[amp_mcq option1=”\\[4{\\rm{\\hat i}} – {\\rm{\\hat j}}\\]” option2=”\\[4{\\rm{\\hat i}} – {\\rm{\\hat k}}\\]” option3=”\\[{\\rm{\\hat i}} – 4{\\rm{\\hat j}}\\]” option4=”\\[{\\rm{\\hat i}} – 4{\\rm{\\hat k}}\\]” correct=”option4″]<\/p>\n
The correct answer is $\\boxed{\\rm{\\hat i} – 4{\\rm{\\hat j}}}$.<\/p>\n
The vorticity vector is defined as the curl of the velocity vector, which is given by:<\/p>\n
$$\\omega = \\nabla \\times \\overrightarrow V$$<\/p>\n
In Cartesian coordinates, the curl can be written as:<\/p>\n
$$\\omega = \\left( \\dfrac{\\partial Q}{\\partial x} – \\dfrac{\\partial P}{\\partial y} \\right) {\\rm{\\hat i}} + \\left( \\dfrac{\\partial P}{\\partial z} – \\dfrac{\\partial Q}{\\partial x} \\right) {\\rm{\\hat j}} + \\left( \\dfrac{\\partial R}{\\partial y} – \\dfrac{\\partial Q}{\\partial z} \\right) {\\rm{\\hat k}}$$<\/p>\n
where $P$, $Q$, and $R$ are the $x$, $y$, and $z$ components of the velocity vector, respectively.<\/p>\n
In this case, the velocity vector is given by:<\/p>\n
$$\\overrightarrow V = 2xy\\hat i – {{\\rm{x}}^2}{\\rm{z\\hat j}}$$<\/p>\n
Therefore, the components of the vorticity vector are:<\/p>\n
$$\\begin{align} Therefore, the vorticity vector is given by:<\/p>\n $$\\omega = 2y{\\rm{\\hat i}} – 2z{\\rm{\\hat j}}$$<\/p>\n Evaluating the vorticity vector at the point $(1, 1, 1)$ gives:<\/p>\n $$\\omega = (2)(1){\\rm{\\hat i}} – (2)(1){\\rm{\\hat j}} = {\\rm{\\hat i}} – 2{\\rm{\\hat j}}$$<\/p>\n","protected":false},"excerpt":{"rendered":" [amp_mcq option1=”\\[4{\\rm{\\hat i}} – {\\rm{\\hat j}}\\]” option2=”\\[4{\\rm{\\hat i}} – {\\rm{\\hat k}}\\]” option3=”\\[{\\rm{\\hat i}} – 4{\\rm{\\hat j}}\\]” option4=”\\[{\\rm{\\hat i}} – 4{\\rm{\\hat k}}\\]” correct=”option4″]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"class_list":["post-20273","post","type-post","status-publish","format-standard","hentry","category-calculus","no-featured-image-padding"],"yoast_head":"\n
\n\\dfrac{\\partial Q}{\\partial x} – \\dfrac{\\partial P}{\\partial y} &= 2y – 0 = 2y \\
\n\\dfrac{\\partial P}{\\partial z} – \\dfrac{\\partial Q}{\\partial x} &= -2xz = -2z \\
\n\\dfrac{\\partial R}{\\partial y} – \\dfrac{\\partial Q}{\\partial z} &= 0 – 0 = 0
\n\\end{align<\/em>}$$<\/p>\n