Let $f ( x , y , z ) = e ^ { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ , find curl $( \nabla f )$ at the point $( 1,1,1 )$ .
A) $e ^ { 3 } \mathbf { i }$
B)$e ^ { 3 } 3 \mathrm { j }$

C) $e ^ { 3 } \mathbf { k }$
D)$e ^ { 3 } ( \mathbf { i } + \mathbf { j } )$
E)$e ^ { 3 } ( \mathbf { i } + \mathbf { j } )$.
F)$e ( \mathbf { i } + \mathbf { j } + \mathbf { k } )$
G)$e ^ { 6 } ( \mathbf { i } + \mathbf { j } + \mathbf { k } )$
H) $0$

Question

Let $f ( x , y , z ) = e ^ { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ , find curl $( \nabla f )$ at the point $( 1,1,1 )$ .
A) $e ^ { 3 } \mathbf { i }$ 
B)$e ^ { 3 } 3 \mathrm { j }$

C) $e ^ { 3 } \mathbf { k }$
D)$e ^ { 3 } ( \mathbf { i } + \mathbf { j } )$
E)$e ^ { 3 } ( \mathbf { i } + \mathbf { j } )$.
F)$e ( \mathbf { i } + \mathbf { j } + \mathbf { k } )$ 
G)$e ^ { 6 } ( \mathbf { i } + \mathbf { j } + \mathbf { k } )$ 
H) $0$

Quizplus · Accepted Answer

The curl of the gradient of any scalar field, including $f(x, y, z) = e^{x^2 + y^2 + z^2}$, is always zero. This is a direct consequence of the properties of curl and gradient in vector calculus.

Let f(x,y,z)=ex2+y2+z2f ( x , y , z ) = e ^ { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }f(x,y,z)=ex2+y2+z2

Let $f ( x , y , z ) = e ^ { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$