I am currently enrolled in a university level Multivariable Calculus course in which my professor taught finding the tangent plane at a point on a surface the following way:
To find the tangent plane at point $(x_0,y_0,z_0)$ on the surface $ z=x^2+y^2 $ set $f(x,y,z)=x^2+y^2-z=0$ and then find $\nabla f(x_0,y_0,z_0) $.
According to his method, $\nabla f $ would be the following…
$$
\nabla f = \begin{bmatrix} \frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \\ \frac{\partial f}{\partial z} \end{bmatrix} = \begin{bmatrix} 2x \\ 2y \\ -1 \end{bmatrix}
$$
While this indeed is an expression for the tangent plane's normal vector, I don't understand how this is a mathematically valid statement. If $f(x,y,z)=0$, than how can the rates of instantaneous change (i.e.$\frac{\partial f}{\partial x}$) ever be anything other than zero?
Is $f(x,y,z)=x^2+y^2-z=0$ a valid function? Why would the partials not always be equal to zero? Am I thinking about partial derivatives incorrectly? Is this just a mathematical "hack" to serve as a shortcut? If this is the case, is there a more mathematically rigorous method to find the tangent plane of a surface?
Thank you in advance.
Best Answer
You have quite a few questions there. To begin with, $f:(x,y,z)\mapsto x^2+y^2-z$ is certainly a valid function from $\mathbb R^3$ to $\mathbb R$. Its partial derivatives clearly don’t all vanish everywhere. The expression $x^2+y^2-z=0$, on the other hand, describes one of $f$’s level (hyper)surfaces. It’s a basic property of the gradient operator that the gradient of a function is always normal to its level surfaces. Your professor is taking advantage of this property.