The function f , g : R 3 → R {\displaystyle \displaystyle f,\ g:\mathbb {R} ^{3}\rightarrow \mathbb {R} } are
f ( x , y , z ) = 2 ( x + 1 ) 2 + y 2 + 2 ( z − 1 ) 2 g ( x , y , z ) = x 2 + y 2 + z 2 {\displaystyle \displaystyle {\begin{aligned}f(x,y,z)&=2(x+1)^{2}+y^{2}+2(z-1)^{2}\\g(x,y,z)&=x^{2}+y^{2}+z^{2}\end{aligned}}}
Hence, ∇ f = ( 4 ( x + 1 ) 2 y 4 ( z − 1 ) ) {\displaystyle \displaystyle \nabla f={\begin{pmatrix}4(x+1)\\2y\\4(z-1)\end{pmatrix}}} and ∇ g = ( 2 x 2 y 2 z ) {\displaystyle \displaystyle \nabla g={\begin{pmatrix}2x\\2y\\2z\end{pmatrix}}}
To be parallel, ∇ f {\displaystyle \displaystyle \nabla f} and ∇ g {\displaystyle \displaystyle \nabla g} must satisfy
∇ f = λ ∇ g {\displaystyle \displaystyle \nabla f=\lambda \nabla g}
for a λ ∈ R {\displaystyle \displaystyle \lambda \in \mathbb {R} } .
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