An intuition to an tangent plane to a surface

To me: Page 126 of Cracking the GRE Math Subject Test.

A rigorous proof can be found here (Calculus, Howard Anton). A note of stress should be on “any smooth curve C on the surface”. Below is my attempt interpretation before looking at any proofs. While it cannot be considered as a proof, it shows my intuition which can be fallacious.

Let P = (x_{0}, y_{0}) and z = f(x, y), \left. \frac{\partial f}{\partial x} \right |_{x_{0}} measures how z changes as x changes for x which is close to x_{0}. The same goes for y. Now let us consider a point P_{1} = (x_{1}, y_{0}) with x_{1} approaches x_{0}, we get:

    \[ z_{1} - z_{0} \approx \left. \frac{\partial f}{\partial x} \right |_{(x_{0}, y_{0})} (x_{1} - x_{0}) \]

Next, we consider a point P_{2} = (x_{1}, y_{1}) with y_{1} approaches y_{0}, we get:

    \[ z_{2} - z_{1} \approx \left. \frac{\partial f}{\partial y} \right |_{(x_{1}, y_{0})} (y_{1} - y_{0}) \]

However, if f is smooth at x = x_{0}:

    \[ \lim_{x_{1} \approx x_{0}} {\left. \frac{\partial f}{\partial y} \right |_{(x_{1}, y_{0})}} = \left. \frac{\partial f}{\partial y} \right |_{(x_{0}, y_{0})} \]

We then get the below formula as x_{1} approaches x_{0}:

    \[ z_{2} - z_{1} \approx \left. \frac{\partial f}{\partial y} \right |_{(x_{0}, y_{0})} (y_{1} - y_{0}) \]


    \[ z_{2} - z_{1} + z_{1} - z_{0} \approx \left. \frac{\partial f}{\partial y} \right |_{(x_{0}, y_{0})} (y_{1} - y_{0}) + \left. \frac{\partial f}{\partial x} \right |_{(x_{0}, y_{0})} (x_{1} - x_{0})\\ \Leftrightarrow z_{2} - z_{0} \approx \left. \frac{\partial f}{\partial x} \right |_{(x_{0}, y_{0})} (x_{1} - x_{0}) + \left. \frac{\partial f}{\partial y} \right |_{(x_{0}, y_{0})} (y_{1} - y_{0}) \]

So we can use the above formula to easily compute an approximation of f(x_{1}, y_{1}), for (x_{1}, y_{1}) is near to (x_{0}, y_{0}). The formula also represents a tangent plane.


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