Who has potential?

Calculus Level 4

True or False?

(I) The vector field F ⃗ ( x , y , z ) = 1 x 2 + y 2 ( − y , x , 0 ) \vec{F}(x,y,z)=\frac{1}{x^2+y^2}(-y,x,0) has a vector potential G ⃗ \vec{G} , defined when x 2 + y 2 > 0 x^2+y^2>0 , meaning that F ⃗ = ∇ × G ⃗ \vec{F}=\nabla \times \vec{G} .

(II) The vector field F ⃗ ( x , y , z ) = ( x 2 + y 2 + z 2 ) − 3 / 2 ( x , y , z ) \vec{F}(x,y,z)=(x^2+y^2+z^2)^{-3/2}(x,y,z) has a vector potential, defined when ( x , y , z ) ≠ ( 0 , 0 , 0 ) (x,y,z)\neq(0,0,0) .

(I) is false, (II) is true (I) is true, (II) is false both are true both are false Cannot be determined

Otto Bretscher by Otto Bretscher , 65, USA

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