Question

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

Answer 1

By definition, $curl(x² - y² -z²-2)=∇×(x² - y²-z²-2)$ or, equivalently,

$∇×(x² - y²-z²-2)= \begin{vmatrix} i & j & k\\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ x^2 & y^2 & z^2-2 \end{vmatrix}$

$curl(x^2,y^2,z^2-2) =(\frac{∂}{∂y}(y^2)−\frac{∂}{∂z}(z^2-2),\frac{∂}{∂z}(z^2-2)−\frac{∂}{∂x}(x^2),\frac{∂}{∂x}(x^2)−\frac{∂}{∂y}(y^2))\\$

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