Question

Find the curl of ~V
~V
= sin(x) cos(y) tan(z) i + x^2y^2z^2 j + x^4y^4z^4 k

Answer 1

Given

`\vec v = f(x,y,z)\,\vec\imath+g(x,y,z)\,\vec\jmath+h(x,y,z)\,\vec k \\\\ \vec v = \sin(x)\cos(y)\tan(z)\,\vec\imath + x^2y^2z^2\,\vec\jmath+x^4y^4z^4\,\vec k`

the curl of
`\vec v` is

`\displaystyle \\abla*\vec v = \left((\partial h)/(\partial y)-(\partial g)/(\partial z)\right)\,\vec\imath - \left((\partial h)/(\partial x)-(\partial f)/(\partial z)\right)\,\vec\jmath + \left((\partial g)/(\partial x)-(\partial f)/(\partial y)\right)\,\vec k`

`\\abla*\vec v = \left(4x^4y^3z^4-2x^2y^2z\right)\,\vec\imath \\\\ - \left(4x^3y^4z^4-\sin(x)\cos(y)\sec^2(z)\right)\,\vec\jmath \\\\ + \left(2xy^2z^2+\sin(x)\sin(y)\tan(z)\right)\,\vec k`

`\\abla*\vec v = \left(4x^4y^3z^4-2x^2y^2z\right)\,\vec\imath \\\\ + \left(\sin(x)\cos(y)\sec^2(z)-4x^3y^4z^4\right)\,\vec\jmath \\\\ + \left(2xy^2z^2+\sin(x)\sin(y)\tan(z)\right)\,\vec k`