Question #229231

If Ø(x, y, z) = x² - y²-2²-2, then ∇Ø at (2,1,-1) is ,,,choose the correct answer?

A) 4i +2j-2k

B)4i-2j-2k

C) 4i + 2j + 2k

D)4i - 2j +2k

Expert's answer

By definition, "curl(x\u00b2 - y\u00b2 -z\u00b2-2)=\u2207\u00d7(x\u00b2 - y\u00b2-z\u00b2-2)" or, equivalently,

"\u2207\u00d7(x\u00b2 - y\u00b2-z\u00b2-2)= \\begin{vmatrix}\n i & j & k\\\\\n \\frac{\\partial }{\\partial x} & \\frac{\\partial }{\\partial y} & \\frac{\\partial }{\\partial z}\\\\\nx^2 & y^2 & z^2-2\n\\end{vmatrix}"

"curl(x^2,y^2,z^2-2) =(\\frac{\u2202}{\u2202y}(y^2)\u2212\\frac{\u2202}{\u2202z}(z^2-2),\\frac{\u2202}{\u2202z}(z^2-2)\u2212\\frac{\u2202}{\u2202x}(x^2),\\frac{\u2202}{\u2202x}(x^2)\u2212\\frac{\u2202}{\u2202y}(y^2))\\\\"

Now, just plug in the found partial derivatives to get the curl:

"curl(x^2,y^2,z^2-2)=(2y-2z,2z-2x,2x-2y)."

Finally, find the curl at the specific point.

"(curl(x^2,y^2,z^2-2))|((x_0,y_0,z_0)=(2,1,-1))=(4,-6,2)"

The answer should be 4i - 6j+2k

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