Topic 5 Curl

Curl So far: 𝛻 𝑓(𝑥,𝑦,𝑧)= 𝜕𝑓 𝜕𝑥 𝑖 + 𝜕𝑓 𝜕𝑦 𝑗 + 𝜕𝑓 𝜕𝑧 𝑘 𝛻 ∙ 𝐹 = 𝜕 𝐹 1 𝜕𝑥 + 𝜕 𝐹 2 𝜕𝑦 + 𝜕 𝐹 3 𝜕𝑧

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

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 …

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 …

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 …

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

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 …

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 …

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

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

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

Curl The curl is a vector result that describes the rotation locally of a vector field. At every point in the field, the curl of that point is represented by a vector. The attributes of this vector (magnitude and direction) characterize the rotation at that point.

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

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.

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 related to the magnitude of rotation.

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

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

Deeper into Curl

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→0 1 ℎ 𝐹 2 𝑥+ℎ,𝑦,𝑧 − 𝐹 2 𝑥,𝑦,𝑧 − 𝐹 1 𝑥,𝑦+ℎ,𝑧 + 𝐹 1 (𝑥,𝑦,𝑧)

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

Curl 𝛻 × 𝐹 =0 𝛻 · 𝐹 =0

Curl 𝛻 × 𝐹 =0 𝛻 · 𝐹 >0

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

Curl 𝛻 × 𝐹 >0 𝛻 · 𝐹 =0

Curl 𝛻 × 𝐹 >0 𝛻 · 𝐹 >0

Curl

Curl 𝛻 × 𝐹 >0 𝛻 · 𝐹 >0

Curl 𝛻 × 𝐹 >0 𝛻 · 𝐹 >0

Curl 𝛻 × 𝐹 >0 𝛻 · 𝐹 >0

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

Curl 𝛻 × 𝐹 <0 𝛻 · 𝐹 =0

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

Curl 𝐹 𝑥,𝑦 = −𝑦,𝑥 𝐶𝑢𝑟𝑙 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 −𝑦 𝑥 0 𝐹 𝑥,𝑦 = −𝑦,𝑥 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

Curl 𝐹 𝑥,𝑦 = 𝑥−𝑦,𝑥−𝑦

Curl Curl doesn’t have to mean circulation round a point

Curl 𝐹 𝑥,𝑦 = 𝑥−𝑦,𝑥−𝑦 𝑑𝑖𝑣 𝐹 = 𝛻 ∙ 𝐹 = 𝜕 𝐹 1 𝜕𝑥 + 𝜕 𝐹 2 𝜕𝑦 =1−1=0 𝐹 𝑥,𝑦 = 𝑥−𝑦,𝑥−𝑦 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

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.

Curl 𝐹 𝑥,𝑦 = 𝑥−𝑦,𝑥+𝑦 𝐶𝑢𝑟𝑙 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝑥−𝑦 𝑥+𝑦 0 𝐹 𝑥,𝑦 = 𝑥−𝑦,𝑥+𝑦 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

Curl 𝐹 𝑥,𝑦 = 𝑥− 𝑦 2 +2,𝑥𝑦+𝑦 Red point Orange point 𝐹 𝑥,𝑦 = 𝑥− 𝑦 2 +2,𝑥𝑦+𝑦 Red 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. 

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

Curl Example question: A vector is given by the following function 𝐹 𝑥,𝑦,𝑧 = 𝑥 𝑒 𝑥 −4𝑧 𝑖 −𝑦𝑧 𝑗 − 𝑥 3 3 +4𝑧 𝑘 Evaluate the curl of 𝐹 and identify the locations where there is no rotation.

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 3 3 +4𝑧 𝑘 Evaluate the curl of 𝐹 and find where no rotation.

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

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.

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

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 and an essay on the subject that won her the Paris Academy of Sciences Prize. During Napoleonic wars, she had military sent to ensure Gauss’ safety during the siege of his village.

Sophie Germain The taste for the abstract sciences in general and, above all, for the mysteries of numbers, is very rare: this is not surprising, since the charms of this sublime science in all their beauty reveal themselves only to those who have the courage to fathom them. But when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarizing herself with their knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius.