The domain of the vector field $\mathbf { F } ( x , y , z ) = \ln x \mathbf { i } + \ln y \mathbf { j } + \ln z \mathbf { k }$ is

A) $\{ ( x , y , z ) : x y z 0 \}$
B) $\{ ( x , y , z ) : x y 0 \}$
C) $\{ ( x , y , z ) : x z 0 \}$
D) $\{ ( x , y , z ) : y z 0 \}$
E) $\{ ( x , y , z ) : x 0 , y 0 , z 0 \}$

Question

The domain of the vector field $\mathbf { F } ( x , y , z ) = \ln x \mathbf { i } + \ln y \mathbf { j } + \ln z \mathbf { k }$ is

A) $\{ ( x , y , z ) : x > y > z > 0 \}$
B) $\{ ( x , y , z ) : x y > 0 \}$
C) $\{ ( x , y , z ) : x z > 0 \}$
D) $\{ ( x , y , z ) : y z > 0 \}$
E) $\{ ( x , y , z ) : x > 0 , y > 0 , z > 0 \}$

Quizplus · Accepted Answer

The logarithm function $\ln(x)$ is defined for $x > 0$. Therefore, for the vector field to be defined, $x$, $y$, and $z$ must all be greater than 0.

The Domain of the Vector Field F(x,y,z)=ln⁡xi+ln⁡yj+ln⁡zk\mathbf { F } ( x , y , z ) = \ln x \mathbf { i } + \ln y \mathbf { j } + \ln z \mathbf { k }F(x,y,z)=lnxi+lnyj+lnzk

The Domain of the Vector Field $\mathbf { F } ( x , y , z ) = \ln x \mathbf { i } + \ln y \mathbf { j } + \ln z \mathbf { k }$