Approximation of a function of two variables

[AAMAIK]

TL;DR Summary

Why the assumptions, that the tangent plane must pass through the point we wish to approximate the function nearby and the plane has the same slopes in the i and j directions as the surface does will make the tangent plane a good approximation to the surface?

In a manner analogues to the linearization of functions of a single variable to approximate the value of a function of two variables in the neighbourhood of a given point (x0,y0,z0) where z=f(x,y) using a tangent plane. The tangent plane must pass through the point we wish to approximate z nearby and the second condition which I don't quite understand why it makes the tangent plane a good approximate is that it must have the same slopes as the surface in the i and j directions. Therefore Z≈ z0+fx(x-x0)+fy(y-y0)
I can understand that the equation of the tangent plane will serve as a good approximation if we are changing x and keeping y=y0 or vice versa. Namely, if we are varying y and keeping x=x0. Then the lines z=z0+fx(x-x0) and z=z0+fy(y-y0) of the line will serve as a good approximation to the curve traced in the vertical plane y=0 and x=x0. But since x and y are free to take any values what if we decided to change both x and y how can I prove that the tangent plane with these set of conditions will be a good approximation to the curve.

 

[jedishrfu]

Here's a discussion on it:

http://tutorial.math.lamar.edu/Classes/CalcIII/TangentPlanes.aspx

and Khan Academy on it:

https://www.khanacademy.org/math/mu...local-linearization/v/what-is-a-tangent-plane

An aside: The Khan video is by Grant Sanderson aka 3blue1brown on Youtube. His youtube channel has some great math videos on the Essence of Calculus.

The way I look at it is the tangent plane at some point x,y is the plane that holds the tangent line of the curve in the $XZ$ plane and holds the tangent line of the curve in the $YZ$ plane at a given x and y value respectively for the function $z=f(x,y)$

 

[Svein]

In matrix notation: $\Delta z(x_{0}, y_{0})\approx \begin{pmatrix} \frac{\partial z}{\partial x}\ & \frac{\partial z}{\partial y} \ \end{pmatrix}_{x_{0},y_{0}}\begin{pmatrix} \Delta x \\ \Delta y \ \end{pmatrix}$

 

[AAMAIK]

Here's a discussion on it:

http://tutorial.math.lamar.edu/Classes/CalcIII/TangentPlanes.aspx

and Khan Academy on it:

https://www.khanacademy.org/math/mu...local-linearization/v/what-is-a-tangent-plane

An aside: The Khan video is by Grant Sanderson aka 3blue1brown on Youtube. His youtube channel has some great math videos on the Essence of Calculus.

The way I look at it is the tangent plane at some point x,y is the plane that holds the tangent line of the curve in the $XZ$ plane and holds the tangent line of the curve in the $YZ$ plane at a given x and y value respectively for the function $z=f(x,y)$

I read the first link but they don't answer my question which is that x and y are free to take any values and that I can understand why the tangent plane approximation would serve as a good approximation to the surface at some specific point if we were to vary x or y holding the other as constant.
The second link video series also laid down the criteria to approximation and how the criteria (slopes in the i and j direction=slope of the surface) narrows down the many candidates for the tangent plane approximation and that the tangent plane passing through the point of approximation alone is not enough.