Which of the following vector fields has the following properties: 1)the largest value of its circulation density at (1,2,1)is 50
2)the largest value of its circulation density at (1,2,1)occurs around the direction $\frac { 3 } { 5 } \vec { i } + \frac { 4 } { 5 } \vec { j }$
A) $\vec { F } _ { 1 } = - y ^ { 2 } x \vec { i } - z ^ { 3 } \vec { j } + z ^ { 2 } \vec { k }$
B) $\vec { F _ { 2 } } = \frac { 1 } { 100 } \left( - y ^ { 2 } \times \vec { i } - z ^ { 3 } \vec { j } + z ^ { 2 } \vec { k } \right)$
C) $ { \vec { F } _ { 3 } } = \frac { 1 } { 10 } \left( - y ^ { 2 } x \vec { i } - z ^ { 3 } \vec { j } + z ^ { 2 } \vec { k } \right)$
D) $ { \vec { F } _ { 4 } } = 10 \left( - y ^ { 2 } x \vec { i } - z ^ { 3 } \vec { j } + z ^ { 2 } \vec { k } \right)$
E) $\vec { F _ { 5 } } = 100 \left( - y ^ { 2 } x \vec { i } - z ^ { 3 } \vec { j } + z ^ { 2 } \vec { k } \right)$

Question

Which of the following vector fields has the following properties: 1)the largest value of its circulation density at (1,2,1)is 50
2)the largest value of its circulation density at (1,2,1)occurs around the direction $\frac { 3 } { 5 } \vec { i } + \frac { 4 } { 5 } \vec { j }$
A) $\vec { F } _ { 1 } = - y ^ { 2 } x \vec { i } - z ^ { 3 } \vec { j } + z ^ { 2 } \vec { k }$
B) $\vec { F _ { 2 } } = \frac { 1 } { 100 } \left( - y ^ { 2 } \times \vec { i } - z ^ { 3 } \vec { j } + z ^ { 2 } \vec { k } \right)$
C) $  { \vec { F } _ { 3 } } = \frac { 1 } { 10 } \left( - y ^ { 2 } x \vec { i } - z ^ { 3 } \vec { j } + z ^ { 2 } \vec { k } \right)$
D) $  { \vec { F } _ { 4 } } = 10 \left( - y ^ { 2 } x \vec { i } - z ^ { 3 } \vec { j } + z ^ { 2 } \vec { k } \right)$
E) $\vec { F _ { 5 } } = 100 \left( - y ^ { 2 } x \vec { i } - z ^ { 3 } \vec { j } + z ^ { 2 } \vec { k } \right)$

Quizplus · Accepted Answer

The circulation density of a vector field around a point in a specific direction is given by the curl of the vector field at that point dotted with the unit vector in the specified direction. The curl of $\vec{F} = -y^2x\vec{i} - z^3\vec{j} + z^2\vec{k}$ is a vector that represents the rotation of $\vec{F}$ at a point. The given direction is $\frac{3}{5}\vec{i} + \frac{4}{5}\vec{j}$, which is a unit vector. To find the vector field with the largest value of circulation density at (1,2,1) in this direction, we need to calculate the curl of each vector field and then dot it with the given direction vector. However, since all options are scalar multiples of the same vector field, the one with the largest scalar multiple will have the largest circulation density, assuming the direction of maximum circulation density aligns with the given direction. Option D, being scaled by 10, will have the largest circulation density at the specified point and direction compared to the other options, which are scaled by smaller factors or, in the case of option E, a larger factor but not relevant to the maximum condition specified.

Which of the Following Vector Fields Has the Following Properties 35i⃗+45j⃗\frac { 3 } { 5 } \vec { i } + \frac { 4 } { 5 } \vec { j }53​i+54​j​

Which of the Following Vector Fields Has the Following Properties $\frac { 3 } { 5 } \vec { i } + \frac { 4 } { 5 } \vec { j }$