What is the curl of the vector field 2x²yi + 5z²j - 4yzk?

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What is the curl of the vector field 2x²yi + 5z²j - 4yzk?

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$- 14 z i - 2 x^{2} k$
$6 z i + 4 x j - 2 x^{2} k$
$6 z i + 8 x y + 2 x^{2} y k$
$- 14 z i + 6 y j + 2 x^{2} k$

Answer 1

Concept:

Consider the vector field

$\vec{V} = 2 x^{2} y i + 5 z^{2} j - 4 y z k$

$c u r l \vec{V} = \nabla \times \vec{V} = | \begin{matrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 2 x^{2} y & 5 z^{2} & - 4 y z \end{matrix} |$

$\Rightarrow C u r l \vec{V} = i [\frac{\partial}{\partial y} (- 4 y z) - \frac{\partial}{\partial z} (5 z^{2})] - j [\frac{\partial}{\partial x} (- 4 y z) - \frac{\partial}{\partial z} (2 x^{2} y)] + k [\frac{\partial}{\partial x} (5 z^{2}) - \frac{\partial}{\partial y} (2 x^{2} y)]$

$\Rightarrow C u r l \vec{V} = i [- 4 z - 10 z] - j [0] + k [- 2 x^{2}]$

\Rightarrow C u r l \vec{V} = - 14 z i - 2 x^{2} k

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What is the curl of the vector field 2x²yi + 5z²j - 4yzk?

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Question

What is the curl of the vector field 2x2yi + 5z2j - 4yzk?

Answer (Detailed Solution Below)

Detailed Solution

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What is the curl of the vector field 2x²yi + 5z²j - 4yzk?