Remarks by Fortney Following Theorems on Directional Derivative ....

[Math Amateur]

I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Chapter 2: An Introduction to Differential Forms ...I need help to understand some remarks by Fortney following Theorems 2.1 and 2.2 on the directional derivative ...

Those remarks by Fortney read as follows: (for more text showing the context of the remarks including the two Theorems ... see scanned text below ...)View attachment 8776
In the above text by Fortney we read the following:

" ... ... From this it is straightforward to show that the equation of a plane through the point \(\displaystyle (x_0, y_0, f( x_0, y_0) )\) on the graph of \(\displaystyle f\) is given by

\(\displaystyle z - f( x_0, y_0) = m_x ( x - x_0 ) + m_y ( y - y_0 )\)
... ... "Can someone please explain/demonstrate exactly how the above equation arises or comes about ...?
Help will be appreciated ...

Peter========================================================================================So as to make clear the context of the above qestion I am providing Fortney's text before and after the text provided above ... as follows:View attachment 8777
View attachment 8776
View attachment 8778Hope that helps ...Peter

 

[joypav]

In the above text by Fortney we read the following:

" ... ... From this it is straightforward to show that the equation of a plane through the point \(\displaystyle (x_0, y_0, f( x_0, y_0) )\) on the graph of \(\displaystyle f\) is given by

\(\displaystyle z - f( x_0, y_0) = m_x ( x - x_0 ) + m_y ( y - y_0 )\)
... ... "Can someone please explain/demonstrate exactly how the above equation arises or comes about ...?

Do you already understand where the equation given for a plane through the origin comes from? If so, it's just a translation!
Fortney gives that the equation for a plane through the origin is...
$z = m_xx + m_yy$
Note that what we have here is...
$(z-z_0) = m_x(x-x_0) + m_y(y-y_0)$ where $(x_0, y_0, z_0) = (0, 0, 0)$ (the origin).

So in order to get a plane through our point $(x_0, y_0, f(x_0, y_0))$, we can translate the origin to this point by simply replacing $x$ with $x-x_0$, $y$ with $y-y_0$, and $z$ with $z-z_0$, giving us
$z - f(x_0, y_0) = m_x(x-x_0) + m_y(y-y_0)$

 

[Math Amateur]

Do you already understand where the equation given for a plane through the origin comes from? If so, it's just a translation!
Fortney gives that the equation for a plane through the origin is...
$z = m_xx + m_yy$
Note that what we have here is...
$(z-z_0) = m_x(x-x_0) + m_y(y-y_0)$ where $(x_0, y_0, z_0) = (0, 0, 0)$ (the origin).

So in order to get a plane through our point $(x_0, y_0, f(x_0, y_0))$, we can translate the origin to this point by simply replacing $x$ with $x-x_0$, $y$ with $y-y_0$, and $z$ with $z-z_0$, giving us
$z - f(x_0, y_0) = m_x(x-x_0) + m_y(y-y_0)$

Thanks joypav ...

Yes ... follow that ... straightforward when you see how ...

Appreciate your help ...

Peter