梯度几何与切平面

The gradient is perpendicular to level sets

Tangent lines to implicit curves, revisited

Tangent planes in both descriptions

For the level curve $x^2 + 3y^2 = 4$ at the point $(1, 1)$, which line is tangent to the curve there?

The surface $z = f(x, y)$ can be rewritten as the level set $F(x, y, z) = 0$ with $F(x, y, z) = f(x, y) - z$. What is $\nabla F$ at a point on the surface?

A surface is given as the level set $F(x, y, z) = c$. Which vector is normal to the tangent plane at a point on the surface?

The surface $(x-1)^2 + y^2 + z^2 = 9$ passes through the point $(3, 2, 2)$. The tangent plane there is $a(x - 3) + b(y - 2) + c(z - 2) = 0$. Give the value of $a$, the coefficient of the $x$ term, as a plain number.

A hill has height $h(x, y) = 100 - x^2 - 2y^2$ and you stand at the point $(2, 5)$, where the height is $100 - 4 - 50 = 46$. Give the first component of the gradient $\nabla h$ at your position, as a plain number.