# 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 and , measures how changes as changes for which is close to . The same goes for . Now let us consider a point with approaches , we get:

Next, we consider a point with approaches , we get:

However, if is smooth at :

We then get the below formula as approaches :

Therefore:

So we can use the above formula to easily compute an approximation of , for is near to . The formula also represents a tangent plane.