 𝜕𝑧 − 𝑗 𝜕 𝐹 3 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑧 + 𝑘 𝜕 𝐹 2 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑦

11. Curl
The curl is a vector operator that describes the infinitesimal rotation of a vector field. At every point in the field, the curl of that point is represented by a vector. The attributes of the vector (length and direction) relate to the direction of rotation and strength of rotation.

12. Curl
𝛻 × 𝐹 = 𝜕 𝐹 3 𝜕𝑦 − 𝜕 𝐹 2 𝜕𝑧 𝑖 − 𝜕 𝐹 3 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑧 𝑗 + 𝜕 𝐹 2 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑦 𝑘

13. Curl
curl 𝐹 is a vector. The direction of curl 𝐹 is the axis of rotation (as shown by the right hand rule) and the magnitude of curl 𝐹 is the magnitude of rotation.

14. Curl
What is the curl for a vector field 𝐹 𝑥,𝑦 instead of 𝐹 𝑥,𝑦,𝑧 ? It becomes?
𝛻 × 𝐹 = 𝜕 𝐹 3 𝜕𝑦 − 𝜕 𝐹 2 𝜕𝑧 𝑖 − 𝜕 𝐹 3 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑧 𝑗 + 𝜕 𝐹 2 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑦 𝑘

15. Curl
For a vector field 𝐹 𝑥,𝑦 , 𝐹 3 =0 and 𝜕 𝐹 1 𝜕𝑧 = 𝜕 𝐹 2 𝜕𝑧 =0 therefore: becomes
𝛻 × 𝐹 = 𝜕 𝐹 3 𝜕𝑦 − 𝜕 𝐹 2 𝜕𝑧 𝑖 − 𝜕 𝐹 3 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑧 𝑗 + 𝜕 𝐹 2 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑦 𝑘
𝛻 × 𝐹 = 𝜕 𝐹 2 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑦 𝑘

16. Deeper into Curl

17. Deeper into Curl 𝛻 × 𝐹 = 𝜕 𝐹 2 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑦 𝑘
𝛻 × 𝐹 = 𝜕 𝐹 2 𝜕𝑥 − 𝜕 𝐹 1 𝜕𝑦 𝑘
= lim ℎ→0 𝐹 2 𝑥+ℎ,𝑦,𝑧 − 𝐹 2 (𝑥,𝑦,𝑧) ℎ − 𝐹 1 𝑥,𝑦+ℎ,𝑧 − 𝐹 1 (𝑥,𝑦,𝑧) ℎ
= lim ℎ→ ℎ 𝐹 2 𝑥+ℎ,𝑦,𝑧 − 𝐹 2 𝑥,𝑦,𝑧 − 𝐹 1 𝑥,𝑦+ℎ,𝑧 + 𝐹 1 (𝑥,𝑦,𝑧)

18. Curl
𝛻 × 𝐹 =0 No curl in the field means that there is no rotational component.

19. Curl
𝛻 × 𝐹 =0
𝛻 • 𝐹 =0

20. Curl
𝛻 × 𝐹 =0
𝛻 • 𝐹 >0

21. Curl
𝛻 × 𝐹 >0 Positive curl in the vector field shows an anti-clockwise rotation.

22. Curl
𝛻 × 𝐹 >0
𝛻 • 𝐹 =0

23. Curl
𝛻 × 𝐹 >0
𝛻 • 𝐹 >0

24. Curl

25. Curl
𝛻 × 𝐹 >0
𝛻 • 𝐹 >0

26. Curl
𝛻 × 𝐹 >0
𝛻 • 𝐹 >0

27. Curl
𝛻 × 𝐹 <0 Negative curl in the vector field shows a clockwise rotation.

28. Curl
𝛻 × 𝐹 <0
𝛻 • 𝐹 =0

29. Curl
𝐹 𝑥,𝑦 = −𝑦,𝑥
Determine 𝛻∙𝐹
𝜕 𝐹 1 𝜕𝑥 + 𝜕 𝐹 2 𝜕𝑦 =0+0
𝛻∙𝐹 =0

30. Curl 𝐹 𝑥,𝑦 = −𝑦,𝑥 𝐶𝑢𝑟𝑙 𝐹= 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 −𝑦 𝑥 0
𝐶𝑢𝑟𝑙 𝐹= 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 −𝑦 𝑥 0
= 𝜕 𝜕𝑦 (0)− 𝜕 𝜕𝑧 (𝑥) 𝑖 − 𝜕 𝜕𝑥 (0)− 𝜕 𝜕𝑧 (−𝑦) 𝑗 + 𝜕 𝜕𝑥 (𝑥)− 𝜕 𝜕𝑦 (−𝑦) 𝑘
=2 𝑘

31. Curl
𝐹 𝑥,𝑦 = 𝑥−𝑦,𝑥−𝑦

32. Curl

33. Curl 𝐹 𝑥,𝑦 = 𝑥−𝑦,𝑥−𝑦 𝑑𝑖𝑣 𝐹= 𝛻∙𝐹 = 𝜕 𝐹 1 𝜕𝑥 + 𝜕 𝐹 2 𝜕𝑦 =1−1=0
𝐶𝑢𝑟𝑙 𝐹= 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝑥−𝑦 𝑥−𝑦 0
= 0 𝑖 − 0 𝑗 + 𝜕 𝜕𝑥 (𝑥−𝑦)− 𝜕 𝜕𝑦 (𝑥−𝑦) 𝑘
=2 𝑘
𝑑𝑖𝑣 𝐹= 𝛻∙𝐹
= 𝜕 𝐹 1 𝜕𝑥 + 𝜕 𝐹 2 𝜕𝑦
=1−1=0

34. Curl
𝐹 𝑥,𝑦 = 𝑥−𝑦,𝑥+𝑦
Here is a vector field with both divergence and curl
Anti-clockwise curl = positive curl
Outward streaming = positive divergence.

35. Curl 𝐹 𝑥,𝑦 = 𝑥−𝑦,𝑥+𝑦 𝐶𝑢𝑟𝑙 𝐹= 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝑥−𝑦 𝑥+𝑦 0
𝐶𝑢𝑟𝑙 𝐹= 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝑥−𝑦 𝑥+𝑦 0
= 0 𝑖 − 0 𝑗 + 𝜕 𝜕𝑥 (𝑥+𝑦)− 𝜕 𝜕𝑦 (𝑥−𝑦) 𝑘
=2 𝑘
𝑑𝑖𝑣 𝐹= 𝛻∙𝐹
= 𝜕 𝐹 1 𝜕𝑥 + 𝜕 𝐹 2 𝜕𝑦
=1+1=2

36. Curl 𝐹 𝑥,𝑦 = 𝑥− 𝑦 2 +2,𝑥𝑦+𝑦 Red point Orange point
positive divergence: pointing out
negative curl: vectors rotated clockwise
Orange point
Positive divergence: pointing out
Positive curl: vectors rotated clockwise
Curl of this vector field depends on the sign of its y-coordinate at any point. 

37. Curl
𝐹 𝑥,𝑦 = 𝑥− 𝑦 2 +2,𝑥𝑦+𝑦
𝐶𝑢𝑟𝑙 𝐹= 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝑥− 𝑦 2 +2 𝑥𝑦+𝑦 0
= 0 𝑖 + 0 𝑗 + 𝜕 𝜕𝑥 (𝑥𝑦+𝑦)− 𝜕 𝜕𝑦 (𝑥− 𝑦 2 +2) 𝑘
=(𝑦+2𝑦) 𝑘 =(3𝑦) 𝑘

38. Curl Example question: A vector is given by the following function
𝐹 𝑥,𝑦,𝑧 = 𝑥 𝑒 𝑥 −4𝑧 𝑖 −𝑦𝑧 𝑗 − 𝑥 𝑧 𝑘
Evaluate the curl of 𝐹 and identify the locations where there is no rotation.

39. Curl 𝐹 𝑥,𝑦,𝑧 = 𝑥 𝑒 𝑥 −4𝑧 𝑖 −𝑦𝑧 𝑗 − 𝑥 3 3 +4𝑧 𝑘
𝐹 𝑥,𝑦,𝑧 = 𝑥 𝑒 𝑥 −4𝑧 𝑖 −𝑦𝑧 𝑗 − 𝑥 𝑧 𝑘
Evaluate the curl of 𝐹 and find where no rotation.

40. Curl
𝛻 × 𝐹 =𝑦 𝑖 +( 𝑥 2 −4) 𝑗
𝛻 × 𝐹 =0 at (2,0) and (-2,0)

41. Curl
div 𝐹 and curl 𝐹 give us the mathematical tools to describe any vector field in terms of the extent to which it behaves like a source/sink and its rotation.

42. Sophie Germain Born in Paris, France, 1776 Wealthy silk merchants
Aged 13 the French Revolution was in full swing
House bound during revolution so delved into parents library and took an interest in mathematics
Parents didn’t want her to study mathematics.
She used to wrap up in blankets at night and study

43. Sophie Germain Born in Paris, France, 1776 Wealthy silk merchants
Aged 13 the French Revolution was in full swing
House bound during revolution so delved into parents library and took an interest in mathematics
Parents didn’t want her to study mathematics.
She used to wrap up in blankets at night and study
Corresponded with Gauss as M. LeBlanc
Contributed to seeking a solution to Fermat’s last theorem and important work on elasticity.
During Napoleonic wars, she had military sent to ensure Gauss was okay during the siege of his village.