Question

Find the unit upward normal vector to the ellipsoid 4x^2 + y^2 + z^2 = 9 at the point P(1,2,-1)

Answer 1

i somehow know the steps to solve this question, but i couldn't understand it completely. any kind soul can explain it to me?\[\nabla f(1,2,-1)=<\frac{\delta f}{\delta x}, \frac{\delta f}{\delta y}, \frac{\delta f}{\delta z}> |_{(1,2,-1)}\]\[\nabla f(1,2,-1)=<8,4,-2>\]right after this im confused on the solution.

Answer 2

how does the statement \['\nabla f(1,2,-1) is perpendicular \to f=0 at P(1,2,-1)'\]related to the drawing?

Answer 3

gradient is defined as the rate of maximum change .... along the maximum change

Answer 4

so the gradient of f at (1,2,-1) is the arrow pointing outwards?